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fb299f5f78169de94c54c41b329e77e5cf539e84 | subsection | 13 | 62 | Strichartz Estimates | Furthermore, let (q,p) be a sharp wave-admissible Strichartz pair, i.e., 2 \le q,p < \infty and\frac{1}{q}+ \frac{1}{r} = \frac{1}{2}~.Then, it holds for all T>0 that\Big \Vert P_{M;k} F \Big \Vert _{L_t^q L_x^p([0,T]\times )} \lesssim \Big ( \frac{M}{N} \Big )^{\frac{1}{2}-\frac{1}{p}} N^{-1} N^{\frac{3}{2}-\frac{1}{q... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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e52c65c5a0dde2bf910ad7de95a1c36d3af8e391 | subsection | 14 | 62 | The truncated equations | Recall from the introduction that u_n , F_n^\omega , and w_n are supposed to solve (REF ), (REF ), and (REF ). However, we cannot directly work with the weak formulation of these equations. The problem is unrelated to any estimates in the deterministic part argument, and comes only from the moments with respect to \ome... | {
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"raw": "A. de Bouard and A. Debussche. A stochastic nonlinear Schrödinger equation with multiplicative noise. Comm. Math. Phys., 205(1):161–181, 1999.",
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equation | [
"Bjoern Bringmann"
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fa88c530c6b146fc8167bad037a1ab43d2db9ceb | subsection | 15 | 62 | The truncated equations | We now define the cutoff functions\theta _{F,w;\le n-1}(s)
&:= \theta \left(\sum _{m=0}^{n-1} \Big (\Vert \langle \nabla \rangle ^{\sigma ^\prime } F_{m}^\omega \Vert _{([0,s])} +\Vert \langle \nabla \rangle ^{\sigma } w_m \Vert _{([0,s])}+ \Vert \langle \nabla \rangle ^{\nu } w_m \Vert _{ ([0,s])}\Big ) \right)~, \\
\... | {
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} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
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372666ec0661dd9f589e3b01369e2992328938df | subsection | 16 | 62 | The truncated equations | \end{aligned}Then, we let F_n^\omega be a solution of the truncated equationF_n^\omega (t)= W(t) (Q_N f_0^\omega , Q_N f_1^\omega ) + 2 \theta _{F,w;\le n-1}(s) P_{\le N^\gamma } \nabla u_{n-1}(s) \nabla F_n^\omega (s) ~.In Section , it will be useful to decompose F_n^\omega into a superposition of the solutions corres... | {
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"Bjoern Bringmann"
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a83b4e3ee5aff0ab98d097601f441f7f1e3a577a | subsection | 17 | 62 | The adapted linear evolution | In this section, we study the adapted linear evolution F_n^\omega . Our main objective is to understand the frequency localization of the functions F_{n,k} and F_n^\omega , which we then use to prove probabilistic Strichartz estimates. In order to avoid continually interrupting the main argument, we deal with any issue... | {
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equation | [
"Bjoern Bringmann"
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1d7f1740d31559ece37eaa2ebf32ca62a7428c21 | subsection | 18 | 62 | The adapted linear evolution | We setE_q(F(t)) := \frac{1}{2} \Vert q(|\nabla | ) \partial _t F(t) \Vert _{L_x^2()}^2 + \frac{1}{2} \Vert q(|\nabla |) \nabla F(t) \Vert _{L_x^2()}^2~.For any \widetilde{D}> 0 , constructs a multiplier q satisfying the growth conditions (see )q(|\xi |) ~~ {\left\lbrace \begin{array}{ll}
\begin{}{ll}
> N^{(1-\gamma )\... | {
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... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
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] | 2,018 | en | Mathematics | [
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7baa72f0a6e9e110e0cfcfde592599075877280a | subsection | 19 | 62 | The adapted linear evolution | Putting everything together, we obtain that\Vert \langle \nabla \rangle ^s P_M F_n^\omega \Vert _{L_t^\infty L_x^2([0,1]\times )} &\lesssim N^{-(1-\gamma ) \widetilde{D} } \Vert (Q_N f_0^\omega , Q_N f_1^\omega )\Vert _{H^s\times H^{s-1}} \qquad \qquad &&\text{if} ~ 1\le M\ll N~, \\
\Vert \langle \nabla \rangle ^s P_M ... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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6cfe7916887b4b69b0d6948b64ce7f50aae49e25 | subsection | 20 | 62 | The adapted linear evolution | Furthermore, let F_k be the solution of-\partial _{tt} F_k + \Delta F_k = 2 ~ \nabla \phi \cdot \nabla F_k~, \qquad (F_k,\partial _t F_k )|_{t=0}= ( P_k f_0,P_k f_1) ~.Then, we have for all 0< T \le 1 that&\Vert \nabla F_k \Vert _{L_t^\infty ([0,T]\times )}+ \Vert \partial _t F_k \Vert _{L_t^\infty ([0,T]\times )}+ \Ve... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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b326a66104d1ddec20279567ff1c838e674e0047 | subsection | 21 | 62 | The adapted linear evolution | From Duhamels formula, it follows that&\Vert P_{M;k} \nabla _{x,t} F_k \Vert _{L_t^\infty L_x^2([0,T]\times )} \\
&\lesssim \Vert P_{M;k} \nabla _{x,t} W(t) (P_kf_0,P_k f_1) \Vert _{L_t^\infty L_x^2([0,T]\times )} + \Vert P_{M;k} \left( \nabla F_k \cdot \nabla \phi \right) \Vert _{L_t^1 L_x^2([0,T]\times )} \\
&\lesssi... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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c86f2b55096d759c1f0a133060b190f0a646b7da | subsection | 22 | 62 | The adapted linear evolution | For the inhomogeneous term, we also use the fundamental theorem of calculus.We remark that the definition of c(M) for M \gtrsim N^\gamma does not enter in a significant way. The weight only needs to grow in M and satisfy a local constancy condition.
Under the same conditions as in Proposition , we have that\begin{ali... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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a16734b72dc2ad0d4c405336e212f0e74279f2e4 | subsection | 23 | 62 | The adapted linear evolution | Furthermore, we assume that\sigma < \frac{3}{2}~.Then, it holds for all 0 < T \le 1 and all r \ge 1 that\begin{aligned}&\Vert \langle \nabla \rangle ^{\sigma ^\prime } F_n^\omega \Vert _{L_\omega ^r (\Omega \times [0,T])} \\
&\lesssim \sqrt{r} T^\frac{1}{2} N^{2\delta } \Vert (P_N f_0, P_N f_1) \Vert _{H_x^{\sigma ^\pr... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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ada8a58d5579bab72c25eb74a42e457a808ec938 | subsection | 24 | 62 | The adapted linear evolution | To this end, we prove that if F_k is a solution of (REF ) and D^\prime > 0 , then there exists a D^{\prime \prime }>0 s.t.\begin{aligned}\Vert \langle \nabla \rangle ^{\sigma ^\prime } F_k \Vert _{([0,T])}
&\lesssim T^{\frac{1}{q}} \Vert (P_k f_0,P_k f_1) \Vert _{H_x^{\sigma ^\prime } \times H_x^{\sigma ^\prime -1}} \\... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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1b91a24ebd1ddb898601d0a275571ff27057735a | subsection | 25 | 62 | The adapted linear evolution | Using the refined Strichartz estimates (see Lemma REF ), we have that&\Vert \langle \nabla \rangle ^{\sigma ^\prime } P_M \int _0^t \frac{\sin ((t-s)|\nabla |)}{|\nabla |} \nabla \phi \cdot \nabla F_k \Vert _{L_t^q L_x^p} \\
&\lesssim \sum _{1\le L \le N^\gamma } \sum _{1\le K \ll N} \Vert \langle \nabla \rangle ^{\sig... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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b82e81c86613c4017ccd39871cb06f20428fdc60 | subsection | 26 | 62 | The adapted linear evolution | \\
&\lesssim M^{\sigma ^\prime } \sum _{1\le L \le N^\gamma } \sum _{1\le K \ll N} \left( \frac{\max (L,K)}{N} \right)^{\frac{1}{2}-} \Vert P_M ( P_L \nabla \phi \cdot P_{K;k} \nabla F_{k} )\Vert _{L_t^1L_x^2} \\
&~~+ M^{\sigma ^\prime } \sum _{K\sim N} \Vert P_M( \nabla \phi \cdot \nabla F_k ) \Vert _{L_t^1L_x^2} \\
&... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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d3bc19777b1f0a8107853bc20ef37e1915bb0676 | subsection | 27 | 62 | The adapted linear evolution | After multiplying with c_{N,D^\prime }(M) and summing over M this completes the proof of (REF ).
We now apply (REF ) to the functions F_{n,k} . Due to the cutoff, \sigma ^\prime > \sigma , and (REF ), we have that\Vert \langle \nabla \rangle ^{\sigma } \left( \theta _{F,w;\le n-1}(t) u_{n-1}(t,x) \right) \Vert _{L_t^2 ... | {
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"raw": "Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu. On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on {R}^d, d\\ge 3. Trans. Amer. Math. Soc. Ser. B, 2:1–50, 2015.",
... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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3819102d435309307aa3d88c333dab012def6fe7 | subsection | 28 | 62 | The adapted linear evolution | Since we cannot use Minkowski's integral inequality to switch \ell _M^1 \ell _k^2 into \ell _k^2 \ell _M^1 , we use (REF ) with a slightly larger D^\prime .In this step, we crucially rely on the independence of the individual (Q_N f_0^\omega , Q_N f_1^\omega )_N .
Recall that the functions F_{n,k} are measurable with r... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0480b2a20fc4ed45ef0395770747b0779daf6f22 | subsection | 29 | 62 | The adapted linear evolution | We have for all T>0 and all r \ge \max (q,p) that&\Vert \langle \nabla \rangle ^{\sigma ^\prime } F_n^\omega \Vert _{L_\omega ^r }\\
&= \mathbb {E}\Big [ \mathbb {E} \Big [ \Vert c_{N,D^\prime }(M) \sum _{k} g_k \langle \nabla \rangle ^{\sigma ^\prime } P_M F_{n,k} \Vert _{\ell _M^1 L_t^q L_x^p}^r\Big | {F}_{n-1} \Big ... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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d7b3ae41f8f092c6d1744339d1f27f276637f22a | subsection | 30 | 62 | The nonlinear evolution | Recall that the nonlinear evolution w_n solves the truncated equationw_n(t) &= \theta _{F;n}(s) |\nabla F_n^\omega |^2\\
&~~~+ 2 \theta _{F;n}(s) \nabla F_n^\omega \nabla w_n \\
&~~~+ \theta _{w;n}(s) |\nabla w_n|^2\\
&~~~+ 2 \theta _{F,w;\le n-1}(s) \nabla F_{\le n-1}^\omega \nabla w_n \\
&~~~+ 2 \theta _{F,w;\le n-1}... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0e523227a9fecf8aa45e4e8164e23de89ebba237 | subsection | 31 | 62 | Bilinear Estimates | In this section we prove the main bilinear estimates for the Duhamel terms in (). In order to group similar estimates together, we work with a paraproduct decomposition. We define(v,w) &:= \sum _{L,K\colon L \ll K} \nabla P_L v \cdot \nabla P_K w ~~,\\
(v,w) &:=\sum _{L,K\colon L \gg K } \nabla P_L v \cdot \nabla P_K w... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
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-0.018448861315846443,
0.002151604276150465,
-0.0045130280777812,
0.009880593977868557,
0.033845800906419754,
0.01017052587121725,
0.006809598300606012,
0.038789913058280945,
0.023423492908477783,
0.013703124597668648,
0.00026084386627189815,
0.01... | |
20b086616dbfb1b607c57c8a093cc81e49792f85 | subsection | 32 | 62 | Bilinear Estimates | Then, we have for any 0 < T \le 1 that\Vert (G,F) \Vert _{} &\lesssim T^{\frac{1}{2}} N^{\nu -s+1-\sigma ^\prime } \Vert \langle \nabla \rangle ^{\sigma ^\prime } G \Vert _{} \Vert \langle \nabla \rangle ^s F \Vert _{}~, \\
\Vert (P_{>N^\gamma } G,F) \Vert _{} &\lesssim T^{\frac{1}{2}} N^{\nu -s+\gamma (1-\sigma ^\prim... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.007020783144980669,
0.02471010386943817,
-0.014736012555658817,
-0.0015625057276338339,
0.0027777880895882845,
-0.009821465238928795,
0.0066697439178824425,
0.0031078411266207695,
0.0003610552230384201,
0.022405454888939857,
0.011935330927371979,
-0.01353790145367384,
-0.01584254950284958... | |
c12c886c13aa4891d33010cb8c6f760965fc0fb8 | subsection | 33 | 62 | Bilinear Estimates | Then, we have for any 0 < T \le 1 that\Vert (G,F) \Vert _{} &N^{\nu -s+1-\sigma ^\prime } \Vert \langle \nabla \rangle ^{\sigma ^\prime } G \Vert _{} \Vert \langle \nabla \rangle ^s F\Vert _{}~~, \\
\Vert (v,F) \Vert _{} &N^{1-\sigma ^\prime } \Vert \langle \nabla \rangle ^{\nu } v \Vert _{} \Vert \langle \nabla \rangl... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.015087735839188099,
0.01543861348181963,
-0.011495054699480534,
0.009168583899736404,
0.004363086074590683,
-0.005053399596363306,
0.011441660113632679,
0.004664383362978697,
-0.0028184622060507536,
0.024546174332499504,
0.025919172912836075,
-0.01845921203494072,
-0.011868814937770367,
... | |
b58195cba1003688a39b1e5623882eba6e060e82 | subsection | 34 | 62 | Bilinear Estimates | Then, we have for any 0 < T \le 1 that\Vert (G,F) \Vert _{} &N^{\nu -s+1-\sigma ^\prime } \Vert \langle \nabla \rangle ^{\sigma ^\prime } G \Vert _{} \Vert \langle \nabla \rangle ^s F \Vert _{}~, \\
\Vert (v,F) \Vert _{} &N^{1-\sigma ^\prime } \Vert \langle \nabla \rangle ^{\nu } v \Vert _{} \Vert \langle \nabla \rangl... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.012262471951544285,
0.03186563774943352,
-0.008454771712422371,
-0.005902315955609083,
-0.0025143027305603027,
-0.002069816691800952,
0.005921392701566219,
0.019824456423521042,
0.00831742025911808,
0.0481647290289402,
-0.013880171813070774,
-0.005887054838240147,
-0.021793168038129807,
... | |
328aab37ebe6a585cc7cfa291f3bd682f99746e4 | subsection | 35 | 62 | Bilinear Estimates | Then, we have for any 0 < T \le 1 that\begin{aligned}&\Big \Vert \theta _{F,w;\le n-1}(s) \nabla P_{>N^\gamma } u_{n-1} \nabla F_n^\omega \Big \Vert _{} \\
&\left( N^{\nu -s+\gamma (1-\sigma ^\prime )} + N^{(1-\gamma ) (\nu -1)+1-\sigma ^\prime } \right) \left( \Vert ^s F_n^\omega \Vert _{} + \Vert ^{\sigma ^\prime } F... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.006480558775365353,
0.05184447020292282,
-0.03265072777867317,
-0.029111020267009735,
-0.005885522346943617,
-0.0436055026948452,
0.045527927577495575,
0.018385104835033417,
0.019147971644997597,
0.006343242712318897,
-0.0027978161815553904,
0.013266264460980892,
-0.04653491452336311,
-... | |
fc8578a17c985e4d39cceb2c4615261a372b7df6 | subsection | 36 | 62 | Bilinear Estimates | It is bounded by\Vert \langle \nabla \rangle ^{\nu } \nabla P_{N^\gamma } F_{\le n-1}^\omega \cdot \nabla F_n^\omega \Vert _{L_t^\infty L_x^2} \lesssim T^{\frac{1}{2}} N^{\nu -1} \Vert \nabla P_{N^\gamma } F_{\le n-1}^\omega \Vert _{L_t^2L_x^\infty } \Vert \nabla F_n^\omega \Vert _{L_t^\infty L_x^2}~.Thus, the resultin... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04378727078437805,
0.059807002544403076,
-0.0036196967121213675,
-0.0282099861651659,
-0.0030494704842567444,
-0.039362773299217224,
0.043085452169179916,
0.02428896725177765,
0.01010768860578537,
0.014844952151179314,
-0.04766251891851425,
-0.0016410690732300282,
-0.040918976068496704,
... | |
2d3a187a5edf2be23d4f6443256a3fece5f1af90 | subsection | 37 | 62 | Bilinear Estimates | Then, we for all M \ge 1 that&\Vert ^\nu P_M (H,F) \Vert _{L_t^\infty L_x^2} + \Vert ^{\nu -1} \partial _t P_M (H,F) \Vert _{L_t^\infty L_x^2}+ \Vert ^\sigma P_M (H,F) \Vert _{L_t^2L_x^\infty } \\
&\lesssim M^{\nu -1} \sum _{1\le L \ll M} \sum _{K\sim M} \Vert \nabla P_L H \cdot \nabla P_K F \Vert _{L_t^1 L_x^2} \\
&M^... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.0012310475576668978,
0.03228376805782318,
-0.009779723361134529,
-0.004458852112293243,
-0.03798987716436386,
0.004218554589897394,
-0.0070754243060946465,
0.03335175663232803,
0.006549058482050896,
-0.007208922877907753,
-0.033778950572013855,
0.0010679885745048523,
0.013685510493814945,... | |
0d2c69ed4b5dc886fdfcb80c767a6aea8e8175e0 | subsection | 38 | 62 | Bilinear Estimates | If H= G , then&\sum _{M\ge 1} \sum _{1\le L \ll M} M^{\nu -s} L^{1-\sigma ^\prime } \max \left( \frac{N}{M} , \frac{M}{N} \right)^{-D} \Vert ^{\sigma ^\prime } P_L H \Vert _{L_t^2 L_x^\infty }\\
&\lesssim \sum _{M\ge 1} \sum _{1\le L \ll M} M^{\nu -s} L^{1-\sigma ^\prime } \max \left( \frac{N}{M} , \frac{M}{N} \right)^... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02081206627190113,
0.020079676061868668,
-0.02792234532535076,
-0.051572419703006744,
-0.039060767740011215,
-0.012618458829820156,
0.008269896730780602,
0.02513011172413826,
-0.02123929187655449,
0.008681866340339184,
-0.034208688884973526,
-0.0149682080373168,
0.009452400729060173,
0.... | |
de2e0c7a1beb0a61c61ae7f1552f8a0cc72b1c7b | subsection | 39 | 62 | Bilinear Estimates | For any M \ge 1 , we have that&~\Vert ^\nu P_M (P_{>N^\gamma }v,F) \Vert _{L_t^\infty L_x^2} +\Vert ^{\nu -1} \partial _t P_M (P_{>N^\gamma }v,F) \Vert _{L_t^\infty L_x^2} \\
&+ \Vert ^\sigma P_M (P_{>N^\gamma } v,F) \Vert _{L_t^2L_x^\infty } \\
&\lesssim M^{\nu -1} \sum _{N^\gamma \le L \ll M} \sum _{K\sim M} \Vert \n... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.006963795050978661,
0.036123257130384445,
-0.04012000188231468,
-0.04768636077642441,
-0.035299498587846756,
-0.00759686715900898,
0.014850502833724022,
0.0256279855966568,
-0.007657886482775211,
0.017436183989048004,
-0.0022863061167299747,
0.0010258822003379464,
-0.008268076926469803,
... | |
1544db073b40bd7eda3f36b860b0f9a2189c7155 | subsection | 40 | 62 | Bilinear Estimates | For any M \ge 1 , we have that&\Vert ^\nu (G,w) \Vert _{L_t^\infty L_x^2} +\Vert ^{\nu -1} \partial _t (G,w) \Vert _{L_t^\infty L_x^2}+ \Vert ^\sigma (G,w) \Vert _{L_t^2 L_x^\infty } \\
&\lesssim M^{\nu -1} \sum _{1\le L \ll M} \sum _{K\sim M} \Vert \nabla P_L G \cdot \nabla P_K w \Vert _{L_t^1L_x^2}\\
&\sum _{L\ge 1} ... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.047171082347631454,
0.05839938670396805,
-0.021419210359454155,
-0.03176267445087433,
-0.02649940922856331,
-0.010724861174821854,
0.005869687534868717,
0.011930073611438274,
-0.012929331511259079,
0.003085495438426733,
-0.02172432839870453,
-0.01861213520169258,
-0.014104031957685947,
... | |
7d5bae217c0ef38cb8721463494b5c6bc8aec96d | subsection | 41 | 62 | Bilinear Estimates | For any M \ge 1 , we have for all sufficiently large D^\prime >0 that&\Vert \langle \nabla \rangle ^{\nu } P_M (G,F) \Vert _{L_t^\infty L_x^2}+\Vert \langle \nabla \rangle ^{\nu -1} \partial _t P_M (G,F) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (G,F) \Vert _{L_t^2 L_x^\infty }\\
&\lesssim... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.005013902671635151,
0.04731536656618118,
-0.03022078238427639,
-0.03281549736857414,
-0.03214392438530922,
-0.006837833672761917,
0.010508590377867222,
-0.006708097644150257,
-0.007936771027743816,
0.004697194788604975,
-0.027534490451216698,
-0.010439906269311905,
-0.001778141944669187,
... | |
45578d9d15ee5089954dd6971b38aaca9c26f893 | subsection | 42 | 62 | Bilinear Estimates | For any M \ge 1 , we have that&\Vert \langle \nabla \rangle ^{\nu } P_M (v,F) \Vert _{L_t^\infty L_x^2}+\Vert \langle \nabla \rangle ^{\nu -1} P_M \partial _t (v,F) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (v,F) \Vert _{L_t^2 L_x^\infty }\\
&\lesssim M^{\nu -1} \sum _{L\sim M} \sum _{K\ll... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.014314013533294201,
0.045200541615486145,
-0.029177391901612282,
-0.027941320091485977,
-0.02897901087999344,
0.005371570121496916,
0.011094124056398869,
0.010147994384169579,
-0.002172662876546383,
-0.0001472365256631747,
-0.013169502839446068,
-0.01173505000770092,
-0.00415075896307826,... | |
84b81360624a3eec8a74f9c4ed914aacf3dbe0db | subsection | 43 | 62 | Bilinear Estimates | For any M \ge 1 , it follows from \eta < \nu -1 that&\Vert \langle \nabla \rangle ^{\nu } P_M (G,w) \Vert _{L_t^\infty L_x^2} +\Vert \langle \nabla \rangle ^{\nu -1} P_M \partial _t (G,w) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (G,w) \Vert _{L_t^2 L_x^\infty }\\
&\lesssim M^{\nu -1} \sum... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.008924124762415886,
0.06034843996167183,
-0.010655557736754417,
-0.034079477190971375,
-0.03368284925818443,
-0.026284215971827507,
0.005998994689434767,
0.005354474764317274,
-0.00863428134471178,
-0.005590925924479961,
-0.009130066260695457,
-0.010602165013551712,
-0.00996908638626337,
... | |
2d91743d438f2ceb08e10e9761e2c990316b548c | subsection | 44 | 62 | Bilinear Estimates | For any M \ge 1 , it holds that&\Vert \langle \nabla \rangle ^{\nu } P_M (v,w) \Vert _{L_t^\infty L_x^2}+\Vert \langle \nabla \rangle ^{\nu -1} P_M \partial _t(v,w) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (v,w) \Vert _{L_t^2 L_x^\infty }\\
&\Big ( \sum _{K\ll M} K^{1-\sigma } \max \left(... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.029156263917684555,
0.04436756670475006,
-0.038661420345306396,
-0.018247459083795547,
-0.02645576372742653,
-0.0011366518447175622,
0.020734360441565514,
-0.003951581660658121,
0.008688902482390404,
-0.004744949284940958,
-0.025509824976325035,
0.01573004201054573,
-0.016447123140096664,... | |
7f212c8dca5cc527607c2da9a883583d4e9b684b | subsection | 45 | 62 | Bilinear Estimates | For any M \ge 1 , we have that&\Vert \langle \nabla \rangle ^{\nu } P_M (G,F) \Vert _{L_t^\infty L_x^2}+\Vert \langle \nabla \rangle ^{\nu -1} \partial _t P_M (G,F) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (G,F) \Vert _{L_t^2 L_x^\infty }\\
&\lesssim M^{\nu -1} \sum _{L\sim K \gg M} \Vert... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.014004810713231564,
0.053059618920087814,
-0.022578561678528786,
-0.03209817036986351,
-0.014859134331345558,
-0.008733936585485935,
0.016506759449839592,
0.01612536422908306,
-0.01421076338738203,
-0.007631706073880196,
-0.02483641728758812,
-0.012227511033415794,
-0.001702926936559379,
... | |
369988a2b057fb263e3735b3e2ff33cd29267105 | subsection | 46 | 62 | Bilinear Estimates | For any M \ge 1 , we have that&\Vert \langle \nabla \rangle ^{\nu } P_M (v,F) \Vert _{L_t^\infty L_x^2}+ \Vert \langle \nabla \rangle ^{\nu -1} P_M \partial _t (v,F) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (v,F) \Vert _{L_t^2 L_x^\infty }\\
&\lesssim M^{\nu -1} \sum _{L\sim K \gg M} \Ver... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.006651284173130989,
0.042409565299749374,
-0.023554088547825813,
-0.026086458936333656,
-0.017818119376897812,
-0.00551858264952898,
0.027108559384942055,
0.01853511482477188,
-0.01071680523455143,
-0.0019755535759031773,
-0.02143361046910286,
-0.010182872414588928,
-0.0059800539165735245... | |
626d4f451c9fc958541ae785e00678ec86b8a04e | subsection | 47 | 62 | Bilinear Estimates | For any M \ge 1 , we have that&\Vert \langle \nabla \rangle ^{\nu } P_M (G,w) \Vert _{L_t^\infty L_x^2}+\Vert \langle \nabla \rangle ^{\nu -1} P_M \partial _t (G,w) \Vert _{L_t^\infty L_x^2} + \Vert \langle \nabla \rangle ^{\sigma } P_M (G,w) \Vert _{L_t^2 L_x^\infty }\\
&\lesssim M^{\nu -1} \sum _{L\sim K \gg M} \Vert... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.017482437193393707,
0.04719952866435051,
-0.02968658320605755,
-0.03160873427987099,
-0.021769143640995026,
-0.0075208041816949844,
0.0024789669550955296,
0.004496464505791664,
-0.011754116974771023,
0.004397306125611067,
-0.014370380900800228,
0.0019488494144752622,
-0.008253052830696106... | |
79b5e2faf30f1169095528df52eaef445c7fb8c3 | subsection | 48 | 62 | Control of the nonlinear component | [Proof of Proposition :]We begin by showing the a-priori estimate for w_n , which forms the main part of the proof. Afterwards, we will use contraction mapping to prove the existence and uniqueness of w_n . This step could potentially be replaced by a soft argument, since all involved functions are smooth (with norms g... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.008133835159242153,
0.035801082849502563,
0.003689220640808344,
-0.022661810740828514,
-0.03531274572014809,
-0.00612708181142807,
0.006016443483531475,
0.017854759469628334,
0.032901592552661896,
0.023516396060585976,
0.014726361259818077,
0.007050340995192528,
-0.01249070093035698,
0.0... | |
60ca10823181481290e5f0ac70abe646ff7def7e | subsection | 49 | 62 | A-priori bounds: | We separate the proof into six cases, corresponding to the different terms in ().Case 1: Contribution of |\nabla F_n^\omega |^2 . Using (REF ), (REF ), and (REF ), we have that\Big \Vert \theta _{F;n}(s) \nabla F_n^\omega \cdot \nabla F_n^\omega \Big \Vert _{}
N^{\nu -s+1-\sigma ^\prime } \Vert ^s F_n^\omega \Vert _{} ... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.010180259123444557,
0.03458540886640549,
0.0009381826384924352,
-0.010752612724900246,
-0.014072262682020664,
-0.01796426624059677,
0.010363412089645863,
0.03064761683344841,
0.013728850521147251,
0.005849451757967472,
-0.015476436354219913,
0.006925476249307394,
-0.03504329174757004,
-0... | |
284a3cc4c6331eb809ed6ff4637163f8c206e310 | subsection | 50 | 62 | A-priori bounds: | We have that&\Big \Vert \theta _{F,w;\le n-1}(s) \nabla P_{>N^\gamma } u_{n-1} \nabla F_n^\omega \Big \Vert _{} \\
&\left( N^{\nu -s+\gamma (1-\sigma ^\prime )} + N^{(1-\gamma ) (\nu -1)+1-\sigma ^\prime } \right) \left( \Vert ^s F_n^\omega \Vert _{} + \Vert ^{\sigma ^\prime } F_n^\omega \Vert _{} \right)~.Combining th... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.037083204835653305,
0.03595392405986786,
0.0004506639379542321,
-0.01777857355773449,
-0.02086121030151844,
-0.02209731750190258,
0.0021574641577899456,
0.015069820918142796,
0.01373452041298151,
0.03174200281500816,
-0.01311646681278944,
0.012414480559527874,
-0.0062263160943984985,
-0... | |
2f29fd54b7b0159e61d51d9dc8c1fcaee1515063 | subsection | 51 | 62 | Contraction Mapping: | Due to the cutoffs, we may work on the whole space . We set\Gamma w(t)&:=
\theta _{F;n}(s) |\nabla F_n^\omega |^2+2 \theta _{F;n}(s) \nabla F_n^\omega \nabla w \\
&~~+ \theta _{w}(s) |\nabla w|^2+ 2 \theta _{F,w;\le n-1}(s) \nabla F_{\le n-1}^\omega \nabla w \\
&~~+ \theta _{F,w;\le n-1}(s) \nabla w_{\le n-1} \nabla w ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1314,
"openalex_id": "",
"raw": "A. de Bouard and A. Debussche. A stochastic nonlinear Schrödinger equation with multiplicative noise. Comm. Math. Phys., 205(1):161–181, 1999.",
"source_ref_id": "6c001de4c3fea21ec0082d4f4a8f... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.022861674427986145,
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... | |
1f4a48aeb2ce8106656ac4ca78f4c4fdf307db14 | subsection | 52 | 62 | Contraction Mapping: | Due to the continuity statement (REF ), we have that\Vert 1_{[0,t_v]} ~ \langle \nabla \rangle ^\nu v \Vert _{(\bigcap )([0,T])}+ \quad \Vert 1_{[0,t_v]} ~ \langle \nabla \rangle ^\sigma v \Vert _{(\bigcap )([0,T])} \le 2~.To avoid confusion, we point out that the continuity statement (REF ) is not enforced solely by t... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
94fca4362bdc5da9a9ddd989c62d212047eeb0ef | subsection | 53 | 62 | Contraction Mapping: | Using (), (), and (), we have that&\Big \Vert \big ( \theta _{v}(s) \ |\nabla v|^2- \theta _{w}(s) |\nabla w|^2 \big ) \Big \Vert _{}\\
&\le \Big \Vert 1_{[0,t_v]}(s) (\theta _v(s)-\theta _w(s)) |\nabla v|^2 \Big \Vert _{} \\
&+ \Big \Vert 1_{[0,t_v]}(s) \theta _w(s) \left( |\nabla v|^2 - |\nabla w|^2 \right) \Big \Ver... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
12be325d891d81480dfa0234e36fb2bd07e3b1ab | subsection | 54 | 62 | Proof of the Main Theorem | As in Section , any question regarding the (strong) measurability of the solutions is addressed in the appendix.
Before we begin with the proof of the main theorem, we collect all conditions on the parameters. | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0... | |
1273a697e03e5fc0082b5a2f9102032b37c0e893 | subsection | 55 | 62 | Parameter Conditions: | First, we have the basic conditions\nu > 2 > s > 1~, \quad \sigma = \nu - 1 -, \quad \sigma ^\prime > \sigma , \quad \text{and} ~ \quad \gamma \in (0,1)~.In order to use Proposition , Proposition , and Corollary REF , we require the major conditions\begin{aligned}\sigma ^\prime -s + 1 - \gamma (\sigma -1) - \frac{1}{2}... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
bce1ba25bca611a6d4891a06afbc639a0d7ddcc1 | subsection | 56 | 62 | Parameter Conditions: | Using Lemma and writing M=2^m , we have that&\Big \Vert \sum _{m=n_-}^{n_+} \langle \nabla \rangle ^s F_m^\omega \Big \Vert _{L_\omega ^2 L_t^\infty L_x^2}
\lesssim \Big \Vert \sum _{m=n_-}^{n_+} \langle \nabla \rangle ^s P_N F_m^\omega \Big \Vert _{L_\omega ^2 L_t^\infty \ell _N^2 L_x^2}
\lesssim \Big \Vert \sum _{m=n... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.01835673116147518,
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... | |
e65c24cf58fe3a00ed57b9982fd0503d393ca4a2 | subsection | 57 | 62 | Parameter Conditions: | From Proposition , we have that\Vert \langle \nabla \rangle ^{\sigma } F_m^\omega \Vert _{L_\omega ^2 L_t^2 L_x^\infty } \lesssim \Vert \langle \nabla \rangle ^{\sigma ^\prime } F_m^\omega \Vert _{L_\omega ^2 } \lesssim M^{-\epsilon } \Vert (\widetilde{P}_M f_0, \widetilde{P}_M f_1) \Vert _{H_x^s\times H_x^{s-1}}~.This... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
8fde72a7018eda87c5f3f8401bbe37d5dbbd597d | subsection | 58 | 62 | Parameter Conditions: | To control the nonlinear components w_m , we recall from Proposition that\Vert w_m \Vert _{([0,T])}\lesssim T^{\frac{1}{2}} M^{-\epsilon } \left( \Vert \langle \nabla \rangle ^s F_m^\omega \Vert _{([0,T])}+ \Vert \langle \nabla \rangle ^{\sigma ^\prime } F_m^\omega \Vert _{([0,T])}\right)Using Lemma and Proposition , w... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-... | |
959e502138b72b4cc25a9269b6c99e23f94b5f22 | subsection | 59 | 62 | Strong measurability of | In this section, we prove the strong measurability of the iterates. As before, let (\Omega , {F},\mathbb {P} ) be the given probability space. We recall the following definition from the theory of Bochner-integration.
Let E be a Banach space. A function v\colon \Omega \rightarrow E is called simple if there exist meas... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0089... | |
30eb67ada86b7e16e54ffeaaca94013f6d718546 | subsection | 60 | 62 | Strong measurability of | Then, the maps \omega \mapsto u\in C_t^0 H_x^s L_t^2 W_x^{\sigma ,\infty } and \omega \mapsto \partial _t u \in C_t^0 H_x^{s-1} are strongly \mathbb {P}-measurable.Before we prove the proposition, we need the following lemma which proves the measurability of the cutoff.If \omega \in \Omega \mapsto v^\omega \in ([0,T]) ... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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002c980f562f267ff2939c48e64a2190a3a3a236 | subsection | 61 | 62 | Strong measurability of | SinceF_n^\omega = \sum _{N/2 \le \Vert k\Vert _2 < N} g_k(\omega ) F_{n,k}~,this proves REF . Since the proof of Proposition consists of a contraction mapping argument, w_n \in depends continuously on F_m^\omega , w_m , where m=0,\hdots , n-1, and F_n^\omega , all in their respective norms. Thus, REF follows from REF w... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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49ea2de983c22ec5addb3433b80fcdc0ac5a2a06 | abstract | 0 | 167 | Abstract | This paper presents an accelerated quadrature scheme for the evaluation of
layer potentials in three dimensions. Our scheme combines a generic, high order
quadrature method for singular kernels called Quadrature by Expansion (QBX)
with a modified version of the Fast Multipole Method (FMM). Our scheme extends
a recently... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.007266426458954811,
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-0.02065877430140972,... |
56a175d16387986e3838a768376c5ec60c917014 | subsection | 1 | 167 | Introduction | Integral equation methods are an attractive approach for the solution of
boundary value problems of elliptic partial differential equations (PDEs). The
mathematical features that make integral equation methods attractive in two
dimensions also hold in the three dimensional case, where their impact is felt
even more dra... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... |
ce0f4c1e1e2a35756a2b735d451abcfc3d88b0e5 | subsection | 2 | 167 | Introduction | A variety of other
acceleration methods have been utilized, such as fast direct solvers
(e.g. ), recursive compressed inverse
preconditioning , particle-mesh Ewald summation
(e.g. ), or methods based on the Fast Fourier Transform (e.g. ).Quadrature by Expansion (QBX, ) is a quadrature method that has been recently deve... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.006890913937240839,
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... |
4560c47816faf2181671d73eb63c85e89d2515f0 | subsection | 3 | 167 | Introduction | The contribution
develops a three-dimensional local QBX algorithm with optimizations to decrease
the cost of applying the QBX expansions.This paper describes an accelerated global QBX scheme in three dimensions which
builds and extends on GIGAQBX, our previous scheme for two dimensions featuring rigorous
error bounds ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
44653e1038d18bc8615ba3f2f88bb633a83c0e34 | subsection | 4 | 167 | Mathematical Preliminaries | As a model problem, consider the exterior Neumann problem for the Laplace
equation in three dimensions, for a smooth bounded domain \Omega . Given
continuous Neumann boundary data g, the problem is to find u such that\triangle u &= 0 & \quad & \text{in } \mathbb {R}^3 \setminus \Omega , \\
\partial _n u &= g & \quad & ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.070878766477108,
0.04587889462709427,
0.0108744... |
1db87318448aabf45ab4e10861787c2af5d63285 | subsection | 5 | 167 | QBX Discretization | The idea of QBX is to use the smoothness of the potential for purposes of close and on-surface evaluation
to recover a
high-order accurate approximation everywhere in the domain. This is accomplished
through formation of a local expansion of the potential near the source geometry
and analytic continuation of the local ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.049501631408929825,
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0.008613955229520798,
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0.009979675523936749,
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0.059145599603652954,
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0.0... |
5351b301b8974185bb9059269e9a3f94208cd349 | subsection | 6 | 167 | QBX Discretization | (Some authors,
e.g. , , follow the convention of
defining the local coefficient (REF ) using the
Y^m_n, the complex conjugate
of Y^{-m}_n. Both (REF ) and the latter definition
yield equivalent expansions, since the outer partial sums
of (REF ) are real .)Next, we describe the details of QBX. The QBX-based approximatio... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0006032749079167843,
0.029102172702550888,
-0.01922849379479885,
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-0.004860535729676485,
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0.07288514822721481,
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0.03... |
082b02f34d6ff69da184049acb127fec2f64e1b5 | subsection | 7 | 167 | First Approximation Step: Truncation | In the first stage, a local expansion of the potential is formed and truncated.
For a selection of points \lbrace x_i\rbrace _{i=1}^{N_C/2} on the surface
\Gamma , we define a collection of N_C expansion centers c_{i}^{\pm }c_{i}^{\pm }x_i\pm r(x_i)\hat{n}(x_i)where \hat{n}(x) is a unit-length normal vector to the surf... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.01881353370845318,
0.06689256429672241,
-0.04870462045073509,
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0.02397085167467594,
-0.0103451544418931,
0.00... |
007bb698e902450258605750d2ccc8b2e54fe651 | subsection | 8 | 167 | First Approximation Step: Truncation | Then
for each p > 0 and \delta > 0, there is a constant M_{p,\delta } such that\left| \mathcal {S}\mu (x_i) -
\sum _{n=0}^p \sum _{m=-n}^n L^m_n
\vert x_i - c \vert _2^n Y^m_n(\theta _{x_i-c}, \phi _{x_i-c})
\right| \le M_{p, \delta } r^{p+1}
\Vert \mu \Vert _{W^{3 + p + \delta , 2}(\Gamma )}. | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
6e4bc8bb1467a3a1fa493a70120735f35ff74a4d | subsection | 9 | 167 | Second Approximation Step: Quadrature | In the second stage, we apply numerical quadrature to discretize the
integrals for the computation of the expansion coefficients in (REF ). We assume that
the smooth, non-self-intersecting surface \Gamma is tessellated into individual,
disjoint surface elements \Gamma _k so that\Gamma =\bigcup _{k=1}^{K} \Gamma _k.Each... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.027114802971482277,
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0.06420882791280746,
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0.0... |
eb0dca046e1a4a586fd5d46bc75da89eb631924c | subsection | 10 | 167 | Second Approximation Step: Quadrature | (In practice, our implementation uses a triangular reference element
with nodes and weights based on .)
A tensor product rule is based on iterated evaluation of a
one-dimensional q-point quadrature ruleQ_q\left\lbrace \int _{-1}^1 f(y) \, dy \right\rbrace = \sum _{j=1}^q w_j f(y_j).After repeated application of (REF ),... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.03259306773543358,
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0.0... |
b31bde8711ffa5e0454268b9fa965226adb742b8 | subsection | 11 | 167 | Second Approximation Step: Quadrature | Then there is a constant C_{p,q} > 0 such that for all
h > 0 and r > 0\left|
\sum _{j_1=1}^q \sum _{j_2=1}^q w_{j_1} w_{j_2}
\frac{\mu (\Psi _k(y_{j_1},y_{j_2}))}{\vert \Psi _k(y_{j_1}, y_{j_2}) - c_{i}^{\pm } \vert _2^{n+1}}
\tilde{Y}^{-m}_{n}(\Psi _k(y_{j_1},y_{j_2}) - c_{i}^{\pm })
\vert \partial _{s_1} \Psi _k(y_{j... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.032723553478717804,
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0.004323202650994062,
-0.0001874469016911462... |
2c53f2db82b95d325ff158bb60734b59e1e21dc6 | subsection | 12 | 167 | Third Approximation Step: Acceleration | The third approximation applied in the rapid, QBX-based evaluation of
layer potentials like (REF ) arises due to acceleration.
The formation of local expansions (REF ) at all centers
covering a neighborhood of \Gamma requires O(NM) operations, where N is the number
of source points and M is the number of target points.... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
7d8354e56b33a4b0041b26cead237ee916480c8c | subsection | 13 | 167 | Third Approximation Step: Acceleration | The procedure suggested in is to
set the FMM order to {p_\mathrm {fmm}}^{\prime } = {p_\mathrm {fmm}}+ {p_\mathrm {add}}, where {p_\mathrm {fmm}} is the FMM order
required for the point FMM to achieve a specified tolerance, and {p_\mathrm {add}}> 0 is
an empirically determined quantity that depends on {p_\mathrm {fmm}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
1c22d2003c5183d791483f58169e94ef0e8f594d | subsection | 14 | 167 | Accuracy Control for QBX on Surfaces | Since the cost of computational methods dealing with
three-dimensional geometries is typically far greater than that of methods
applied to two-dimensional geometries, and since that cost is directly
related to the resolution supplied, it is not surprising that careful control of
resolution and accuracy plays an importa... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00350481946952641,
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0.01281348243355751,
0.... |
ce970ca875b9bc62f4ce9686cb40674f14db11f6 | subsection | 15 | 167 | Overview | We commence our discussion with an outline of a procedure for efficiently
detecting and remedying potential sources of truncation and quadrature
inaccuracy in arbitrary smooth geometries. From an initial, user-supplied,
unstructured mesh, the process creates a set of three related, unstructured
discretizations satisfyi... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0041885981336236,
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0.0060501969419419765,
-0.010948339477181435,
0.03... |
add32d762a9cce646e0f4b1152bed6195c46041b | subsection | 16 | 167 | Stage-1 Discretization. | Algorithm REF of
Section REF produces the stage-1 discretization from the user-supplied mesh. The
stage-1 discretization is a locally refined mesh fitting the geometry
description which ensures that [(a)]sufficient resolution to represent the density and the geometry
is available, and thatthe assumptions of Lemma REF a... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.004376161843538284,
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0.032292794436216354,
-0.023303920403122902,
... |
01093a656fc1a76de98c94aa1dd591a1679d32ff | subsection | 17 | 167 | Stage-1 Discretization. | In our
case, these are quadrature rules based on . To
accomplish this measurement, we define a modified element mapping
\tilde{\Psi }_k:\tilde{E} \rightarrow \mathbb {R}^3, where \tilde{E} is the
`bi-unit' equilateral triangle with vertices
v_1=\begin{bmatrix}
-1\\ -1/\sqrt{3}
\end{bmatrix},\quad v_2=\begin{bmatrix}
1... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.041587427258491516,
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... |
ee3dae5f51df18818fe1b25938b8040307b562dc | subsection | 18 | 167 | Stage-1 Discretization. | To prevent the source point that spawned the center from being found and
causing refinement, we reduce the size of the queried area by a factor of
\varepsilon _{\text{exp-disturb}}. In practice, we choose \varepsilon _{\text{exp-disturb}}=0.025.
The discretization appears to be fairly insensitive to the choice
of this ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04649128019809723,
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-0.015629373490810394,
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0.03956184908747673,
-0.039592377841472626,
0... |
935e7da4b9141af2f34c5a6ffadd44c5771f77f9 | subsection | 19 | 167 | Stage-1 Discretization. | We use the complex-valued logarithm, which satisfies \mathop {\mathrm {Re}}\log y = \log |y| for all |y| > 0, to
rewrite the above as
S\sigma (z)
= - \frac{1}{2 \pi } \mathop {\mathrm {Re}}\int _0^L \sigma (w(t)) \log \left( 1 - \frac{z}{w(t)} \right) \, dt
+ S \sigma (0).
By expanding the kernel \log (1 - z/w(t)) in... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03254026174545288,
0.026603037491440773,
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0.013568617403507233,
-0.010783158242702484,
... |
174f6ff610131c0d739722556b2da07b09eecd4f | subsection | 20 | 167 | Stage-1 Discretization. | The next higher derivative w^{\prime \prime }(t), whose magnitude represents the
curvature at parameter t, is the first derivative whose magnitude is not controlled.
However, the contribution of the term
w^{\prime \prime }(t) to the truncation error may be dampened by ensuring that r is chosen
to locally enforce that |... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.029311802238225937,
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0.020187148824334145,
0.0023956026416271925,
... |
a77ab6bb3517c00a6fec49c1c4707b66fc99c60f | subsection | 21 | 167 | Stage-1 Discretization. | We leave a detailed discussion and potential proofs of
its properties for future work.
[Figure: 2D QBX geometric evaluation scenario for the single-layer potential\mathcal {S}\sigma in Section , for a segment ofthe closed curve \Gamma .]The steps in this section work together to manage truncation error under the
assump... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.017551658675074577,
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0.015689658001065254,
-0.004429886117577553,
... |
f709374a36a49b365bf8f6e946cfb3adc07372fc | subsection | 22 | 167 | Stage-1 Discretization. | That is, the main factors governing
the error are the ratio of the source panel size h^{\text{stage-2},1}_j to the center
distance, the density norm \Vert \mu \Vert (with the choice of norm depending on the quadrature rule),
and the order of quadrature accuracy Q. For simplicity, we may
consider h^{\text{stage-2},1}_{j... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.007542537059634924,
0.005299233365803957,
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... |
e22e7227fc6c54e50e5943d614ce23890e286e1f | subsection | 23 | 167 | Stage-1 Discretization. | Instead, for the benefit of the treatment of
the `non-self interaction' from other elements whose resolution may differ, we
choose a higher value of Q so that the coefficient integrals for a
hypothetical source element larger by a factor of (4/3)
would still attain the required level of accuracy in its
coefficient inte... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02362990379333496,
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0.049029380083084106,
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0... |
874ac82487d15ccb68375fd8c112431b09909701 | subsection | 24 | 167 | Stage-1 Discretization. | The halving of \eta ^{{\text{stage-}2}}_{} through bisection implies that the set of
`endangered' centers found in the current iteration will be equal to or a
superset of that found in the following iteration. For smooth,
non-self-intersecting geometries, the associated procedure, detailed in
Algorithm REF , is guarant... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... |
430177fe5196a487996437ab35d5cb20600aaa31 | subsection | 25 | 167 | Stage-1 Discretization. | Section REF implies that a `danger
zone' of radius {r^\mathrm {danger}_{s}} = \eta _k/2 exists around each source particle
s\in \Gamma _k. Using area queries around each source point of size
{r^\mathrm {danger}_{s}}, every target that some source endangers can be identified
efficiently.
In the second stage, an area que... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0... |
64ce42f6273ffe84ca54170290608c0141e15988 | subsection | 26 | 167 | Stage-1 Discretization. | Flag targets that could not be associatedendangered targets t
t is not associated to a center
Flag t.
Error Estimates for FMM Translations
In , error estimates were presented for the GIGAQBX FMM that
applied to the 2D Laplace kernel with complex Taylor expansions. In this
section, we present their analogs in thre... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.015944046899676323,
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0.028561968356370926,
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... |
bd3a1d36d3f3d6fcce5f114a5773c5301771f12c | subsection | 27 | 167 | Stage-1 Discretization. | The expression M_p(t) for a p-th order multipole expansion evaluated at t \in \mathbb {R}^3 takes the form
M_p(t) = \sum _{n=0}^p \sum _{m=-n}^n \frac{M^m_n}{\vert t-c \vert _2^{n+1}} Y^{-m}_n(\theta _{t-c}, \phi _{t-c}).
The multipole expansion converges for \vert t - c \vert _2 > \vert s - c \vert _2.
Translation o... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.040216799825429916,
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0.008307301439344883,
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0.010778895579278469,
-0.002479222137480974,
-... |
ffc5d72be4f2d80b079a4a538f47834a935d51f6 | subsection | 28 | 167 | Stage-1 Discretization. | If r < \rho , then
\left|
\mathcal {G}(s, t) - \sum _{n=0}^p \sum _{m=-n}^n L^m_n \vert t-c \vert _2^n Y^m_n(\theta _{t-c}, \phi _{t-c})
\right| \le \frac{1}{4 \pi }
\frac{1}{\rho - r} \left( \frac{r}{\rho } \right)^{p+1}.
Next, consider the multipole expansion of \mathcal {G}(s, \cdot ) centered at c and
evaluated ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.021456750109791756,
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0.014223108999431133,
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0.008233226835727692,
0.02568400464951992,
-0.04257776215672493,
0.008782616816461086,
-0.03308551385998726,
... |
91be45ccdaae8e48ef6a738454302df1039a9504 | subsection | 29 | 167 | Stage-1 Discretization. | Then the accuracy
may be measured by
\text{GIGAQBX~FMM Accuracy} = \left| \sum _{n=0}^q \sum _{m=-n}^n
(\tilde{L}_p)^m_n \vert x-c \vert _2^n Y^m_n(\theta _{x-c}, \phi _{x-c}) - \sum _{n=0}^q
\sum _{m=-n}^n L^m_n \vert x-c \vert _2^n Y^m_n(\theta _{x-c}, \phi _{x-c}) \right|.
The formulas (REF ) and (REF ) are
relat... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.023408031091094017,
-0.0034314740914851427,
-0.04419143497943878,
-0.007690774276852608,
0.013329150155186653,
-0.0019303233129903674,
0.029466042295098305,
-0.0019169712904840708,
0.01568673737347126,
-0.005100478883832693,
-0.0433674231171608,
0.02485768124461174,
-0.03192281723022461,
... |
b30cab58bd982224378e92b3e0fcdf074520bb54 | subsection | 30 | 167 | Stage-1 Discretization. | Suppose a p-th order multipole expansion M_p with coefficients
\langle (T_p^c)^m_n \rangle is formed at c due to the source s. Next
suppose that this is translated to a q-th order local expansion with
coefficients \langle (T_q^{c^{\prime }})^m_n \rangle = \mathrm {M2L}_{{p} \rightarrow {q}}^{{c} \rightarrow {c^{\prime ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.028795504942536354,
0.02119605429470539,
-0.022645749151706696,
-0.013711054809391499,
0.010346238501369953,
-0.01250552013516426,
0.005680512171238661,
0.00929330289363861,
0.05072403699159622,
-0.008591345511376858,
-0.023912323638796806,
0.025285718962550163,
-0.03799724578857422,
-0... |
bb71827f1e88c3635ba563221a451c1565489e54 | subsection | 31 | 167 | Stage-1 Discretization. | Given this asymptotic behavior
and the fact that it holds for q = 0, it is at
least plausible that the bound E_M(q) \le C ((4\pi )^{-1}/(\rho - r)) (r/\rho )^{p+1}, for
some C > 0, should hold for all q. We formulate this statement as the
following hypothesis.
Hypothesis 1 (Source \rightarrow Multipole(p) \rightarrow L... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03995407000184059,
0.008836289867758751,
-0.028935415670275688,
0.029255902394652367,
0.0018561549950391054,
-0.03818375989794731,
0.03629136085510254,
0.01739787682890892,
0.032506559044122696,
-0.015963314101099968,
-0.019000312313437462,
0.03952675312757492,
-0.02963743545114994,
0.0... |
f7ec389cacd7c1e9cc4ff159f828614ba553c406 | subsection | 32 | 167 | Stage-1 Discretization. | The potential due to the source s can be described in a q-th order local expansion
centered at c with coefficients \langle (L_q^{c})^m_n \rangle . Consider a p-th
order local expansion of the potential centered at the origin with coefficients
\langle (T_p^0)^m_n \rangle . Suppose this expansion is translated to a q-th
... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02342217229306698,
0.04583726450800896,
-0.02444450743496418,
0.0033206839580088854,
-0.019042912870645523,
-0.02326958440244198,
0.008560155518352985,
0.0049209450371563435,
0.02436821348965168,
-0.0009279223158955574,
-0.004417406395077705,
0.053008876740932465,
-0.04830918088555336,
... |
27a053e2ff7f1926b8dd55d4274db84e2eec3832 | subsection | 33 | 167 | Stage-1 Discretization. | Hypothesis 2 (Source \rightarrow Local(p) \rightarrow Local(q))
For the situation described above, there exists a constant C > 0 independent
of p, q, \rho , r, s, c, and t such that the error in the
local-mediated approximation to the local expansion of the potential satisfies
the bound
\left|
\sum _{n=0}^q \sum _{m=... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.016490798443555832,
0.028221994638442993,
-0.03295108675956726,
0.019023150205612183,
-0.003909127786755562,
0.0034133358858525753,
0.017055237665772438,
0.018931617960333824,
0.0424092672765255,
-0.030372967943549156,
-0.021250398829579353,
0.030235672369599342,
-0.015636511147022247,
... |
a144e809505c52eb4ecfd4407a573a05850e5e83 | subsection | 34 | 167 | Stage-1 Discretization. | The error is given by
E_{\mathit {M2L}}(q) = \left|
\sum _{n=0}^q \sum _{m=-n}^n
(T_q^{c^{\prime }})^m_n \vert t-c^{\prime } \vert _2^n Y^m_n(\theta _{t-c^{\prime }}, \phi _{t-c^{\prime }})
-
\sum _{n=0}^q \sum _{m=-n}^n
(L_q^{c^{\prime }})^m_n \vert t-c^{\prime } \vert _2^n Y^m_n(\theta _{t-c^{\prime }}, \phi _{t-c^{... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0695972666144371,
0.013728671707212925,
0.0003913415421266109,
-0.021489683538675308,
0.0008070078329183161,
0.00036773228202946484,
-0.005582280922681093,
0.02512217126786709,
-0.0023351714480668306,
-0.012728974223136902,
-0.05821140110492706,
0.034951258450746536,
-0.0391942523419857,
... |
f5c5570a8ac8b566706d6628a75b9bcbcfc39e51 | subsection | 35 | 167 | Stage-1 Discretization. | Hypothesis 3 (Source \rightarrow Multipole(p) \rightarrow Local(p) \rightarrow Local(q))
For the situation described above, there exists a constant C > 0 independent
of R, p, q, \rho , r, s, c, c^{\prime }, and t such that the error in the multipole and
local mediated approximation to the local expansion of the potent... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.023119576275348663,
-0.012551717460155487,
-0.04706321656703949,
-0.010071894153952599,
0.013871746137738228,
-0.026827864348888397,
0.061194390058517456,
0.028414951637387276,
0.019960664212703705,
0.003458398627117276,
-0.027606148272752762,
-0.010392364114522934,
-0.017229044809937477,... |
2bfbb193d1b1abeb72a4cfee9b10df637dd137f8 | subsection | 36 | 167 | Stage-1 Discretization. | The most important implication of this design is that interaction lists
involving direct evaluations at particles (List 1 and List 3), as well as the
FMM step of evaluation of far-field local expansions, must be redefined to
incorporate the possibility of evaluation at non-leaf boxes.
Two-Away Near Neighborhood.
To ob... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05534491315484047,
-0.015697330236434937,
-0.04018150642514229,
-0.013416717760264874,
-0.004446814302355051,
-0.01986192911863327,
0.035147376358509064,
-0.016536351293325424,
0.034384630620479584,
0.062484223395586014,
-0.002978526521474123,
0.016231253743171692,
-0.01993820257484913,
... |
633dd079bfd9265e7e96db073d6f17ea4512bbb0 | subsection | 37 | 167 | Stage-1 Discretization. | We say that two boxes are k-well-separated if they are on the same
level and are not k-colleagues.
The parent of b is denoted \mathsf {Parent}(b). The set of ancestors is
\mathsf {Ancestors}(b). The set of descendants is \mathsf {Descendants}(b). \mathsf {Ancestors} and
\mathsf {Descendants} are also defined in the nat... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.009392818436026573,
-0.016679691150784492,
-0.06562001258134842,
-0.027804572135210037,
0.006588707212358713,
-0.0395856536924839,
0.03189437836408615,
-0.014550856314599514,
0.01895350217819214,
0.02994103915989399,
-0.026705820113420486,
-0.00018205262313131243,
-0.0396161749958992,
0... |
1de5dda980003e3524766fadee6594b7ed46dd41 | subsection | 38 | 167 | Stage-1 Discretization. | Definition 2 (List 1, {1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3far}{W^\mathrm {far}_{b}}{{1}{4}{X_{b}}{{1}{4close}{X^\mathrm {close}_{b}}{{1}{4far}{X^\mathrm {far}_{b}}{}}}}}}}})
For a target box b, {1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.012174873612821102,
-0.009199810214340687,
-0.055839646607637405,
0.004241371992975473,
0.0037550635170191526,
-0.004794428590685129,
0.04842487350106239,
0.030132049694657326,
0.015333017334342003,
-0.011084017343819141,
0.011831597425043583,
0.01872001215815544,
0.005377998575568199,
... |
04f2fc25a15263a08f58cabd6d3339d60d2d5743 | subsection | 39 | 167 | Stage-1 Discretization. | Definition 4 (List 3, {3}{1}{U_{b}}{{3}{2}{V_{b}}{{3}{3}{W_{b}}{{3}{3close}{W^\mathrm {close}_{b}}{{3}{3far}{W^\mathrm {far}_{b}}{{3}{4}{X_{b}}{{3}{4close}{X^\mathrm {close}_{b}}{{3}{4far}{X^\mathrm {far}_{b}}{}}}}}}}})
For a target box b, a box d \in \mathsf {Descendants}(T_b) is in {3}{1}{U_{b}}{{3}{2}{V_{b}}{{3}{3}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0028478484600782394,
0.03047369420528412,
-0.05966557189822197,
-0.005825404543429613,
0.011292205192148685,
-0.00039031545748002827,
0.05347011610865593,
0.01152873132377863,
0.03729479759931564,
0.0020924913696944714,
0.007362823002040386,
0.010239283554255962,
0.0022984978277236223,
... |
53ea469859c0f87ba5936ab22ffe63c51f03d40c | subsection | 40 | 167 | Stage-1 Discretization. | Definition 5 (List 4, {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{W^\mathrm {close}_{b}}{{4}{3far}{W^\mathrm {far}_{b}}{{4}{4}{X_{b}}{{4}{4close}{X^\mathrm {close}_{b}}{{4}{4far}{X^\mathrm {far}_{b}}{}}}}}}}})
For a target or target-ancestor box b, a source box d is in List 4 of b if
d is a 2-colleague of so... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.004835828207433224,
0.006441414821892977,
-0.05674852430820465,
-0.01639910228550434,
0.005076094530522823,
-0.018641585484147072,
0.03316432610154152,
0.027672532945871353,
0.04335465282201767,
-0.010876799933612347,
0.029915014281868935,
-0.008153785951435566,
0.003992990590631962,
0.0... |
02fdd4a1743c9516ce76252fc42818cc4d79c205 | subsection | 41 | 167 | Stage-1 Discretization. | For any d \in {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{W^\mathrm {close}_{b}}{{4}{3far}{W^\mathrm {far}_{b}}{{4}{4}{X_{b}}{{4}{4close}{X^\mathrm {close}_{b}}{{4}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}, either b \in {3}{1}{U_{d}}{{3}{2}{V_{d}}{{3}{3}{W_{d}}{{3}{3close}{W^\mathrm {close}_{d}}{{3}{3far}{W^\mathr... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03140251711010933,
0.002094264142215252,
-0.058441027998924255,
-0.018279988318681717,
-0.013298004865646362,
-0.01158139482140541,
0.028243955224752426,
-0.014167753979563713,
-0.012603730894625187,
0.012870759703218937,
-0.015823328867554665,
0.008186321705579758,
0.013740508817136288,
... |
80aa6accdf0cd7dc3ce27aef78a1a62fa3340d56 | subsection | 42 | 167 | Stage-1 Discretization. | Interactions from these source boxes must be accumulated directly. The `far' list
is the smallest possible `complement' of this list in the sense that the close
and far lists must cover the entire near field mediated by List 3, and the `far' lists contains
boxes that are as large as permissible given the target confine... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.032623205333948135,
0.01969904452562332,
-0.042632944881916046,
-0.024505550041794777,
-0.004852282349020243,
-0.031524576246738434,
0.03570547327399254,
0.01673884689807892,
-0.005191789474338293,
0.005638107657432556,
-0.0021896304097026587,
0.016281085088849068,
-0.016021685674786568,
... |
41c07282a34bb82cafcb98b6867c4f54ba8029ea | subsection | 43 | 167 | Stage-1 Discretization. | Definition 7 (List 3 far, {3far}{1}{U_{b}}{{3far}{2}{V_{b}}{{3far}{3}{W_{b}}{{3far}{3close}{W^\mathrm {close}_{b}}{{3far}{3far}{W^\mathrm {far}_{b}}{{3far}{4}{X_{b}}{{3far}{4close}{X^\mathrm {close}_{b}}{{3far}{4far}{X^\mathrm {far}_{b}}{}}}}}}}})
For a target box b, a box d is in {3far}{1}{U_{b}}{{3far}{2}{V_{b}}{{3f... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02303655631840229,
0.03414292261004448,
-0.0542503260076046,
-0.031305305659770966,
-0.0002858115767594427,
0.005743883084505796,
0.03003905899822712,
0.016247637569904327,
0.0185512937605381,
-0.00426786532625556,
0.02158723585307598,
-0.008390798233449459,
0.006666870787739754,
0.0270... |
9103933db3a57a9c8014d5f2f2e5e173e32a9965 | subsection | 44 | 167 | Stage-1 Discretization. | While {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4}{X_{b}}{{3close}{4close}{X^\mathrm {close}_{b}}{{3close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}\cup {3far}{1}{U_{b}}{{3far}{2}{V_{b}}{{3far}{3}{W_{b}}{{3far}{3close}{W^\math... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03304358199238777,
0.03139597922563553,
-0.07762037962675095,
-0.013989364728331566,
-0.03429453819990158,
0.012631618417799473,
-0.000037543020880548283,
0.04286817088723183,
-0.017177779227495193,
-0.02324949949979782,
0.002091921167448163,
0.010625509545207024,
0.02186124213039875,
0... |
30c0d9836fdd3993e4663d02b3ab83aa149375d6 | subsection | 45 | 167 | Stage-1 Discretization. | For the case of List 4, the `close' list consists of source boxes from which the
TCR of the target box is not adequately separated. List 4 close is evaluated
directly only at the targets in the box, and is not
downward-propagating. List 4 far is downward-propagating. It consists of boxes
form which the TCR of the targe... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06453010439872742,
0.015216772444546223,
-0.04774128645658493,
0.013530259020626545,
0.000522742688190192,
-0.047313936054706573,
0.0264805406332016,
0.02510691061615944,
-0.005944767035543919,
0.007829693146049976,
-0.004887834656983614,
0.003458877559751272,
-0.0017551943892613053,
0.... |
b278800db1c6c47e4c9b2b8d9da6554591486f63 | subsection | 46 | 167 | Stage-1 Discretization. | A box d \in {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{W^\mathrm {close}_{b}}{{4}{3far}{W^\mathrm {far}_{b}}{{4}{4}{X_{b}}{{4}{4close}{X^\mathrm {close}_{b}}{{4}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} is in
List 4 far if \mathsf {TCR}(b) \prec d. Furthermore, if b has a parent, a
box d \in {4close}{1}{U_{\maths... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02731196954846382,
0.03545978665351868,
-0.04507237672805786,
-0.0037668393924832344,
-0.022200511768460274,
0.005683635827153921,
0.035856496542692184,
0.024718094617128372,
-0.0015963769983500242,
0.010261060670018196,
0.0209493488073349,
-0.015258083119988441,
0.011680062860250473,
-... |
a52ea57415b75285fbd422af5eb2d32028005b69 | subsection | 47 | 167 | Stage-1 Discretization. | However, {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} \cup {4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}{{4far}{3close}{W^\... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.022466493770480156,
0.0191087294369936,
-0.05131273716688156,
0.014369703829288483,
-0.0058760871179401875,
-0.02261911891400814,
0.06349226087331772,
0.010538799688220024,
0.028434155508875847,
0.00022345346224028617,
0.011027202010154724,
-0.00452916556969285,
0.03192928433418274,
0.0... |
c5cfd963c190e2b1d665f2b5015933459ee45489 | subsection | 48 | 167 | Stage-1 Discretization. | The following notation refers to `point'
potentials evaluated at a target not requiring QBX owned by a box b:
[(a)]
\mathsf {P}^{\text{near}}_b(t) denotes the potential at a target point t due to all
sources in {1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3far}{W^\mathrm {far}_{b}}{{... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.04239572212100029,
-0.06727154552936554,
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c4345d3d6516fa8f9a52f9e48de78b29edcd4608 | subsection | 49 | 167 | Stage-1 Discretization. | The following notation refers to
potentials evaluated with QBX mediation:
[(a)]
\mathsf {L}^{\text{qbx},\text{near}}_{c}(t) denotes the (QBX) local expansion of the potential at
the center c, evaluated at target t, due to all sources in {1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.02250153385102749,
0.006430099252611399,
0.01... |
3945b8d5746def17dee2588a61f386ee24b14dac | subsection | 50 | 167 | Stage-1 Discretization. | Lastly, given a box b, \mathsf {M}_b and \mathsf {L}^{\text{far}}_b refer respectively to the
multipole and local expansions associated with the box.
Algorithmic Parameters
The parameters to the algorithm are {p_\mathrm {fmm}}, the FMM order; {p_\mathrm {qbx}}, the QBX
order; {p_\mathrm {quad}}, the upsampled quadratur... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
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