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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
63063801fffe89a234ccb3984b62ae5928dca792 | subsection | 75 | 121 | Conclusion and open problems | Similar to the previous proof, we have the following decomposition for the error term:
R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \nabla f_{\sigma _t\left(i\right)}\left(x^t_{0}\right)\right]\\
&= \sum _{i=1}^n \left[\int _{x^t_0}^{x^t_{i-1}} H_{\sigma _t\left(i\right)}\left... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
0.012566659599542618,
0.017900049686431885,
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0.021699804812669754,
0.003008139319717884,
-0.004059175960719585,
0.0009094041888602078,
0.01918189600110054,
-0.00764528987929225,
0.008957655169069767,
-0.0319240465760231,
-0.018815653398633003,
-0.010186090134084225,
0... | |
e3d5b5177cf181792afbcb8154381639b681d1bb | subsection | 76 | 121 | Conclusion and open problems | Here we define random variables
A^t = -\gamma \sum _{i=1}^n \left[H_{\sigma _t\left(i\right)}\sum _{j=1}^{i-1} \nabla f_{\sigma _t\left(j\right)} \left(x^t_0\right)\right],
B^t = - \gamma \sum _{i=1}^n \left\lbrace H_{\sigma _t\left(i\right)}\sum _{j=1}^{i-1}\left[\nabla f_{\sigma _t\left(j\right)} \left(x^t_{j-1}\righ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
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] | 2,018 | en | Mathematics | [
-0.04345737397670746,
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0.006137896329164505... | |
4f528573952f582134c2e7e4619fe167d7cf4d66 | subsection | 77 | 121 | Conclusion and open problems | \left\Vert C^t\right\Vert &\le \sum _{i=1}^n \left[\int _{0}^{\left\Vert x^t_{i-1}-x^t_0\right\Vert } \left\Vert H_{\sigma _t\left(i\right)}\left(x^t_0+t\frac{x^t_{i-1}-x^t_0}{\left\Vert x^t_{i-1}-x^t_0\right\Vert }\right)-H_{\sigma _t\left(i\right)}\right\Vert dt\right]\\
&\le \sum _{i=1}^n \left[L_H\max \left\lbrace ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
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] | 2,018 | en | Mathematics | [
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0.0025444147177040577,
0.0017690738895907998,
... | |
ccd1ed928559b2ff756890cbf776220ed84501aa | subsection | 78 | 121 | Conclusion and open problems | Using (REF ) and (REF ), we can decompose the inner product of x^t_0 - x^* and \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] into:
-2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] \right\rangle &= -2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\lef... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.016512149944901466,
0.01646636798977852,
-0.020342601463198662,
0.017885617911815643,
0.001797909033484757,
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0.03268856182694435,
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0.022891148924827576,
-0.01587119698524475,
-0.0... | |
c5689eadb55256165d3d097d8464ed802efc76f8 | subsection | 79 | 121 | Conclusion and open problems | For the first term in (REF ), there is
&\ \ \ \ \gamma ^2 n\left(n-1\right) \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i \nabla f_j\left(x^t_0\right)\right\rangle \\
&= \gamma ^2 n\left(n-1\right)\operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} \left\langle H_i \left(x^t_0 -... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.057683516293764114,
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-0.031252868473529816,
... | |
aed4d90f2a7166d874e7bbc7ea35ee5530187062 | subsection | 80 | 121 | Conclusion and open problems | The last inequality is because of
\left\Vert H_i\left(x^t_0 - x^*\right) - \left(\nabla f_i\left(x^t_0\right) - \nabla f_i\left(x^*\right)\right)\right\Vert &=\left\Vert H_i\left(x^t_0 - x^*\right) - \int _{x^*}^{x^t_0} H_i\left(x\right) dx\right\Vert \\
&=\left\Vert \int _{x^*}^{x^t_0} \left(H_i - H_i\left(x\right)\r... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.02051360346376896,
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-0.01897202990949154,
... | |
6725a260ce0c651e0d982da1a935789c2f9f84eb | subsection | 81 | 121 | Conclusion and open problems | Furthermore, we have
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert R^t\right\Vert ^2\right] &\le \left[3n^2L\gamma (\left\Vert \nabla F(x^t_0)\right\Vert + \delta )\right]^2\\
&\le 18n^4L^2\gamma ^2 (\left\Vert \nabla F(x^t_0)\right\Vert ^2 + \delta ^2).
Inequality (REF ) can be simplified to:
\operatorname{\... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.04038400202989578,
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0.03159292787313461,
-0.03647685796022415,
-0.... | |
8fb7310bae906509343ec2c6ebddd7a5133aeffd | subsection | 82 | 121 | Conclusion and open problems | So there is
\operatorname{\mathrm {\mathbb {E}}}[\left\Vert x^{t}_n - x^*\right\Vert ^2] &\le \left\Vert x^t_0-x^*\right\Vert ^2 - (2n\gamma -16n^2\gamma ^2L)\left[F(x^t_0) - F(x^*)\right] + \gamma ^2n\left(n-1\right)D\left\Vert \Delta \right\Vert \\
&\ \ \ \ +\gamma ^2n^2L_H (LD^3 + 2D^2G)+ 12\gamma ^4n^4L^4D^2 + 6\g... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.033298999071121216,
0.004849109333008528,
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-0.04422570765018463,
... | |
353be87860ffdcd0d646a7ae8bb824fbc510472d | subsection | 83 | 121 | Conclusion and open problems | Substituting the step size into (REF ), we have
F(\bar{x}) - F(x^*) &\le \frac{D^2}{T}\max \left\lbrace 16nL, \sqrt{\frac{ Tn\left(\left\Vert \Delta \right\Vert +L_H LD^2 + 2L_HDG\right)}{D}}, \left(\frac{Tn^2L^2\delta }{D}\right)^\frac{1}{3}, (Tn^3L^4)^\frac{1}{4}\right\rbrace \\
&\ \ \ \ + \frac{D\sqrt{nD\left(\left... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.0427895151078701,
0.02198990248143673,
-0.023988986387848854,
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0.041904423385858536,
-0.040347885340452194,
... | |
c2285bfa4305bd5507ce2748302623eacb838841 | subsection | 84 | 121 | Conclusion and open problems | We have the following inequality
||x_{t} - x^*||^2 &= ||x_{t-1} - x^*||^2 -2\gamma \langle x_{t-1} - x^*, \nabla f_{s(t)}(x_{t-1}) \rangle + \gamma ^2 ||\nabla f_{s(t)} (x_{t-1})||^2\\
&= ||x_{t-1} - x^*||^2 -2\gamma \langle x_{t-1} - x^*, \nabla f_{s(t)}(x_{t-1}) - \nabla f_{s(t)}(x^*) \rangle + \gamma ^2 ||\nabla f_... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.03716812655329704,
0.012358860112726688,
-0.038815975189208984,
-0.0037305448204278946,
-0.0031145091634243727,
0.00660283537581563,
0.016737863421440125,
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0.006560876499861479,
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-0.01890447922050953,
-0.03115653432905674,
... | |
c49c77e62d28ccb14ce4e86d9e74a3abe705962d | subsection | 85 | 121 | Conclusion and open problems | By the AM-GM inequality, we know the term \operatorname{\mathrm {\mathbb {E}}}[\prod _{i=1}^T (1-\gamma \mu _{s(t)})^2] for RandomShuffle is no larger than that of Sgd. Also, this bound is tight when we consider f_i(x) = \frac{\mu _i}{2}||x-x^*||^2, which completes the proof.
Sgd under Polyak-Łojasiewicz condition
For... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06336595863103867,
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0.03333122655749321,
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-0.004605173133313656,
-0.011636932380497456,... | |
ee4da081d3ee6eed0f0b04cfeec8a04bd3d9b08d | subsection | 86 | 121 | Proof of Theorem | Assume T = nl where l is positive integer. Notate x^t_i as the ith iteration for tth epoch. There is x_0^1 = x_0, x^t_n = x^{t+1}_0, x^l_n = x_T. Assume the permutation used in tth epoch is \sigma _t\left(\cdot \right). | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.0607658326625824,
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0.03454893082380295,
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-0.014428451657295227,
0.0... | |
b9545810b6bae2031d838f037348627890e136c0 | subsection | 87 | 121 | Proof of Theorem | Define error termR^t = \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)}\left(x^t_0\right).For one epoch of RandomShuffle, We have the following inequality\left\Vert x^{t}_n - x^*\right\Vert ^2 &= \left\Vert x^t_0 - x^*\right\Vert ^2 - 2\gamma \le... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1518,
"openalex_id": "https://openalex.org/W3141595720",
"raw": "Y. Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013.",
"source_ref_id": "ac12aab6cc5b... | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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... | |
4da5f65589392f9073e266ed3752c160cd119820 | subsection | 88 | 121 | Proof of Theorem | Firstly, we give a bound on the norm of R^t:\left\Vert R^t\right\Vert &= \left\Vert \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)}\left(x^t_0\right)\right\Vert \\
&\le \sum _{i=1}^n \left\Vert \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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0.01600242406129837,
-0.0014349169796332717,... | |
021c57dca3dec5e113be0a5f0ce3b24f2fc0cb04 | subsection | 89 | 121 | Proof of Theorem | We begin with the following decomposition:R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \nabla f_{\sigma _t\left(i\right)}\left(x^t_{0}\right)\right] \\
&= \sum _{i=1}^n \left[H_{\sigma _t\left(i\right)}\left(x^t_{i-1} - x^t_0\right)\right] \\
&= \sum _{i=1}^n \left\lbrace H_{\s... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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0.005... | |
2287647631a9db45c33f1a502334dafe757e63d9 | subsection | 90 | 121 | Proof of Theorem | The first inequality is by \left\langle x^t_0 - x^*, H_iH_i\left(x^t_0 - x^*\right)\right\rangle \ge 0 and AM–GM inequality, where \lambda _1 is any positive number. | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.01607002690434456,
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... | |
29f37c2b9cdd6ee35f6bb0f4349d14657117ee86 | subsection | 91 | 121 | Proof of Theorem | The second inequality comes from noticing that \operatorname{\mathrm {\mathbb {E}}}\limits _{i, j}H_iH_j = H^2 (with i, j uniformly sampled from all pairs of indices), and let \lambda _1 = \frac{1}{2}\mu \gamma ^{-1} n^{-1}.For the second term in (REF ), we use the bound-2\gamma \left\langle x^t_0 - x^*, \operatorname{... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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0... | |
d49c4f1df0bb6bcbc2fe44524bfc3a3918799bc3 | subsection | 92 | 121 | Proof of Theorem | Toward this end, we use the following important fact:\left\Vert \Delta \right\Vert &= \left\Vert \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i \nabla f_j\left(x^*\right)\right\Vert \\
&= \left\Vert \frac{1}{n\left(n-1\right)} \sum _{i\ne j} H_i \nabla f_j\left(x^*\right)\right\Vert \\
&= \left\Vert \frac{-1... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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0.... | |
8c9f7423513173530e247dbd449c2e4111beaede | subsection | 93 | 121 | Proof of Theorem | Substituting (REF ) (REF ) back to (REF ) and using (REF ) , we finally get a recursion bound for one epoch:&\ \ \ \ \operatorname{\mathrm {\mathbb {E}}}\left\Vert x^t_n - x^*\right\Vert ^2 \\
&\le \left(1-2n\gamma \frac{L\mu }{L+\mu } +\frac{1}{2}\gamma \mu \left(n-1\right)\right)\left\Vert x^t_0 - x^*\right\Vert ^2-\... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.03484300896525383,
0.023157481104135513,
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0.038534779101610184,
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... | |
5f48043731ab88eda60c1d876b5ea88768eb441e | subsection | 94 | 121 | Proof of Theorem | Expanding (REF ) over all epochs leads to a final bound of RandomShuffle:\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] \le \left(1-n\gamma \frac{L\mu }{L+\mu }\right)^{\frac{T}{n}}\left\Vert x_0 - x^*\right\Vert ^2 + \frac{T}{n} \left(\gamma ^3nC_1 + \gamma ^{5}n^{5}C_2 + \gamma ^4... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.056931525468826294,
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0.003995127510279417,
-0.029396813362836838,
... | |
ef20abc028cc5ffc60410da129ed04dab4752b2b | subsection | 95 | 121 | Proof of Theorem | Or in the expanding version with constant dependence, we have\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] \le \frac{\left(\log T\right)^2}{T^2}\left(D^2 + 128\frac{L^2G^2}{\mu ^4}\right) + \frac{n^3\left(\log T\right)^4}{T^3}128\frac{L^2G^2}{\mu ^4} + \frac{n^4\left(\log T\right)^... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05425288900732994,
0.007826696150004864,
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0.037928201258182526,
0.0051110005006194115,
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0.02016938105225563,
... | |
a4cb57cae9f81d537477e63b8c3859be69d7938b | subsection | 96 | 121 | Proof of Theorem | Again, define error termR^t = \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)}\left(x^t_0\right).We have the following decomposition for the error term:R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \nabla ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.01809144951403141,
-0.004511422012001276,
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0.022911768406629562,
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0.0006368617760017514,
-0.008832166902720928,
... | |
ea0e14e66efdd80bd5331052ef6cdb0ff09f27d8 | subsection | 97 | 121 | Proof of Theorem | There is\operatorname{\mathrm {\mathbb {E}}}\left[A^t\right] = -\frac{n\left(n-1\right)}{2} \gamma \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j}\left[ H_{i} \nabla f_{j}\left(x^t_0\right)\right],\left\Vert B^t\right\Vert &\le \gamma \sum _{i=1}^n H_{\sigma _t\left(i\right)} \sum _{j=1}^{i-1} \left(\nabla f_{\si... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.025946762412786484,
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0.026664113625884056,
-0.00153391144704073... | |
ed2e6f4f3f312f4c04db519629c2628119cb36bc | subsection | 98 | 121 | Proof of Theorem | The last inequality is because of\left\Vert H_i\left(x^t_0 - x^*\right) - \left(\nabla f_i\left(x^t_0\right) - \nabla f_i\left(x^*\right)\right)\right\Vert &=\left\Vert H_i\left(x^t_0 - x^*\right) - \int _{x^*}^{x^t_0} H_i\left(x\right) dx\right\Vert \\
&=\left\Vert \int _{x^*}^{x^t_0} \left(H_i - H_i\left(x\right)\rig... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.04299446567893028,
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0.014662058092653751,
-0.0075675141997635365,
... | |
019881a5fc7aa2ae02a6a521c1ff0b173d96e23b | subsection | 99 | 121 | Proof of Theorem | Further assume n\gamma \frac{L\mu }{L+\mu }<1, which we call assumption 3, expanding (REF ) over all the epochs we finally get a bound for RandomShuffle:\operatorname{\mathrm {\mathbb {E}}}\left\Vert x_T - x^*\right\Vert ^2 \le \left(1-\frac{1}{2}n\gamma \frac{L\mu }{L+\mu }\right)^{\frac{T}{n}}\left\Vert x_0 - x^*\rig... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06987273693084717,
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0.007589887827634811,
-0.032098740339279175,
... | |
e0bb60f8008789179841bc0b917f81f46621fa89 | subsection | 100 | 121 | Proof of Theorem | The first is satisfied whenn\gamma \frac{L\mu }{L+\mu }>\frac{1}{2}\gamma \mu \left(n-1\right),which is naturally satisfied and\frac{1}{2}n\gamma \frac{L\mu }{L+\mu }>\gamma ^2n^2\left(L_H LD + 3L_H G\right),which is equivalent to\frac{T}{\log T} > 16\frac{L+\mu }{L\mu ^2}\left(L_H LD + 3L_H G\right)n,which is obviousl... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.03234345465898514,
0.022442085668444633,
-0.031092435121536255,
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0.010595533065497875,
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0.018246591091156006,
0.0027499564457684755,
0.016705699265003204,
-0.023098109290003777... | |
9e1945ad2ca2167681fae5cfdaa639c358b815c0 | subsection | 101 | 121 | Proof of Theorem | In this setting, we have:x_{t} &= x_{t-1} - \gamma A(x_{t-1} + (-1)^{\sigma (t)}b)\\
&= (I - \gamma A) x_{t-1} -(-1)^{\sigma (t)} \gamma A b.Expanding (REF ) over iterations leads to:x_T &= (I-\gamma A)^T x_0 - \sum _{t=1}^T (-1)^{\sigma (t)} \gamma (I-\gamma A)^{T-t} A b.Taking expectation of (REF ) over the randomnes... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.008133991621434689,
0.022860636934638023,
-0.030033200979232788,
-0.015146316960453987,
0.006577393040060997,
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0.024463018402457237,
-0.00516958674415946,
... | |
1f01de5bad6e997b882c3583129ab0fde02e1e0b | subsection | 102 | 121 | Proof of Theorem | We can writeb = \sum _{i=1}^d b_i e_i.Since \left\langle e_i, e_j\right\rangle = 0 for i\ne j, we can simplify the last term in (REF ):\left\Vert \sum _{t=1}^T (-1)^{\sigma (t)} \gamma (I-\gamma A)^{T-t} A b\right\Vert ^2 &= \left\Vert \sum _{t=1}^T (-1)^{\sigma (t)} \gamma (I-\gamma A)^{T-t} A (\sum _{i=1}^d b_ie_i)\r... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.0374201275408268,
0.02827874943614006,
-0.014726925641298294,
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0.02647794410586357,
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0.045416925102472305,
0.009034549817442894,
0.003553... | |
4996e76281ef03eb7254b43c5dbd29cc9a822638 | subsection | 103 | 121 | Proof of Theorem | Then for any index pair t\ne u, over randomness of \sigma , there is\operatorname{\mathrm {\mathbb {E}}}\left[s_ts_u\right] &= \frac{2\frac{(\frac{T}{2})(\frac{T}{2}-1)}{2}}{\frac{T(T-1)}{2}} - \frac{(\frac{T}{2})(\frac{T}{2})}{\frac{T(T-1)}{2}}\\
&= -\frac{1}{T-1}.Using this fact, we can simplify the last term in (REF... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.055794112384319305,
0.030949249863624573,
-0.013086282648146152,
0.008660601451992989,
0.040716271847486496,
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0.006920850370079279,
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0.032444823533296585,
-0.01880151592195034,
0.01... | |
5fd4564fdb362a2f935b93baa660b3ed0e3896e6 | subsection | 104 | 121 | Proof of Theorem | Now for the faster convergence rate (REF ) to hold, from (REF ) we know there must be(1-\gamma \lambda _i)^{2T} = o(\frac{1}{T}),\gamma ^2 \operatorname{\mathrm {\mathbb {E}}}\left[\left[ \sum _{t=1}^T (-1)^{\sigma (t)} (1-\gamma \lambda _i)^{T-t}\right]^2\right] = o(\frac{1}{T}),hold for any i.However with (REF ), we ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.043806418776512146,
0.03577778488397598,
-0.038799967616796494,
-0.026405958458781242,
0.004113529343158007,
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0.024238532409071922,
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0.007975210435688496,
-0.01830100826919079,
... | |
ae828129ca343080b33771db417505d8661caee7 | subsection | 105 | 121 | Proof of Theorem | Since (REF ), there is (1-\gamma \lambda _i)^{T} = o(1) , so\gamma ^2\left[- \frac{1}{T-1} \frac{-2(1-\gamma \lambda _i)^T+(1-\gamma \lambda _i)^{2T}}{\gamma ^2\lambda _i^2}\right] = o(\frac{1}{T}).Again, since |1-\gamma \lambda _1|<1, for i=2,3 there is|\frac{\gamma ^2}{2\gamma \lambda _i - \gamma ^2\lambda _i^2}|\le ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06059154123067856,
0.0376790314912796,
-0.02365999110043049,
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0.007962936535477638,
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0.004259866196662188,
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0.005914997775107622,
-0.02950252778828144,
0.006... | |
fc36d78bec3eeaea70da7c3aadee47a3c56eb142 | subsection | 106 | 121 | Proof of Theorem | As a result, no step size can leads to convergence of o(\frac{1}{T}).The idea is similar to the proof of theorem REF , with a slightly different analysis on the R^t term adopting the sparsity parameter. For any i, we use H_i to denote H_i(x^*). | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.013721507973968983,
0.03595981374382973,
-0.046063970774412155,
-0.01064599771052599,
-0.0211088377982378,
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0.004323266912251711,
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-0.013599403202533722,
0.01871253363788128,
0.0... | |
e6cec35197dfdae32f37560916e0fa7d901cb2b4 | subsection | 107 | 121 | Proof of Theorem | Again, we have the following decomposition for the error term:R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t(i)} (x^t_{i-1}) - \nabla f_{\sigma _t(i)}(x^t_{0})\right]\\
&= \sum _{i=1}^n \left[\int _{x^t_0}^{x^t_{i-1}} H_{\sigma _t(i)}(x) dx\right]\\
&= \sum _{i=1}^n \left[\int _{x^t_0}^{x^t_{i-1}} H_{\sigma _t(i)} dx\r... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.009118512272834778,
-0.0009786186274141073,
-0.022342262789607048,
0.008500437252223492,
0.013612909242510796,
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0.014078373089432716,
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0.03406279534101486,
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0.023151101544499397,
-0.014009697362780571,
... | |
fb56739cf26dbe7fea44c21fe3c9d23a3b2923ca | subsection | 108 | 121 | Proof of Theorem | The introduction of \rho in (REF ) is similar: if f_{\sigma _t(i)} and f_{\sigma _t(j)} depend on disjoint dimensions of variables and j<i, then there must be \int _{x^t_{j-1}}^{x^t_j} (H_{\sigma _t(i)}(x)-H_{\sigma _t(i)}) dx=0.With (REF ) (REF ), we can decompose the innerproduct of x^t_0 - x^* and \operatorname{\mat... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.03926714509725571,
0.016780830919742584,
-0.02849690243601799,
-0.011319433338940144,
-0.0002996236435137689,
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0.025689927861094475,
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0.012585623189806938,
-0.02022852934896946,
... | |
bc6163ca4e6c3e393fdcd8b37a5f6ced250295f4 | subsection | 109 | 121 | Proof of Theorem | The idea is similar to the proof of theorem REF . For any vector v not being zero, define vector value directional functiondir\left(v\right) = \frac{v}{\left\Vert v\right\Vert },with norm being \ell _2 norm. For the convenience of notation, we define dir\left(\vec{0}\right) = \vec{0}, where \vec{0} is the zero vector. ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06438145041465759,
-0.03383840247988701,
0.004409061744809151,
0.03148893639445305,
0.006365677807480097,
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0.01803290843963623,
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0.0019194292835891247,
0.0300701055675745,
-0.0049... | |
184a6fddb24ea8668d9ee0184f41b47f4f667a90 | subsection | 110 | 121 | Proof of Theorem | For one epoch of RandomShuffle, we haveF(x^{t+1}_0) - F^* &\le F(x^t_0) - F^* - \gamma \left\langle \nabla F(x^t_0), n\nabla F(x^t_0) + R^t\right\rangle + \frac{L}{2}\gamma ^2 \left\Vert n\nabla F(x^t_0) + R^t\right\Vert ^2 \\
&\le (1-2n\mu \gamma )\left[F(x^t_0) - F^* \right] - \gamma \left\langle \nabla F(x^t_0), R^t... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06584832817316055,
0.027843652293086052,
-0.02000165916979313,
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0.017163895070552826,
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0.0049851578660309315,
-0.05364289507269859,
0.009993201121687889,
-0.00945158489048481,
0.00... | |
eb38420fd2533351cdd0ab29debc360374400ab4 | subsection | 111 | 121 | Proof of Theorem | Further assume n\gamma \mu <1, which we call assumption 2, expanding (REF ) over all the epochs we finally get a bound for RandomShuffle:\operatorname{\mathrm {\mathbb {E}}}\left\Vert x_T - x^*\right\Vert ^2 \le \left(1- n\gamma \mu \right)^{\frac{T}{n}}[F(x^t_0) - F^*] + \frac{T}{n} \left(\gamma ^3nC_1 + \gamma ^{4}n^... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06058661267161369,
0.03467324376106262,
-0.007558065466582775,
-0.013941025361418724,
-0.003510055597871542,
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-0.0015547256916761398,
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0.0019066850654780865,
-0.033818621188402176,
... | |
55cda189f3576fa4adc38bf5e4d2541536c28709 | subsection | 112 | 121 | Proof of Theorem | The first is satisfied when\frac{T}{\log T} > 16 \frac{L^2}{\mu ^2} n.The second assumption is satisfied when\frac{T}{\log T} > 2 n.Since 2<\frac{L}{\mu }, the theorem is proved.For one epoch of RandomShuffle, We have the following inequality\left\Vert x^{t}_n - x^*\right\Vert ^2 &= \left\Vert x^t_0 - x^*\right\Vert ^2... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.055557552725076675,
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11bb55c2adb7c420291f6de4116ae394d282acf7 | subsection | 113 | 121 | Proof of Theorem | Obviously R^t_n = R^t. | {
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"Jeff Z. HaoChen",
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942f5c2f929d3d8f62055bbeb21fa7f308dab4db | subsection | 114 | 121 | Proof of Theorem | We firstly show that \left\Vert R^t_k\right\Vert \le 3n^2L\gamma (\left\Vert \nabla F(x^t_0)\right\Vert + \delta ), which is an important fact to be used in further analysis.For any index 1\le id \le n, there is\left\Vert \nabla f_{id}(x^t_1) - \nabla f_{id}(x^t_0)\right\Vert \le L\gamma (\left\Vert \nabla F(x^t_0)\rig... | {
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"Jeff Z. HaoChen",
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e43111f392b4148f4d5e834df9da418cba9891bf | subsection | 115 | 121 | Proof of Theorem | Since \gamma \le \frac{1}{16nL} \le \frac{1}{nL}, there is 1+\gamma L \le \frac{1}{n}. | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
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] | [
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539fb31d343b197efe52efe91a6a718219a2325c | subsection | 116 | 121 | Proof of Theorem | Therefore, we have\left\Vert \nabla f_{id}(x^t_i) - \nabla f_{id}(x^t_0)\right\Vert &\le \left[\sum _{j=0}^{i-1} (1+L\gamma )^j\right] L\gamma (\left\Vert \nabla F(x^t_0)\right\Vert + \delta ) \\
&\le \left[n (1+\frac{1}{n})^n\right] L\gamma (\left\Vert \nabla F(x^t_0)\right\Vert + \delta )\\
&\le 3nL\gamma (\left\Vert... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
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00becf8d2e08cda1029e0b8f1ff9f86380b21616 | subsection | 117 | 121 | Proof of Theorem | The last inequality is because of\left\Vert H_i\left(x^t_0 - x^*\right) - \left(\nabla f_i\left(x^t_0\right) - \nabla f_i\left(x^*\right)\right)\right\Vert &=\left\Vert H_i\left(x^t_0 - x^*\right) - \int _{x^*}^{x^t_0} H_i\left(x\right) dx\right\Vert \\
&=\left\Vert \int _{x^*}^{x^t_0} \left(H_i - H_i\left(x\right)\rig... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
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ff597fcc741cc7072bbdf9b27422a8074cf42331 | subsection | 118 | 121 | Proof of Theorem | \\
&\le \left\Vert x^t_0-x^*\right\Vert ^2 - (2n\gamma -3n^2\gamma ^2L)\left[F(x^t_0) - F(x^*)\right] + \gamma ^2n\left(n-1\right)D\left\Vert \Delta \right\Vert +\gamma ^2n^2L_H (LD^3 + 2D^2G) \\
&\ \ \ \ +12\gamma ^2n^2 \left\Vert \nabla F(x^t_0)\right\Vert ^2 + 12\gamma ^4n^4L^4D^2 + 6\gamma ^3n^3L^2D\delta + 36n^4L^... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
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13da47691a3d7530d6af50aa38a3b69dbfdeb819 | subsection | 119 | 121 | Proof of Theorem | We have the following inequality||x_{t} - x^*||^2 &= ||x_{t-1} - x^*||^2 -2\gamma \langle x_{t-1} - x^*, \nabla f_{s(t)}(x_{t-1}) \rangle + \gamma ^2 ||\nabla f_{s(t)} (x_{t-1})||^2\\
&= ||x_{t-1} - x^*||^2 -2\gamma \langle x_{t-1} - x^*, \nabla f_{s(t)}(x_{t-1}) - \nabla f_{s(t)}(x^*) \rangle + \gamma ^2 ||\nabla f_{s... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
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327c8bd9642f09bdb139211beaa6ef072e0db947 | subsection | 120 | 121 | Proof of Theorem | Also, this bound is tight when we consider f_i(x) = \frac{\mu _i}{2}||x-x^*||^2, which completes the proof. | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
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] | [
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4c8df67f0883fcdbedca16d651ea6f90dadffbd7 | abstract | 0 | 74 | Abstract | In this paper, we derive a theoretical analysis of an interior penalty
discontinuous Galerkin methods for solving the Cahn-Hilliard-Navier-Stokes
model problem. We prove unconditional unique solvability of the discrete
system, obtain unconditional discrete energy dissipation law, and derive
stability bounds with a gene... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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1f1442f353b89b737ff8c71e845337d754fc35a5 | subsection | 1 | 74 | Introduction | The Cahn–Hilliard–Navier–Stokes system
strikes an optimal balance in terms of thermodynamical rigor and computational efficiency for modeling two-component binary flow.
The model that belongs to the class of diffusive interphase or phase-field models, attracts much attention in physics, chemistry, biology, and engineer... | {
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{
"arxiv_id": "",
"doi": "10.1137/050648110",
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"raw": "D. Kay and R. Welford, Efficient numerical solution of Cahn–Hilliard–Navier–Stokes fluids in 2D, SIAM Journal on Scientific Computing, 29 (2007), pp.... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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88a845d78ca1a37fc8ee510e01bf58265ffd5e5d | subsection | 2 | 74 | Introduction | The coupling term in the momentum equation of the Navier–Stokes system may take
several forms, that yield different numerical methods and impact their analysis. We note that
in , , , the coupling term is the product of the chemical potential and the gradient of the order parameter. In the other works , as well as in ou... | {
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"raw": "X. Feng, Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows, SIAM Journal on Numerical Analysis, 44 (2006), pp. 1049–10... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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1ddcf1672c6bf3064e675749895a9bdce0a7f433 | subsection | 3 | 74 | Mathematical Model | Let \Omega \subset {R}^d, where d=2 or 3, be an open bounded polyhedral domain and {{n}} denote the outward normal of \Omega . The unknown variables in Cahn–Hilliard–Navier–Stokes equations are the order parameter c, the chemical potential \mu , the velocity {{v}} and the pressure p, satisfying:\partial _t c - \Delta \... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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c7dfa8205ffb1eb16b1a5577d0914bf6584af725 | subsection | 4 | 74 | Mathematical Model | Under the assumption of the incompressibility constraint (), it is possible to consider employing advection operator {{v}}\cdot \nabla {c} in (REF ) in nonconservative form instead of \nabla \cdot {(c{{v}})} in conservative form. However, for the convenience of proving discrete global mass conservation property, we pro... | {
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"raw": "J. G. Heywood, R. Rannacher, and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations, International Journal for Numerical Methods in Flui... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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5e9761261b1d8ccedbbd30278ab785d1fe3e0ec4 | subsection | 5 | 74 | Well-posedness | A weak formulation of the Cahn–Hilliard–Navier–Stokes system (REF ) is proposed as finding the quaternion (c,\, \mu ,\, {{v}},\, p), wherec \in L^{\infty }\big (0,\, T;\, H^1(\Omega )\big ) \cap L^4\big (0,\, T;\, L^\infty (\Omega )\big ), &&
\mu \in L^2\big (0,\, T;\, H^1(\Omega )\big ), \\
{{v}} \in L^2\big (0,\, T;\... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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13f8473b959863a045848210514f5f50523d7b51 | subsection | 6 | 74 | Well-posedness | Standard notation is used for the Sobolev and Bochner spaces and we recall that L_0^2(\Omega ) denotes the space of L^2 functions with zero average.
The existence of the weak solution to (REF ) follows the argument as in . A generalized version of Cahn–Hilliard–Navier–Stokes model, in which the deformation tensor {{\va... | {
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"arxiv_id": "",
"doi": "10.1007/s00211-017-0887-5",
"end": 222,
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"raw": "A. E. Diegel, C. Wang, X. Wang, and S. M. Wise, Convergence analysis and error estimates for a second order accurate finite element method fo... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0993efff3ab84c9ef3efe0b0e1af396a8cc840ff | subsection | 7 | 74 | Mass conservation | Let \bar{c}_0 denote the mass average at time t_0. The solution of the model problem (REF ) enjoys the global mass conservation property .
The total amount of the order parameter c is preserved, i. e., for any t\in (0,\, T), we have\frac{1}{|\Omega |} \int _\Omega c = \frac{1}{|\Omega |} \int _\Omega c^0 = \bar{c}_0. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
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"raw": "F. Frank, C. Liu, F. O. Alpak, and B. Rivière, A finite volume/discontinuous Galerkin method for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-C... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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84ae8c7640321da50d6cca34ca77b5266d32a027 | subsection | 8 | 74 | Energy dissipation | Benefitting from the boundary conditions (REF -) , the Cahn–Hilliard–Navier–Stokes model (REF ) is an energy dissipative system. Analysis of a similar model can be found in . Define the total energy as followsF(c,{{v}})=\underbrace{\int _\Omega \frac{1}{2}|{{v}}|^2}_{\text{kinetic energy}} + \underbrace{\int _\Omega \B... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/140971154",
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"raw": "J. Shen and X. Yang, Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows, SIAM Journal on Numerical Analysis, 5... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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834d6a851e4fb28f4e766e058412db469938a44d | subsection | 9 | 74 | Chemical energy density | The chemical energy density \Phi may take several forms. Two popular expressions of \Phi are the Ginzburg–Landau double well potential ,\Phi (c) = \frac{1}{4}(1+c)^2(1-c)^2,and the logarithmic potential ,\Phi (c) = \frac{\vartheta }{2}\Big ((1+c)\log {(\frac{1+c}{2})}+(1-c)\log {(\frac{1-c}{2})}\Big )+\frac{\vartheta _... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
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"raw": "A. Novick-Cohen, The Cahn–Hilliard equation, Handbook of Differential Equations: Evolutionary Equations, 4 (2008), pp. 201–228, https://doi.org/10.1016/S1874-5717(08)00004-2.",
"source_ref_id"... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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f259e5cc2bb29d3314f170edfef4a7fe095d87f0 | subsection | 10 | 74 | Convex-concave decomposition | Throughout our analysis, we assume the chemical energy density \Phi \in \mathcal {C}^2, i. e., \Phi is a two times continuously differentiable function with respect to c. Any \mathcal {C}^2 function can be decomposed into the sum of a convex part and a concave part . We write\Phi (c) = \Phi _{+}(c)+\Phi _{-}(c),where \... | {
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"doi": "10.1162/08997660360581958",
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"raw": "A. L. Yuille and A. Rangarajan, The concave-convex procedure, Neural Computation, 15 (2003), pp. 915–936, https://doi.org/10.1162/08997660360... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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094676fae1eb9d4b0708b142f3f8dfa6bfb63cb8 | subsection | 11 | 74 | Numerical Analysis | In this section, we introduce an interior penalty discontinuous Galerkin method for the Cahn–Hilliard–Navier–Stokes system and analyze their numerical properties. These include uniquely solvability of the scheme, discrete mass conservation, energy dissipation, stability and error bounds. Our results are valid
for any g... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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b0e9291c2cdd9464e48a28e3a46afb911f24ec5c | subsection | 12 | 74 | Domain and triangulation | Let \mathcal {T}_h = \lbrace E_k\rbrace be a family of conforming nondegenerate (also called regular) meshes of the domain \Omega . The parameter h denotes the maximum element diameter. Let \Gamma _h denote the set of interior faces. For each interior face e \in \Gamma _h shared by elements E_{k^-} and E_{k^+}, we defi... | {
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"raw": "B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics,... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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2fffd8bf6134b6ee6524b563964a518116f89638 | subsection | 13 | 74 | DG forms | We introduce the formsa_{\mathcal {A}}: H^2(\mathcal {T}_h) \times H^2(\mathcal {T}_h)^d \times H^2(\mathcal {T}_h) &\rightarrow {R}, \\ a_{\mathcal {C}}: H^2(\mathcal {T}_h)^d \times H^2(\mathcal {T}_h)^d \times H^2(\mathcal {T}_h)^d \times H^2(\mathcal {T}_h)^d &\rightarrow {R}, \\
a_{\mathcal {D}}: H^2(\mathcal {T}_... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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6883dc1ca9f2d56421e633796cc530d01eda79b7 | subsection | 14 | 74 | DG forms | \mathrm {ext}) refers to the trace of the function on a face of E coming from the interior of E (resp. coming from the exterior of E on that face), in addition, if the face lies on the boundary of the domain, we take the exterior trace to be zero.
For more details related to (), we refer the reader to .
The derivation ... | {
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"raw": "V. Girault, B. Rivière, and M. Wheeler, A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems, Mathematics of Computation, 74 (2005), pp.... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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c72c999cac49cbc61521e7925c72e9158d826c71 | subsection | 15 | 74 | DG scheme | Uniformly partition [0,\,T] into N subintervals and let \tau be the time step length. For any fixed positive integer q\in {N}_+ , the set {P}_q(E) denotes all polynomials of degree at most q on an element E. Define the following broken polynomial spacesS_h &= \big \lbrace \omega \in L^2(\Omega ):~\forall E \in \mathcal... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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5f76da7b8550d55e7602f96656796e3766aaa99b | subsection | 16 | 74 | DG scheme | The fully discrete mixed convex-concave splitting DG scheme reads:for any 1\le n \le N , given c_h^{n-1} \in S_h and {{v}}_h^{n-1} \in \mathbf {X}_h find (c_h^n, \mu _h^n, {{v}}_h^{n}, p_h^n) \in S_h \times S_h \times \mathbf {X}_h \times Q_h such that(\delta _\tau c_h^n,\chi ) + a_{\mathcal {D}}(\mu _h^n,\chi ) + a_{\... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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93c84effd6b0a287b3caf400a18cf8b346c13ec9 | subsection | 17 | 74 | Operator properties | Throughout this paper, the semi-norms \Vert \cdot \Vert _{\mathrm {DG}} for any scalar quantity c \in H^1(\mathcal {T}_h) and for any vector quantity {{v}} \in H^1(\mathcal {T}_h)^d are defined as follows, respectively\forall c &\in H^1(\mathcal {T}_h), & \Vert c\Vert _{\mathrm {DG}}^2 &= \sum _{E\in \mathcal {T}_h} \V... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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f095f966634adf594cb66b3d6a4b7049bb16bbe0 | subsection | 18 | 74 | Operator properties | Then, we have the following result
[Poincaré's inequality ]
For each p \le p_0 (exclude infinity when d=2), there exists a constant C_P > 0 independent of mesh size h such that\Vert \chi - \frac{1}{|\Omega |}\int _{\Omega }\chi \Vert _{L^p(\Omega )} \le C_P\Vert \chi \Vert _{\mathrm {DG}}, \quad \forall \chi \in S_h.We... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/num.22092",
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"raw": "V. Girault, J. Li, and B. Rivière, Strong convergence of the discontinuous Galerkin scheme for the low regularity miscible displacement equations, Nu... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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10ba11fc1ef3ce7eb23b6d6e8e985b3b3971f84d | subsection | 19 | 74 | Operator properties | Using Hölder's inequality and Cauchy–Schwarz's inequality we have|\sum _{E\in \mathcal {T}_h} \int _E c\,{{v}}\cdot \nabla {\chi }|
\le \Big (\sum _{E\in \mathcal {T}_h} \Vert c\Vert _{L^4(E)}^4\Big )^{\frac{1}{4}}\Big (\sum _{E\in \mathcal {T}_h} \Vert {{v}}\Vert _{L^4(E)}^4\Big )^{\frac{1}{4}}\Big (\sum _{E\in \mathc... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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74727ce20323ffd6827f8cb2357ef97d177544e3 | subsection | 20 | 74 | Operator properties | For the inequality eq:CHNS:boundednessaA2, using similar arguments as above, we have|a_{\mathcal {A}}(c,{{v}},\chi )| \le C \Vert c\Vert _{L^6(\Omega )}\Vert {{v}}\Vert _{L^3(\Omega )}\Vert \chi \Vert _{\mathrm {DG}}.Finally, we conclude our proof by applying Poincaré's inequality and interpolation inequality \Vert {{v... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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390e9f3f6267b68fd47a6165484f5c625d232336 | subsection | 21 | 74 | Operator properties | Then, there exists a constant K_\alpha >0 independent of mesh size h such thata_{\mathcal {D}}(c,c) \ge K_\alpha \Vert c\Vert _{\mathrm {DG}}^2, \quad \forall c \in S_h.[Continuity of a_{{\varepsilon }}]
The bilinear form a_{{\varepsilon }} is continuous on \mathbf {X}_h equipped with the energy norm, i. e., there exis... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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bea2f3cc1190793d21ddcedea6d35a410013ded6 | subsection | 22 | 74 | Discrete mass conservation | The DG scheme (REF ) satisfies the discrete global mass conservation property, i. e., for any 1\le n \le N , we have(c_h^n,1) = (c_h^{0},1) = (c^0,1) = \big (c(t^n),1\big ).The proof for the first equality is straightforward and obtained by choosing \chi = 1 in (REF ) and by using a_{\mathcal {D}}(\mu _h^n,1)=0 and a_{... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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9d01d066de2977fb106ef7fd3fbcd822bd2b5d79 | subsection | 23 | 74 | Existence and uniqueness | Investigating the unique solvability of the fully discrete DG method (REF ) is a complicated task. We will design an equivalent scheme, which is based on an auxiliary flow problem, to overcome this challenge. The existence and uniqueness of the solution for our equivalent scheme can be proved by using nonlinear operato... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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47e19f22942483f1377e1b0d0bb92d59c28d32d3 | subsection | 24 | 74 | Existence and uniqueness | To recover the discrete pressure \tilde{p}_h^n\in Q_h, we then use the inf-sup condition of lem:infsup.Owing to the last result, we can construct the following scheme by employing the unique discrete solution from the auxiliary flow problem: for any 1\le n \le N , given (y_h^{n-1},{{v}}_h^{n-1}) \in M_h\times \mathbf {... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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3f1e84a4494998224e19889452bec14542acae70 | subsection | 25 | 74 | Existence and uniqueness | We also denote
\hat{y}_h^{n-1} \in M_h the solution of(y_h^{n-1},\mathring{\chi }) - \tau a_{\mathcal {A}}(y_h^{n-1}+\bar{c}_0,\tilde{{{v}}}_h^n,\mathring{\chi }) = (\hat{y}_h^{n-1},\mathring{\chi }), && \forall \mathring{\chi } \in M_h,whose existence and uniqueness are asserted by the Riesz representation theorem. Ou... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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eb8d0fec5c21c0d848c143334bede01ddd16c447 | subsection | 26 | 74 | Existence and uniqueness | Due to the translational invariance of the trilinear form a_{\mathcal {A}} with respect to the third argument and using the same techniques as in , we have
The unique solvability of the DG scheme (REF ) is equivalent to the unique solvability of the problem: for any 1\le n \le N , given (y_h^{n-1},{{v}}_h^{n-1}) \in M... | {
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"raw": "C. Liu, F. Frank, and B. Rivière, Numerical error analysis for non-symmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation, Numerical Methods for Partial Differential Equ... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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562388e30f21a774e25fe4826141414d50d2c612 | subsection | 27 | 74 | Existence and uniqueness | (Necessity) If (REF ) has a solution (y_h^n, \mu _h^n, {{v}}_h^{n}, p_h^n). Define w_h^n=\mu _h^n-\frac{1}{|\Omega |}(\mu _h^n,1), then (y_h^n, w_h^n, {{v}}_h^{n}, p_h^n) is a solution of eq:CHNS:solvability:P. If the solution of (REF ) is unique. Assume (y_h^{n,1}, w_h^{n,1},{{v}}_h^{n,1},p_h^{n,1}) and (y_h^{n,2}, w_... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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a20737e8d13cc02cf15ff70dfb90caca1a9828ff | subsection | 28 | 74 | Existence and uniqueness | Assume (y_h^{n,1}, \mu _h^{n,1}, {{v}}_h^{n,1}, p_h^{n,1}) and (y_h^{n,2}, \mu _h^{n,2}, {{v}}_h^{n,2}, p_h^{n,2}) are two different solutions of eq:CHNS:solvability:Paux, then \big (y_h^{n,1}, \mu _h^{n,1}-\frac{1}{|\Omega |}(\mu _h^{n,1},1),{{v}}_h^{n,1},p_h^{n,1}\big ) and \big (y_h^{n,2}, \mu _h^{n,2}-\frac{1}{|\Om... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-0.00427181227132678,
0.... | |
6ad255d0c5c4b1d4bd1b8e54ab558384f078ef85 | subsection | 29 | 74 | Existence and uniqueness | Then (y_h^{n,1},w_h^{n,1},{{v}}_h^{n,1}-\tilde{{{v}}}_h^n,p_h^{n,1}-\tilde{p}_h^n) and (y_h^{n,2},w_h^{n,2},{{v}}_h^{n,2}-\tilde{{{v}}}_h^n,p_h^{n,2}-\tilde{p}_h^n) are two different solutions of (REF ). By contradiction argument, we know the solution of (REF ) is unique.(Sufficiency) If (REF ) has a solution (y_h^n,w_... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.009569760411977768,
-0.0029438077472150326,
-0.01122577115893364,
-0.008661624975502491,
0.0005885707796551287,
-0.02669459395110607,
0.02866349183022976,
0.03458544984459877,
0.048901934176683426,
0.041484225541353226,
-0.03922533243894577,
-0.006482863798737526,
-0.010081063024699688,
... | |
a4c723af34b2a323dfa4d9ca899b17b611dd8d71 | subsection | 30 | 74 | Existence and uniqueness | We first prove the existence of a solution. For each fixed w_h \in M_h , define the mapping \mathcal {F}:~M_h \rightarrow M_h by\big (\mathcal {F}(y_h),\mathring{\varphi }\big ) = \big (\Phi _{+}\,\!^{\prime }(y_h+\bar{c}_0)+\Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0),\mathring{\varphi }\big )\\ + \kappa a_{\mathcal {... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.009154790081083775,
0.002353543881326914,
-0.01748564839363098,
0.005149569362401962,
-0.010245735757052898,
0.008033327758312225,
0.043393705040216446,
0.03948765993118286,
0.051663532853126526,
0.049222253262996674,
-0.053006235510110855,
0.007167437579482794,
-0.020933952182531357,
0... | |
537dbdb9f45266dc2bf2dd3930fe0d518576e618 | subsection | 31 | 74 | Existence and uniqueness | Applying Cauchy–Schwarz's inequality, Young's inequality, and Poincaré's inequality, we have&- \big (\Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0),y_h\big ) + (w_h,y_h)\\
\le & \Vert \Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0)\Vert _{L^2(\Omega )}\Vert y_h\Vert _{L^2(\Omega )} + \Vert w_h\Vert _{L^2(\Omega )}\Vert y_h\... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04276032745838165,
0.019457321614027023,
-0.022463668137788773,
-0.034550100564956665,
-0.00454004155471921,
-0.01570320315659046,
0.012994439341127872,
0.006466698367148638,
0.052099842578172684,
0.006619304418563843,
-0.03799900412559509,
0.009766812436282635,
-0.013368324376642704,
0... | |
b2ea934901d2ad6584c80807debb0e4c68dc9631 | subsection | 32 | 74 | Existence and uniqueness | By Brouwer's fixed point theorem, there exists a function y_h \in M_h such that \mathcal {F}(y_h) = 0. In particular \big (\mathcal {F}(y_h),\mathring{\varphi }\big ) = 0 for all \mathring{\varphi } \in M_h, i. e., the function y_h is a solution of (REF ).
Next, let us prove the solution of (REF ) is unique. Assume y_h... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0007954190950840712,
-0.02075337991118431,
-0.011856894940137863,
-0.03308333083987236,
0.0014182112645357847,
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0.03790543973445892,
0.052982158958911896,
0.07007317990064621,
0.03439567610621452,
-0.046939264982938766,
0.005909372586756945,
0.005550766363739967,
... | |
b4fd84fcaf445642b47fc52dc3538e3dc25e2336 | subsection | 33 | 74 | Existence and uniqueness | Due to the fact that \Vert \cdot \Vert _{\mathrm {DG}} is a norm in M_h, we obtain y_h=\tilde{y}_h, i. e., the solution of (REF ) is unique.
For each fixed w_h \in M_h , given (y_h^{n-1},{{v}}_h^{n-1}) \in M_h \times \mathbf {X}_h and \bar{c}_0 \in S_h , there exists a unique solution ({{v}}_h, p_h) \in \mathbf {X}_h... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/1.9781611972597",
"end": 1168,
"openalex_id": "https://openalex.org/W333643410",
"raw": "P. G. Ciarlet, Linear and nonlinear functional analysis with applications, vol. 130, SIAM, 2013.",
"source_ref_id": "c3b732ba12711d5f8f8b... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02559538371860981,
-0.019612444564700127,
-0.033180899918079376,
-0.019536131992936134,
-0.022359712049365044,
-0.0033043534494936466,
0.0012839664705097675,
-0.0028025954961776733,
0.03333352506160736,
0.040934301912784576,
-0.0319598913192749,
-0.011805622838437557,
0.002016571350395679... | |
d6e59a14869c7af4c7a37f2f224e5560399025ba | subsection | 34 | 74 | Existence and uniqueness | By triangle inequality, Cauchy–Schwarz's inequality, Poincaré's inequality, and the continuity of a_{\mathcal {D}}, we have|\langle \mathcal {G}(w_h),\mathring{\chi }\rangle |
\le & \Vert y_h\Vert _{L^2(\Omega )}\Vert \mathring{\chi }\Vert _{L^2(\Omega )} + \Vert \hat{y}_h^{n-1}\Vert _{L^2(\Omega )}\Vert \mathring{\chi... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05485693737864494,
0.02460629679262638,
-0.008458890952169895,
-0.048663411289453506,
-0.016307583078742027,
0.0025952549185603857,
0.012844700366258621,
0.020762039348483086,
0.012791307643055916,
-0.001350066508166492,
-0.04140203818678856,
0.007734279148280621,
0.02411813661456108,
-... | |
7d7bc77d29b13e421d0dde54d06733d8d8e623a6 | subsection | 35 | 74 | Existence and uniqueness | Since y_h=y_h(w_h) \in M_h is the unique solution of (REF ) which is defined in CHNS:solvability:uniqc, take \mathring{\varphi }=y_h then\big (\Phi _{+}\,\!^{\prime }(y_h+\bar{c}_0)+\Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0),y_h\big ) + \kappa a_{\mathcal {D}}(y_h,y_h) - (w_h,y_h) = 0.Recall the nonnegativity of \big... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008529779501259327,
-0.01980617828667164,
-0.006420223042368889,
-0.029663490131497383,
-0.01169602107256651,
-0.020309725776314735,
0.03222700208425522,
0.03927665948867798,
0.0352788008749485,
0.029037872329354286,
-0.056031037122011185,
0.006012044847011566,
0.0174105167388916,
0.029... | |
08d4ed76cd3d4067e8efd2218c69ab327d714411 | subsection | 36 | 74 | Existence and uniqueness | By the coercivity of a_{\mathcal {D}}, Cauchy–Schwarz's inequality and Poincaré's inequality, we haveK_\alpha \kappa \Vert y_h\Vert _{\mathrm {DG}}^2
\le & \big (\Phi _{+}\,\!^{\prime }(y_h+\bar{c}_0),y_h\big ) + \kappa a_{\mathcal {D}}(y_h,y_h)\\
=& (w_h,y_h) - (\Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0),y_h\big )\\... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03342302888631821,
0.008920438587665558,
-0.016925176605582237,
-0.04218321666121483,
-0.02541065774857998,
-0.002434234134852886,
0.023563997820019722,
0.029989155009388924,
0.03134744241833687,
0.020923729985952377,
-0.04050443693995476,
0.02423550933599472,
0.007069962099194527,
0.00... | |
5ba0d8aff52d3d6a7c27553c9d5e64c8cdd3aa6a | subsection | 37 | 74 | Existence and uniqueness | By the positivity of a_{\mathcal {C}}, the coercivity of a_{{\varepsilon }}, and considering {{v}}_h is discrete divergence-free, we obtain\begin{split}
&a_{\mathcal {A}}\big (y_h^{n-1}+\bar{c}_0,{{v}}_h,w_h\big ) = b_\mathcal {I}\big (y_h^{n-1}+\bar{c}_0,w_h,{{v}}_h\big ) \\
=& \frac{1}{\tau }({{v}}_h,{{v}}_h) + a_{\m... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0026914537884294987,
0.0316794216632843,
-0.009300870820879936,
-0.047458093613386154,
0.002792550250887871,
-0.02577386423945427,
0.008827815763652325,
0.002561745001003146,
0.014542624354362488,
0.021592669188976288,
-0.03961453586816788,
0.003921777941286564,
0.003406758652999997,
-0... | |
da8f30fdc61a8b8d4e00eee51e94b0b78182d1d9 | subsection | 38 | 74 | Existence and uniqueness | \end{split}Taking \mathring{\chi } = w_h in (REF ) and combining the result with (REF ), we obtain the following bound\Vert {{v}}_h\Vert _{\mathrm {DG}} \le \frac{C_\gamma }{K_{{\varepsilon }}\mu _\mathrm {s}}(\Vert y_h^{n-1}+\bar{c}_0\Vert _{\mathrm {DG}} + |\Omega ||\bar{c}_0|) \Vert w_h\Vert _{\mathrm {DG}}.Substitu... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.061921071261167526,
0.017578382045030594,
-0.020706478506326675,
-0.052155300974845886,
-0.007160291541367769,
-0.03494313731789589,
-0.0010290296049788594,
0.00997176393866539,
0.021087953820824623,
0.009857322089374065,
-0.05960170179605484,
-0.0168917253613472,
0.03842219337821007,
-... | |
0826f07ff472ce81299fa1fef073bde1df7dc29f | subsection | 39 | 74 | Existence and uniqueness | By the coercivity of a_{\mathcal {D}}, Cauchy–Schwarz's inequality, Young's inequality, and Poincaré's inequality, we have\begin{split}
-(w_h,y_h)
\le & - \big (\Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0),y_h\big ) - \kappa a_{\mathcal {D}}(y_h,y_h)\\
\le & \Vert \Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0)\Vert _{L^2... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.027068879455327988,
0.02764870785176754,
-0.016509879380464554,
-0.02499369904398918,
-0.004417384508997202,
-0.004062620457261801,
0.020812824368476868,
0.02714517153799534,
0.025939736515283585,
0.0033511852379888296,
-0.047973256558179855,
0.012946981005370617,
0.00010383047629147768,
... | |
3cc67717985b5f837cad37006d6aa0b752c622ca | subsection | 40 | 74 | Existence and uniqueness | \end{split}Using the definition of \mathcal {G}, the coercivity of a_{\mathcal {D}}, the bounds (REF ), (REF ), and (REF ), we obtain\langle \mathcal {G}(w_h),w_h\rangle =& (y_h-\hat{y}_h^{n-1},w_h) + \tau a_{\mathcal {D}}(w_h,w_h) + \tau a_{\mathcal {A}}(y_h^{n-1}+\bar{c}_0,{{v}}_h,w_h)\\
\ge & K_\alpha \tau \Vert w_h... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.028309252113103867,
0.012857436202466488,
0.018710050731897354,
-0.08595024794340134,
-0.0020487962756305933,
-0.04035022109746933,
0.007195586338639259,
0.0027279129717499018,
0.0065240999683737755,
0.027790376916527748,
-0.046149421483278275,
-0.02237270213663578,
0.005318476818501949,
... | |
d219b488d211112fd590a32fa5b78a9b09e2080e | subsection | 41 | 74 | Existence and uniqueness | From CHNS:solvability:uniqc, for any \mathring{\varphi } \in M_h, we obtain(w_h,\mathring{\varphi }) &= \big (\Phi _{+}\,\!^{\prime }(y_h(w_h)+\bar{c}_0)+\Phi _{-}\,\!^{\prime }(y_h^{n-1}+\bar{c}_0),\mathring{\varphi }\big ) + \kappa a_{\mathcal {D}}(y_h(w_h),\mathring{\varphi }),\\
\,\,(s_h,\mathring{\varphi }) &= \bi... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.007096332963556051,
0.010949717834591866,
-0.03198691084980965,
-0.04038042575120926,
-0.022906657308340073,
-0.007516008801758289,
0.03366561606526375,
0.029133116826415062,
0.032536305487155914,
0.03601579740643501,
-0.05130724981427193,
-0.0021994817070662975,
-0.008775035850703716,
... | |
c2a5862075b85a5fd4301ff0195e0d7af7dbd0ff | subsection | 42 | 74 | Existence and uniqueness | \end{split}From CHNS:solvability:uniqvandp, for any {{\theta }} \in \mathbf {X}_h, we obtainb_\mathcal {I}(y_h^{n-1}+\bar{c}_0,w_h,{{\theta }}) =& \frac{1}{\tau }\big ({{v}}_h(w_h),{{\theta }}\big ) + a_{\mathcal {C}}\big ({{v}}_h^{n-1},{{v}}_h^{n-1},{{v}}_h(w_h),{{\theta }}\big ) \\ &+ \mu _\mathrm {s} a_{{\varepsilon... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.013412553817033768,
0.022903576493263245,
-0.04434230178594589,
-0.005077391862869263,
-0.008163499645888805,
-0.01686106063425541,
0.052215881645679474,
0.025711210444569588,
0.01417549792677164,
0.031433288007974625,
-0.03186053782701492,
-0.0018606294179335237,
0.00007957266643643379,
... | |
14624bde25330e0f73aa931829ade813e30ab1f6 | subsection | 43 | 74 | Existence and uniqueness | Using the positivity of a_{\mathcal {C}}, the coercivity of a_{{\varepsilon }}, considering {{v}}_h(w_h) and {{v}}_h(s_h) are discretely divergence-free, and by rem:CHNS:relationaAbI, we obtain\begin{split}
& a_{\mathcal {A}}\big (y_h^{n-1}+\bar{c}_0,{{v}}_h(w_h)-{{v}}_h(s_h),w_h-s_h\big ) \\
=\,& b_\mathcal {I}\big (y... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04712660610675812,
0.01726042479276657,
0.01712307333946228,
-0.05182705819606781,
0.0016062420327216387,
-0.01459734421223402,
0.016619455069303513,
-0.0011941894190385938,
0.03302524983882904,
0.032658983021974564,
-0.04941578954458237,
-0.018923897296190262,
0.005368129350244999,
0.0... | |
76f1c20c81557aad400a4671066ad95926396189 | subsection | 44 | 74 | Existence and uniqueness | For any \mathring{\chi } \in M_h with \Vert \mathring{\chi }\Vert _{\mathrm {DG}}=1, by triangle inequality, Cauchy–Schwarz's inequality, the continuity of a_{\mathcal {D}}, the boundedness of a_{\mathcal {A}}, and Poincaré's inequality, we have\begin{split}
&|\langle \mathcal {G}(w_h)-\mathcal {G}(s_h),\mathring{\chi ... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.043903566896915436,
0.010396206751465797,
-0.013096321374177933,
-0.048937682062387466,
-0.015514221042394638,
-0.013790419325232506,
-0.009366501122713089,
0.04558160528540611,
0.038228750228881836,
0.00795542448759079,
-0.02309590019285679,
-0.01620069146156311,
0.024011194705963135,
... | |
273e09d0adb59f4de8f7ce3e3c0e783bcaabfab6 | subsection | 45 | 74 | Existence and uniqueness | By (REF ), Cauchy–Schwarz's inequality, and Poincaré's inequality, we obtainK_\alpha \kappa \Vert y_h(w_h)-y_h(s_h)\Vert _{\mathrm {DG}}^2 \le & \big (y_h(w_h)-y_h(s_h),w_h-s_h\big )\\
\le & \Vert y_h(w_h)-y_h(s_h)\Vert _{L^2(\Omega )}\Vert w_h-s_h\Vert _{L^2(\Omega )}\\
\le & C_P^2\Vert y_h(w_h)-y_h(s_h)\Vert _{\mathr... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06341157108545303,
0.04830631613731384,
-0.006656231824308634,
-0.05514182522892952,
-0.00834603700786829,
-0.005042715463787317,
0.004085286054760218,
0.022657889872789383,
0.034818384796381,
0.03683242201805115,
-0.029279790818691254,
0.0045277634635567665,
0.014578863978385925,
0.016... | |
915210a70d94549a9c1d953204602cda17d60b1c | subsection | 46 | 74 | Existence and uniqueness | All conditions of the Minty–Browder theorem are satisfied. We conclude that there exists a unique solution w_h^n such that \langle \mathcal {G}(w_h^n),\mathring{\chi }\rangle = 0 for all \mathring{\chi } \in M_h. Recall CHNS:solvability:uniqc and CHNS:solvability:uniqvandp, this implies that \big (y_h(w_h^n),w_h^n,{{v}... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.022076115012168884,
-0.010076900012791157,
0.007212497293949127,
-0.0382632315158844,
0.00031466473592445254,
-0.025554589927196503,
0.009947219863533974,
-0.019711362197995186,
0.015004740096628666,
0.008680932223796844,
-0.033899880945682526,
0.010267606005072594,
0.008802983909845352,
... | |
852c0352881ebf7f405a2a20bfd2bb304d7eb161 | subsection | 47 | 74 | Stability analysis | In this section, we show the discrete solution of (REF ) satisfies the energy dissipation property and we derive stability bounds valid for any chemical energy density \Phi . Analoguously to the energy (REF ) at the continuous level, we define the discrete energy:F_h(c_h,{{v}}_h) = \frac{1}{2}({{v}}_h,{{v}}_h) + \big (... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.016376787796616554,
-0.011866684071719646,
0.004784830380231142,
-0.05131291598081589,
-0.02715219371020794,
-0.06715551018714905,
0.03247884660959244,
-0.007722883950918913,
0.033180925995111465,
0.02568698301911354,
-0.014667374081909657,
0.00018410530174151063,
-0.03208202123641968,
... | |
74fc09853e46253e7135326fd6c0e85b5ff887ff | subsection | 48 | 74 | Stability analysis | Then for any mesh size h, time step size \tau , parameter \kappa , and parameter \mu _\mathrm {s} , the discrete energy (REF ) is non-increasing in time.\forall 1\le n \le N,\quad F_h(c_h^n, {{v}}_h^n) \le F_h(c_h^{n-1}, {{v}}_h^{n-1}).Take \chi =\mu _h^n in (REF ), \varphi =\delta _\tau c_h^n in (), {{\theta }}={{v}}... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.006183673162013292,
0.0003080392489209771,
-0.022934142500162125,
-0.04998818784952164,
-0.03649931401014328,
-0.019531404599547386,
0.02403278276324272,
0.020065465942025185,
0.0300295352935791,
0.017593523487448692,
-0.02444477379322052,
-0.0150071382522583,
-0.04190096631646156,
-0.00... | |
e4df9f103ea3757b48a7be89cb6fed10d99df105 | subsection | 49 | 74 | Stability analysis | \end{split}For the term (\Phi _{+}\,\!^{\prime }(c_h^n)+\Phi _{-}\,\!^{\prime }(c_h^{n-1}),\,\delta _\tau c_h^n) , we utilize Taylor expansions up to the second order. There exist \xi _h and \eta _h between c_h^{n-1} and c_h^n such that\Phi _+\,\!^{\prime }(c_h^n)(c_h^n-c_h^{n-1}) &= \Phi _+(c_h^n) - \Phi _+(c_h^{n-1})... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03708905726671219,
-0.02165817841887474,
-0.03678379952907562,
0.0007412087870761752,
-0.004552185535430908,
0.010561986826360226,
0.02967124618589878,
0.05082574486732483,
0.03144175186753273,
0.016621392220258713,
-0.01953662373125553,
0.017644012346863747,
-0.044964756816625595,
-0.0... | |
a54a5544455d36e64241bf5b9efad8a8bae5e159 | subsection | 50 | 74 | Stability analysis | \end{split}For the terms (\delta _\tau {{v}}_h^n ,{{v}}_h^n) and \kappa a_{\mathcal {D}}(c_h^n,\delta _\tau c_h^n), since the inner product and a_{\mathcal {D}} are both symmetric bilinear forms, we immediately have(\delta _\tau {{v}}_h^n ,{{v}}_h^n)
\ge & \frac{1}{2\tau }({{v}}_h^n,{{v}}_h^n) - \frac{1}{2\tau }({{v}}_... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.007072987966239452,
0.014977643266320229,
-0.022157451137900352,
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0.029695866629481316,
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-0.007156917825341225,
-0.019929498434066772,
... | |
75d8dd6f96e4c5c46e0daf209da0855b59624aaf | subsection | 51 | 74 | Stability analysis | Then for any mesh size h, time step size \tau , parameter \kappa , and parameter \mu _\mathrm {s}, and for any 1 \le \ell \le N we have\frac{1}{2}\Vert {{v}}_h^\ell \Vert _{L^2(\Omega )}^2 + \big (\Phi (c_h^\ell ),1\big ) + \frac{K_\alpha \kappa }{2}\Vert c_h^\ell \Vert _{\mathrm {DG}}^2 \\ + \tau K_\alpha \sum _{n=1}^... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05271025747060776,
0.0008350432617589831,
-0.020418738946318626,
-0.043004490435123444,
-0.030414460226893425,
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0.028186405077576637,
-0.019808312878012657,
-0.008240746334195137,
-0.022921483963727... | |
2ede56f6036b3bf249a13a34cab0903344f61fa9 | subsection | 52 | 74 | Stability analysis | In case \Phi is bounded from below by a constant, since the parameters \kappa , \mu _\mathrm {s} and constants K_\alpha , K_{{\varepsilon }} are all positive, it is straightforward to show (REF ) holds. | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030280187726020813,
0.012424792163074017,
-0.025902146473526955,
0.0005267570377327502,
-0.021066468209028244,
-0.05808914825320244,
0.03368194401264191,
0.022393610328435898,
0.03310227021574974,
0.015079076401889324,
-0.05275006964802742,
0.012356147170066833,
0.014751104637980461,
0.... | |
78c7986a130c3e99a0c86f7427054d3d2cb59ebe | subsection | 53 | 74 | Error analysis | In this section, we derive an optimal error estimate for the fully discrete scheme (REF ) in terms of time and space discretization parameters.
We show that the method (REF ) converges for any general chemical energy density that satisfies Lipschitz continuity constraints on the first order derivative of the convex and... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04382842779159546,
0.013147002086043358,
-0.021609002724289894,
-0.04416416212916374,
-0.008629865944385529,
-0.02601931430399418,
0.03870086744427681,
-0.02212786301970482,
0.04343165084719658,
0.022646723315119743,
-0.021761607378721237,
0.011491227895021439,
-0.015138509683310986,
0.... |
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