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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
b2d025cbce0555d928df987dd31a785339510ed2 | subsection | 71 | 126 | A posteriori error analysis for the Poisson example | In this subsection, we develop a computable quantity that is
equivalent to the function \eta defined above, for the example of the DPG*
method for Poisson equation. We then provide a complete analysis of
reliability and efficiency of the resulting error estimator.Recall the variational formulation derived in eg:poisson... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.00006479136209236458,
0.030258698388934135,
0.003313121385872364,
0.010406672954559326,
-0.004074166528880596,
-0.06866572797298431,
0.024872252717614174,
0.03143364563584328,
0.023163238540291786,
0.03628602251410484,
-0.024811217561364174,
0.0428168959915638,
-0.012268276885151863,
0.0... |
5c0790a86784a0d4bd57ecda5f7f1698ca6af30f | subsection | 72 | 126 | A posteriori error analysis for the Poisson example | If
E\in \mathcal {E}\setminus \mathcal {E}_{{\protect \scalebox {0.6}{\mathrm {int}}}} is an exterior edge on the
boundary of an element K, then with \vec{n} equal to the outward unit
normal on \partial , we simply set
\llbracket \vec{\tau }\cdot \vec{n} \rrbracket = \vec{\tau }_{K}\cdot \vec{n} and
\llbracket \vec{\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.012208187952637672,
0.013016980141401291,
-0.011460435576736927,
-0.014985550194978714,
-0.016252148896455765,
-0.034671250730752945,
0.04581122472882271,
0.015412836335599422,
0.02789570763707161,
0.030093181878328323,
-0.04379687085747719,
0.04227085039019585,
0.024217991158366203,
0.0... |
cc2a4fd82649dff6c57bf5872780ac6a8880dc3f | subsection | 73 | 126 | A posteriori error analysis for the Poisson example | Then\Vert (\vec{p}, v) - (\vec{p}_h, v_h) \Vert _{V}\;\eqsim \;
\eta _i(\vec{p}_h, v_h)for i\in \lbrace 1,2\rbrace , where the computable error estimators \eta _i
are defined by\eta _1(\vec{p}_h, v_h)^2
& =
\big \Vert \mathcal {L}(\vec{p}_h, v_h) - f \big \Vert _{\scriptscriptstyle }^2
+
\sum _{E\in \mathcal {E}_{\prot... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.042503975331783295,
0.024837210774421692,
0.008711331523954868,
0.009702988900244236,
0.004237429704517126,
-0.031916119158267975,
0.0015876058023422956,
0.0271561648696661,
0.018704188987612724,
-0.0034021486062556505,
-0.021801212802529335,
-0.0009682964882813394,
-0.01445531751960516,
... |
852671d2638674aaddf020537d87d38463a89e27 | subsection | 74 | 126 | A posteriori error analysis for the Poisson example | \end{equation}
Similarly, for all \widehat{q}_n\in H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h), the continuous right inverse of \operatorname{\mathrm {tr}}_n: H(\operatorname{div},) \rightarrow H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h) is defined
\begin{equation}
\hspac... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030239492654800415,
0.0341758131980896,
-0.00035043558455072343,
0.028790071606636047,
0.019056066870689392,
-0.045374494045972824,
0.035365864634513855,
0.04873104766011238,
-0.012037820182740688,
0.023404330015182495,
-0.05642060562968254,
-0.021039484068751335,
0.004802159499377012,
... |
5d7806baa2e5bd6b1cf3b9f23303c2345f49a94f | subsection | 75 | 126 | A posteriori error analysis for the Poisson example | Likewise, for any \widehat{\mu }\in \widetilde{P}^0_{p+2}(\partial _h),
the function \hspace{-0.5pt}{E}_{\mathrm {grad}}\hspace{0.5pt}(\widehat{\mu }) is supported only on _E.
}Our first lemma may be
thought of as an inf-sup condition involving the space of edge bubbles \widetilde{P}^0_{p+2}(\partial _h).
}\begin{}
Fo... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04005947336554527,
0.019282246008515358,
-0.004362913314253092,
0.012219207361340523,
0.01478966511785984,
-0.05494829639792442,
0.0003899544244632125,
0.03795429319143295,
-0.0018019898561760783,
0.01978565938770771,
-0.032035376876592636,
-0.008664806373417377,
0.007066851481795311,
-... |
686efaaa635eb8d607706faccf2a985b41827ad2 | subsection | 76 | 126 | A posteriori error analysis for the Poisson example | For any \vec{q}_h \in P_p(_h) with
nontrivial \llbracket \vec{q}_h \cdot \vec{n} \rrbracket _E, we have
\begin{equation}
\begin{aligned}h_E \big \Vert \llbracket \vec{q}_{h} \cdot \vec{n} \rrbracket \big \Vert _{L^2(E)}^2
& \lesssim h_E (b_E \llbracket \vec{q}_h \cdot \vec{n} \rrbracket , \llbracket \vec{q}_h \cdot n ... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.056960877031087875,
0.05818124860525131,
0.015651274472475052,
0.003523824969306588,
-0.018290329724550247,
-0.019800540059804916,
0.012142704799771309,
0.005663290154188871,
0.002061285078525543,
0.002885036403313279,
-0.06480176746845245,
0.014957187697291374,
-0.00986976083368063,
0.... |
2617cc9a287e39a0a0751082982d37764eeb3702 | subsection | 77 | 126 | A posteriori error analysis for the Poisson example | \end{equation}
This can be seen beginning from the linearity of \hspace{-0.5pt}{E}_{\mathrm {grad}}\hspace{0.5pt}
and the fact that \mathcal {E}_K = \lbrace E \in \mathcal {E}:
\mathrm {meas}(E\cap \partial K)\ne 0\rbrace has fixed finite cardinality:
\begin{align}
{|\hspace{-0.5pt}{E}_{\mathrm {grad}}\hspace{0.5pt}(\w... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04156837612390518,
0.054081618785858154,
0.0020219567231833935,
-0.02751387096941471,
-0.01693865656852722,
-0.06879230588674545,
-0.02896357700228691,
0.006336736027151346,
-0.0005040585529059172,
0.0006366302259266376,
-0.026827169582247734,
-0.01754905842244625,
-0.004280444234609604,
... |
0894fe473a2c957d4fd121b6d2cc462d2657b254 | subsection | 78 | 126 | A posteriori error analysis for the Poisson example | Starting from~[{eq:4}]{\textup {{\ref *{eq:4}}}},
\begin{align}
\sum _{E\in \mathcal {E}_{\protect \scalebox {0.6}{\mathrm {int}}}}
h_E &
\Big \Vert \llbracket \vec{q}_h \cdot \vec{n} \rrbracket \Big \Vert _{L^2(E)}^2
\lesssim \sum _{E\in \mathcal {E}_{\protect \scalebox {0.6}{\mathrm {int}}}}
\frac{ (b_E \llbracket \v... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.011008994653820992,
0.042662713676691055,
-0.00197215867228806,
-0.01142860297113657,
0.0032271689269691706,
-0.023742197081446648,
-0.006217831280082464,
0.0480642169713974,
0.000650392787065357,
0.013488497585058212,
-0.03359917551279068,
0.015868820250034332,
-0.006008027121424675,
-0... |
0dd6a2713513d40fd4f34a92cb5d03a4434bfd45 | subsection | 79 | 126 | A posteriori error analysis for the Poisson example | Now, the result follows by noting that the numerator above
equals \langle { \mu , \vec{q}_h \cdot \vec{n}} \rangle _h^2 and by bounding the
denominator using [{eq:BubbleNormBound}]{\textup {{\ref *{eq:BubbleNormBound}}}} and the Poincaré
inequality.
\end{}
}Lemma 4.3
For any degree p\ge 1 and for any w_h \in P_p(_h),\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05341994762420654,
0.02217691019177437,
0.006170003674924374,
-0.028877297416329384,
0.0013240515254437923,
-0.032784584909677505,
-0.016895966604351997,
0.005322915967553854,
0.0025832359679043293,
0.0061509255319833755,
-0.04758954048156738,
0.013637349009513855,
0.019307494163513184,
... |
524660a99418b3fcd4f4aaee8711017e0ae1a6e9 | subsection | 80 | 126 | A posteriori error analysis for the Poisson example | Using
the vector curl of the scalar function \phi _E, by an application
of [eq:BubbleBounds1]eq:BubbleBounds1, we have\big |\llbracket w_h \vec{n} \rrbracket \big |_{H^1(E)}^2
& \lesssim (b_E \llbracket \vec{n}^\perp \cdot \operatorname{grad}w_h \rrbracket ,
\llbracket \vec{n}^\perp \cdot \operatorname{grad}w_h \rrbrac... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.07848268002271652,
0.04531367868185043,
0.00791082251816988,
-0.013792445883154869,
-0.009100878611207008,
-0.008582135662436485,
-0.02181769721210003,
-0.006446137558668852,
0.0014265417121350765,
0.028851233422756195,
-0.03292488679289818,
0.003728461218997836,
0.018308555707335472,
-... |
ad43e1eacc035f3c7b389f706ed0f34c5217b39a | subsection | 81 | 126 | A posteriori error analysis for the Poisson example | Hence
\begin{align}
\big |\llbracket w_h \vec{n} \rrbracket \big |_{H^1(E)}^2
& \lesssim \frac{( \operatorname{curl}\phi _E, \llbracket w_h \vec{n} \rrbracket )_E }{\Vert \operatorname{curl}\phi _E \Vert _{H(\operatorname{div}, _E)}}
\Vert \operatorname{curl}\phi _E \Vert _{H(\operatorname{div}, _E)}
\lesssim \frac{( \... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0519619956612587,
0.04155739024281502,
0.017788516357541084,
-0.005446398165076971,
-0.02077869512140751,
0.030298450961709023,
-0.01960398256778717,
0.034203991293907166,
0.002517242915928364,
0.01624765805900097,
-0.037255194038152695,
-0.0038502374663949013,
-0.0016791154630482197,
0... |
64f3950601fec58762257e46f498af89210447b0 | subsection | 82 | 126 | A posteriori error analysis for the Poisson example | By proving an analogue of
[{eq:BubbleNormBound}]{\textup {{\ref *{eq:BubbleNormBound}}}} for \hspace{-0.5pt}{E}_{\mathrm {div}}\hspace{0.5pt}, and following along the lines of the
proof of {lem:inf-H1-sup}, we finish the proof.
\end{align}
}Since the suprema in {lem:inf-H1-sup,lem:inf-Hdiv-sup} are related to the funct... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.10864151269197464,
0.030852969735860825,
-0.001864415011368692,
-0.0020179550629109144,
0.013900620862841606,
-0.019073300063610077,
-0.03277555853128433,
-0.01710493490099907,
0.005382485222071409,
0.002769443206489086,
-0.06787043064832687,
0.0020904336124658585,
0.010490315034985542,
... |
4ba1afa2a369130d9ae47e5e6f77d76aa3eb03a5 | subsection | 83 | 126 | A posteriori error analysis for the Poisson example | \end{gather}
\end{}
These results also hold with H1() replaced by H10() and H(div,) replaced by H0(div,).Lemma 4.4
For any degree p \ge 0 and any \vec{q}_{h} \in P_p(_h)^2
satisfying \langle \widehat{\mu }_1,\vec{q}_{h}\cdot \vec{n}\rangle _h = 0
for all \widehat{\mu }_1 \in \operatorname{\mathrm {tr}}( P_1(_h) \cap H... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.024922162294387817,
0.016116229817271233,
0.02302972599864006,
-0.004647151567041874,
0.0003722395049408078,
-0.04596788436174393,
-0.017932359129190445,
0.0053453692235052586,
-0.015620228834450245,
0.014429825358092785,
-0.05610157549381256,
-0.009965812787413597,
-0.008950917981564999,... |
cdb1f35194109ba8e933ca14f2ffdd0d9f945269 | subsection | 84 | 126 | A posteriori error analysis for the Poisson example | Since
\langle \mathcal {I}_\mathrm {grad}\mu ,\vec{q}_{h}\cdot \vec{n}\rangle _h = 0,\langle \mu &,\vec{q}_{h}\cdot \vec{n}\rangle _h
=
\langle \mu - \mathcal {I}_\mathrm {grad}\mu ,\vec{q}_{h}\cdot \vec{n}\rangle _h
=
\sum _{E \in \mathcal {E}_{{\protect \scalebox {0.6}{\mathrm {int}}}}}
(\mu - \mathcal {I}_\mathrm {g... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.026269741356372833,
0.03118196874856949,
0.01801658794283867,
-0.05876367166638374,
-0.0030949325300753117,
-0.013455233536660671,
0.007013654801994562,
0.013912893831729889,
-0.006456833798438311,
-0.011113534681499004,
-0.03743666782975197,
0.007364528253674507,
0.026269741356372833,
... |
e95812ff219123ad83e54a51a68baf4e95009848 | subsection | 85 | 126 | A posteriori error analysis for the Poisson example | This completes the proof of~[{eq:FluxEquivalence}]{\textup {{\ref *{eq:FluxEquivalence}}}}.
}
}Lemma 4.5
For any degree p \ge 1 and any w_h \in P_p(_h) satisfying
\int _E \llbracket w_h \vec{n} \rrbracket = 0 on all edges E\in \mathcal {E},\sum _{E\in \mathcal {E}}
h_E
\Big \Vert \llbracket w_h \vec{n} \rrbracket \Big... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03986608237028122,
-0.011324897408485413,
-0.0007092370069585741,
-0.009546796791255474,
-0.009806262329220772,
0.00968416128307581,
0.03968292847275734,
-0.0008451701141893864,
-0.00011655668640742078,
-0.004372753668576479,
-0.012805376201868057,
-0.027243858203291893,
0.029853774234652... |
97dfbe5419e07570888cf99b144921701320b42d | subsection | 86 | 126 | A posteriori error analysis for the Poisson example | To prove the remaining equivalence,
first observe that lem:inf-Hdiv-sup implies\sum _{E \in \mathcal {E}_{{\protect \scalebox {0.6}{\mathrm {int}}}}} h_E \Big | \llbracket w_h \vec{n} \rrbracket \Big |_{H^1(E)}^2
& \lesssim \sup _{\widehat{\sigma }\cdot \vec{n} \in \widehat{P}_p(\partial _h)}
\frac{\langle { \widehat{\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0485934279859066,
0.03497994318604469,
0.03156131133437157,
0.004651019815355539,
0.01514729205518961,
0.008996805176138878,
-0.002149999840185046,
0.023243654519319534,
-0.013918721117079258,
0.008264240808784962,
-0.04270239174365997,
0.003714329795911908,
-0.01893983781337738,
0.0150... |
31ad18095c4f7125f4a6acf9519e13a5f48b0be1 | subsection | 87 | 126 | A posteriori error analysis for the Poisson example | Therefore, by the
commutativity property of the quasi-interpolators,
\langle { {\vec{\sigma }}\cdot \vec{n}, w_h} \rangle _h
=
\langle { \vec{n}\cdot ( \operatorname{curl}\varphi _{\vec{\sigma }}- \mathcal {I}_\mathrm {grad}\varphi _{\vec{\sigma }}),
w_h } \rangle _h
+
\langle {\vec{n}\cdot ({\vec{\psi }}_{\vec{\sigma... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0720910131931305,
-0.0005155216786079109,
0.02510366402566433,
0.020403403788805008,
0.00000524582674188423,
0.0071762907318770885,
0.002724014688283205,
0.024325374513864517,
0.015550537966191769,
0.02118169330060482,
-0.011361506767570972,
0.011620936915278435,
0.02937662973999977,
0.... |
36b8ccc7b2f52400bd5c6cc635430241ad16a6f7 | subsection | 88 | 126 | A posteriori error analysis for the Poisson example | The term t_2 can be estimated
similarly:
\begin{align}
t_2
& =
\sum _{E \in \mathcal {E}} ( {\vec{\psi }}_{\vec{\sigma }}- \mathcal {I}_\mathrm {div}{\vec{\psi }}_{\vec{\sigma }}, \llbracket w_h \vec{n} \rrbracket )_E
\le \sum _{E \in \mathcal {E}} h_E^{-
\Vert {\vec{\psi }}_{\vec{\sigma }}- \mathcal {I}_\mathrm {div}{... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.01855761930346489,
0.008576794527471066,
-0.00682557187974453,
-0.001797006349079311,
-0.007851887494325638,
-0.0551539771258831,
0.01808452233672142,
0.01993112824857235,
0.017733514308929443,
0.01640579104423523,
-0.009248287416994572,
0.010713363066315651,
-0.002090784488245845,
0.024... |
e2b9cc1909d0225c7bc545a215af33951e6b58ab | subsection | 89 | 126 | A posteriori error analysis for the Poisson example | Hence the conditions of {lem:FluxBounds,lem:TraceBounds} are satisfied. The conclusions of these
lemmas show that the \eta in
{thm:DualDPGAPosterioriErrrorEstimator-0} satisfies
\eta ( (\vec{p}_h, v_h) ) \eqsim \sum _{E\in \mathcal {E}_{\protect \scalebox {0.6}{\mathrm {int}}}}
h_E
\big \Vert \llbracket \vec{p}_{h} \c... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.015898050740361214,
0.0039020408876240253,
0.0018213337752968073,
0.0033852015621960163,
-0.002378223231062293,
-0.07091568410396576,
-0.008216027170419693,
0.032192789018154144,
0.00344813778065145,
0.022168779745697975,
-0.022199293598532677,
0.04702283442020416,
-0.014051924459636211,
... |
2b8082997f89edbc23d88f0b7780e135103f9829 | subsection | 90 | 126 | A posteriori error analysis for the Poisson example | Let\llbracket \vec{\tau } \cdot \vec{n} \rrbracket
=
\vec{\tau }_{K^+}\cdot \vec{n}_{K^+} + \vec{\tau }_{K^-}\cdot \vec{n}_{K^-}
\,,
\qquad \llbracket \vec{\tau } \cdot \vec{n}^\perp \rrbracket
=
\vec{n}_{K^+}^\perp \cdot \vec{\tau }_{K^+} + \vec{n}_{K^-}^\perp \cdot \vec{\tau }_{K^-}
\,.Here, \vec{n}_K^\perp is the ... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.011985615827143192,
0.025710709393024445,
-0.012382338754832745,
-0.02139253169298172,
-0.012283158488571644,
-0.020873740315437317,
0.018951158970594406,
0.022597959265112877,
0.01081070490181446,
0.030761301517486572,
-0.04147282615303993,
0.041747480630874634,
0.02818259969353676,
0.0... |
1e14d28ac4b5b31945af8d2c9156636d2ca1ddd3 | subsection | 91 | 126 | A posteriori error analysis for the Poisson example | Then\Vert (\vec{p}, v) - (\vec{p}_h, v_h) \Vert _{V}\;\eqsim \;
\eta _i(\vec{p}_h, v_h)for i\in \lbrace 1,2\rbrace , where the computable error estimators \eta _i
are defined by\eta _1(\vec{p}_h, v_h)^2
& =
\big \Vert \mathcal {L}(\vec{p}_h, v_h) - f \big \Vert _{\scriptscriptstyle }^2
+
\sum _{E\in \mathcal {E}_{\prot... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.042503975331783295,
0.024837210774421692,
0.008711331523954868,
0.009702988900244236,
0.004237429704517126,
-0.031916119158267975,
0.0015876058023422956,
0.0271561648696661,
0.018704188987612724,
-0.0034021486062556505,
-0.021801212802529335,
-0.0009682964882813394,
-0.01445531751960516,
... |
3d82e46651f3e9e027e932dd29c4523c845e2f9a | subsection | 92 | 126 | A posteriori error analysis for the Poisson example | \end{equation}
Similarly, for all \widehat{q}_n\in H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h), the continuous right inverse of \operatorname{\mathrm {tr}}_n: H(\operatorname{div},) \rightarrow H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h) is defined
\begin{equation}
\hspac... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030239492654800415,
0.0341758131980896,
-0.00035043558455072343,
0.028790071606636047,
0.019056066870689392,
-0.045374494045972824,
0.035365864634513855,
0.04873104766011238,
-0.012037820182740688,
0.023404330015182495,
-0.05642060562968254,
-0.021039484068751335,
0.004802159499377012,
... |
1e5806be5e9e1fde05f3994b6b74108619ee1ff7 | subsection | 93 | 126 | A posteriori error analysis for the Poisson example | Likewise, for any \widehat{\mu }\in \widetilde{P}^0_{p+2}(\partial _h),
the function \hspace{-0.5pt}{E}_{\mathrm {grad}}\hspace{0.5pt}(\widehat{\mu }) is supported only on _E.
}Our first lemma may be
thought of as an inf-sup condition involving the space of edge bubbles \widetilde{P}^0_{p+2}(\partial _h).
}\begin{}
Fo... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04005947336554527,
0.019282246008515358,
-0.004362913314253092,
0.012219207361340523,
0.01478966511785984,
-0.05494829639792442,
0.0003899544244632125,
0.03795429319143295,
-0.0018019898561760783,
0.01978565938770771,
-0.032035376876592636,
-0.008664806373417377,
0.007066851481795311,
-... |
88cf61f1dcd2359fad59a87ba6ea12d0f1f3cba0 | subsection | 94 | 126 | A posteriori error analysis for the Poisson example | For any \vec{q}_h \in P_p(_h) with
nontrivial \llbracket \vec{q}_h \cdot \vec{n} \rrbracket _E, we have
\begin{equation}
\begin{aligned}h_E \big \Vert \llbracket \vec{q}_{h} \cdot \vec{n} \rrbracket \big \Vert _{L^2(E)}^2
& \lesssim h_E (b_E \llbracket \vec{q}_h \cdot \vec{n} \rrbracket , \llbracket \vec{q}_h \cdot n ... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.056960877031087875,
0.05818124860525131,
0.015651274472475052,
0.003523824969306588,
-0.018290329724550247,
-0.019800540059804916,
0.012142704799771309,
0.005663290154188871,
0.002061285078525543,
0.002885036403313279,
-0.06480176746845245,
0.014957187697291374,
-0.00986976083368063,
0.... |
7dbe0976fff8617738fc0527007f293ccaf2767b | subsection | 95 | 126 | A posteriori error analysis for the Poisson example | \end{equation}
This can be seen beginning from the linearity of \hspace{-0.5pt}{E}_{\mathrm {grad}}\hspace{0.5pt}
and the fact that \mathcal {E}_K = \lbrace E \in \mathcal {E}:
\mathrm {meas}(E\cap \partial K)\ne 0\rbrace has fixed finite cardinality:
\begin{align}
{|\hspace{-0.5pt}{E}_{\mathrm {grad}}\hspace{0.5pt}(\w... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04156837612390518,
0.054081618785858154,
0.0020219567231833935,
-0.02751387096941471,
-0.01693865656852722,
-0.06879230588674545,
-0.02896357700228691,
0.006336736027151346,
-0.0005040585529059172,
0.0006366302259266376,
-0.026827169582247734,
-0.01754905842244625,
-0.004280444234609604,
... |
99c88c759e68323f8ccdbe955807fe95b23d761a | subsection | 96 | 126 | A posteriori error analysis for the Poisson example | Starting from~[{eq:4}]{\textup {{\ref *{eq:4}}}},
\begin{align}
\sum _{E\in \mathcal {E}_{\protect \scalebox {0.6}{\mathrm {int}}}}
h_E &
\Big \Vert \llbracket \vec{q}_h \cdot \vec{n} \rrbracket \Big \Vert _{L^2(E)}^2
\lesssim \sum _{E\in \mathcal {E}_{\protect \scalebox {0.6}{\mathrm {int}}}}
\frac{ (b_E \llbracket \v... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.011008994653820992,
0.042662713676691055,
-0.00197215867228806,
-0.01142860297113657,
0.0032271689269691706,
-0.023742197081446648,
-0.006217831280082464,
0.0480642169713974,
0.000650392787065357,
0.013488497585058212,
-0.03359917551279068,
0.015868820250034332,
-0.006008027121424675,
-0... |
0506847a7f03603632bcdfc504bc0cf55570b2e7 | subsection | 97 | 126 | A posteriori error analysis for the Poisson example | Now, the result follows by noting that the numerator above
equals \langle { \mu , \vec{q}_h \cdot \vec{n}} \rangle _h^2 and by bounding the
denominator using [{eq:BubbleNormBound}]{\textup {{\ref *{eq:BubbleNormBound}}}} and the Poincaré
inequality.
\end{}
}Lemma 4.3
For any degree p\ge 1 and for any w_h \in P_p(_h),\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05341994762420654,
0.02217691019177437,
0.006170003674924374,
-0.028877297416329384,
0.0013240515254437923,
-0.032784584909677505,
-0.016895966604351997,
0.005322915967553854,
0.0025832359679043293,
0.0061509255319833755,
-0.04758954048156738,
0.013637349009513855,
0.019307494163513184,
... |
22154b40590da63df5330038c3c302900b9fe10a | subsection | 98 | 126 | A posteriori error analysis for the Poisson example | Using
the vector curl of the scalar function \phi _E, by an application
of [eq:BubbleBounds1]eq:BubbleBounds1, we have\big |\llbracket w_h \vec{n} \rrbracket \big |_{H^1(E)}^2
& \lesssim (b_E \llbracket \vec{n}^\perp \cdot \operatorname{grad}w_h \rrbracket ,
\llbracket \vec{n}^\perp \cdot \operatorname{grad}w_h \rrbrac... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.07848268002271652,
0.04531367868185043,
0.00791082251816988,
-0.013792445883154869,
-0.009100878611207008,
-0.008582135662436485,
-0.02181769721210003,
-0.006446137558668852,
0.0014265417121350765,
0.028851233422756195,
-0.03292488679289818,
0.003728461218997836,
0.018308555707335472,
-... |
65c4dacba29a0bf9470630b0a546c426254a8dbd | subsection | 99 | 126 | A posteriori error analysis for the Poisson example | Hence
\begin{align}
\big |\llbracket w_h \vec{n} \rrbracket \big |_{H^1(E)}^2
& \lesssim \frac{( \operatorname{curl}\phi _E, \llbracket w_h \vec{n} \rrbracket )_E }{\Vert \operatorname{curl}\phi _E \Vert _{H(\operatorname{div}, _E)}}
\Vert \operatorname{curl}\phi _E \Vert _{H(\operatorname{div}, _E)}
\lesssim \frac{( \... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0519619956612587,
0.04155739024281502,
0.017788516357541084,
-0.005446398165076971,
-0.02077869512140751,
0.030298450961709023,
-0.01960398256778717,
0.034203991293907166,
0.002517242915928364,
0.01624765805900097,
-0.037255194038152695,
-0.0038502374663949013,
-0.0016791154630482197,
0... |
6539483ddae1f03692d657f99e66ef2200f699d2 | subsection | 100 | 126 | A posteriori error analysis for the Poisson example | By proving an analogue of
[{eq:BubbleNormBound}]{\textup {{\ref *{eq:BubbleNormBound}}}} for \hspace{-0.5pt}{E}_{\mathrm {div}}\hspace{0.5pt}, and following along the lines of the
proof of {lem:inf-H1-sup}, we finish the proof.
\end{align}
}Since the suprema in {lem:inf-H1-sup,lem:inf-Hdiv-sup} are related to the funct... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.10864151269197464,
0.030852969735860825,
-0.001864415011368692,
-0.0020179550629109144,
0.013900620862841606,
-0.019073300063610077,
-0.03277555853128433,
-0.01710493490099907,
0.005382485222071409,
0.002769443206489086,
-0.06787043064832687,
0.0020904336124658585,
0.010490315034985542,
... |
10c896780b89d2cf4c037615cdc76faee0cfa800 | subsection | 101 | 126 | A posteriori error analysis for the Poisson example | \end{gather}
\end{}
These results also hold with H1() replaced by H10() and H(div,) replaced by H0(div,).Lemma 4.4
For any degree p \ge 0 and any \vec{q}_{h} \in P_p(_h)^2
satisfying \langle \widehat{\mu }_1,\vec{q}_{h}\cdot \vec{n}\rangle _h = 0
for all \widehat{\mu }_1 \in \operatorname{\mathrm {tr}}( P_1(_h) \cap H... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.024922162294387817,
0.016116229817271233,
0.02302972599864006,
-0.004647151567041874,
0.0003722395049408078,
-0.04596788436174393,
-0.017932359129190445,
0.0053453692235052586,
-0.015620228834450245,
0.014429825358092785,
-0.05610157549381256,
-0.009965812787413597,
-0.008950917981564999,... |
7de45fd59b77d83345d19ba33c202d1afb650f5d | subsection | 102 | 126 | A posteriori error analysis for the Poisson example | Since
\langle \mathcal {I}_\mathrm {grad}\mu ,\vec{q}_{h}\cdot \vec{n}\rangle _h = 0,\langle \mu &,\vec{q}_{h}\cdot \vec{n}\rangle _h
=
\langle \mu - \mathcal {I}_\mathrm {grad}\mu ,\vec{q}_{h}\cdot \vec{n}\rangle _h
=
\sum _{E \in \mathcal {E}_{{\protect \scalebox {0.6}{\mathrm {int}}}}}
(\mu - \mathcal {I}_\mathrm {g... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.026269741356372833,
0.03118196874856949,
0.01801658794283867,
-0.05876367166638374,
-0.0030949325300753117,
-0.013455233536660671,
0.007013654801994562,
0.013912893831729889,
-0.006456833798438311,
-0.011113534681499004,
-0.03743666782975197,
0.007364528253674507,
0.026269741356372833,
... |
fd5971c97431fc61b3f1ff9857e18cc3174f1689 | subsection | 103 | 126 | A posteriori error analysis for the Poisson example | This completes the proof of~[{eq:FluxEquivalence}]{\textup {{\ref *{eq:FluxEquivalence}}}}.
}
}Lemma 4.5
For any degree p \ge 1 and any w_h \in P_p(_h) satisfying
\int _E \llbracket w_h \vec{n} \rrbracket = 0 on all edges E\in \mathcal {E},\sum _{E\in \mathcal {E}}
h_E
\Big \Vert \llbracket w_h \vec{n} \rrbracket \Big... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03986608237028122,
-0.011324897408485413,
-0.0007092370069585741,
-0.009546796791255474,
-0.009806262329220772,
0.00968416128307581,
0.03968292847275734,
-0.0008451701141893864,
-0.00011655668640742078,
-0.004372753668576479,
-0.012805376201868057,
-0.027243858203291893,
0.029853774234652... |
18b93151dc787cb1a2c28bb42d38316ba4288fdb | subsection | 104 | 126 | A posteriori error analysis for the Poisson example | To prove the remaining equivalence,
first observe that lem:inf-Hdiv-sup implies\sum _{E \in \mathcal {E}_{{\protect \scalebox {0.6}{\mathrm {int}}}}} h_E \Big | \llbracket w_h \vec{n} \rrbracket \Big |_{H^1(E)}^2
& \lesssim \sup _{\widehat{\sigma }\cdot \vec{n} \in \widehat{P}_p(\partial _h)}
\frac{\langle { \widehat{\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0485934279859066,
0.03497994318604469,
0.03156131133437157,
0.004651019815355539,
0.01514729205518961,
0.008996805176138878,
-0.002149999840185046,
0.023243654519319534,
-0.013918721117079258,
0.008264240808784962,
-0.04270239174365997,
0.003714329795911908,
-0.01893983781337738,
0.0150... |
bfb8a6c7dc26e28f55e3eeceff2a22365180c667 | subsection | 105 | 126 | A posteriori error analysis for the Poisson example | Therefore, by the
commutativity property of the quasi-interpolators,
\langle { {\vec{\sigma }}\cdot \vec{n}, w_h} \rangle _h
=
\langle { \vec{n}\cdot ( \operatorname{curl}\varphi _{\vec{\sigma }}- \mathcal {I}_\mathrm {grad}\varphi _{\vec{\sigma }}),
w_h } \rangle _h
+
\langle {\vec{n}\cdot ({\vec{\psi }}_{\vec{\sigma... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0720910131931305,
-0.0005155216786079109,
0.02510366402566433,
0.020403403788805008,
0.00000524582674188423,
0.0071762907318770885,
0.002724014688283205,
0.024325374513864517,
0.015550537966191769,
0.02118169330060482,
-0.011361506767570972,
0.011620936915278435,
0.02937662973999977,
0.... |
2b6b5b10bf4ce80d9901f415e5ccdf2d952708bf | subsection | 106 | 126 | A posteriori error analysis for the Poisson example | The term t_2 can be estimated
similarly:
\begin{align}
t_2
& =
\sum _{E \in \mathcal {E}} ( {\vec{\psi }}_{\vec{\sigma }}- \mathcal {I}_\mathrm {div}{\vec{\psi }}_{\vec{\sigma }}, \llbracket w_h \vec{n} \rrbracket )_E
\le \sum _{E \in \mathcal {E}} h_E^{-
\Vert {\vec{\psi }}_{\vec{\sigma }}- \mathcal {I}_\mathrm {div}{... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.01855761930346489,
0.008576794527471066,
-0.00682557187974453,
-0.001797006349079311,
-0.007851887494325638,
-0.0551539771258831,
0.01808452233672142,
0.01993112824857235,
0.017733514308929443,
0.01640579104423523,
-0.009248287416994572,
0.010713363066315651,
-0.002090784488245845,
0.024... |
c6b3b7b57617b42ed3353ef025bc109ca5d90dfa | subsection | 107 | 126 | A posteriori error analysis for the Poisson example | Hence the conditions of {lem:FluxBounds,lem:TraceBounds} are satisfied. The conclusions of these
lemmas show that the \eta in
{thm:DualDPGAPosterioriErrrorEstimator-0} satisfies
\eta ( (\vec{p}_h, v_h) ) \eqsim \sum _{E\in \mathcal {E}_{\protect \scalebox {0.6}{\mathrm {int}}}}
h_E
\big \Vert \llbracket \vec{p}_{h} \c... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02400832250714302,
-0.02055893838405609,
0.0008852401515468955,
0.019246341660618782,
-0.013217550702393055,
-0.038675837218761444,
-0.02921292372047901,
0.021612068638205528,
-0.00020497411605902016,
-0.003767993999645114,
-0.01637694239616394,
0.03995790705084801,
-0.0001619283575564623... |
8aa31f7e99e2e888be657172152d36f1d4bcc274 | subsection | 108 | 126 | Numerical experiments | In order to verify the mathematical theory developed above, we conducted several standard numerical verification experiments using two finite element software packages which have been used extensively for implementing DPG methods.
In our first set of experiments, we used Camellia , , a user-friendly C++ toolbox develop... | {
"cite_spans": [
{
"arxiv_id": "",
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... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
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3f8e25d2a21c59cc167a3ceb69d4c34680f73893 | subsection | 109 | 126 | Numerical experiments | In our second set of experiments, we used hp2D, a sophisticated suite of Fortan routines with support for 2D local hierarchical and anisotropic h- and p-refinements on hybrid meshes and corresponding oriented embedded shape functions for both quadrilateral and triangular elements in each of the canonical 2D de Rham seq... | {
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f3cb05cd943c1c475fe0d16daf6758e62ebd781e | subsection | 110 | 126 | Set-up | Let \in \lbrace _\square ,_{\begin{}[](0,0) -- (0ex,1ex);(0.5ex,0) -- (0.5ex,1ex);(1ex,0.5ex) -- (1ex,1ex);(0,1ex) -- (1ex,1ex);(0,0.5ex) -- (1ex,0.5ex);(0,0) -- (0.5ex,0ex);\end{}}\rbrace and let _{\protect \scalebox {0.6}{\mathrm {D}}},_{\protect \scalebox {0.6}{\mathrm {N}}} be disjoint and relatively open subsets c... | {
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84268c4732ca84415ffcaceb531791d50f955fb7 | subsection | 111 | 126 | Set-up | For instance, begin with the standard Nédélec spaces of the first type,\mathcal {Q}^{p_K,q_K}(K) \xrightarrow{} \mathcal {Q}^{p_K,q_K-1}\times \mathcal {Q}^{p_K-1,q_K}(K) \xrightarrow{} \mathcal {Q}^{p_K-1,q_K-1}(K)
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d88a6cff145d270c914545cc4051b662fff05a92 | subsection | 112 | 126 | Set-up | With the latter definition, notice that the polynomial order of an hp interface function, when restricted to a single shared edge E\in \mathcal {E}, E= \bigcap _{\hspace{0.83328pt}\overline{\hspace{-0.83328pt}K\hspace{-0.83328pt}}\hspace{0.83328pt}\cap E\ne \emptyset }\hspace{0.83328pt}\overline{\hspace{-0.83328pt}K\hs... | {
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c2b56ce5299889705849897995e1833592589dda | subsection | 113 | 126 | Set-up | All of our experiments investigate some form of Poisson's equation:\left\lbrace
\begin{aligned}-\Delta v &= f &&\text{in } \,,\\
v &= v_0 &&\text{on } _{\protect \scalebox {0.6}{\mathrm {D}}}\,,\\
\frac{\partial v}{\partial n} &= p_n &&\text{on } _{\protect \scalebox {0.6}{\mathrm {N}}}\,,
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56e570d0c218c84c3af9640a558bd4db384c2db8 | subsection | 114 | 126 | Set-up | Now, consider the mesh-dependent sequence {W_{hp} \xrightarrow{} {V}_{\!hp} \xrightarrow{} Y_{hp}}, whereW_{hp}
&=
\lbrace w\in H^1_0() : w|_K \in \mathcal {Q}^{p_K,q_K}(K)~\forall K\in _h\rbrace
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aa1d45265bdcae1d329901aecb8c63ef26058da8 | subsection | 115 | 126 | Set-up | With the latter definition, notice that the polynomial order of an hp interface function, when restricted to a single shared edge E\in \mathcal {E}, E= \bigcap _{\hspace{0.83328pt}\overline{\hspace{-0.83328pt}K\hspace{-0.83328pt}}\hspace{0.83328pt}\cap E\ne \emptyset }\hspace{0.83328pt}\overline{\hspace{-0.83328pt}K\hs... | {
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ed2e0b21250e919d498bbd4f0980a9bb18ef9a2c | subsection | 116 | 126 | Adaptive mesh refinement | In our experiments with h- and hp-adaptive mesh refinement, we used a standard isotropic h-subdivision rule.
Namely, at each refinement step, each element marked for h-refinement was uniformly subdivided into four equal-order quadrilateral elements.
Afterward, a standard so-called “mesh closure” algorithm was called to... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
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4edb04ce251e66f160556e9c2135380ea5f261eb | subsection | 117 | 126 | Adaptive mesh refinement | Alternatively, in the case of hp-adaptive mesh refinement, a common flagging strategy \cite {ainsworth1997aspects} was used to decide whether to h or p refine; see {sub:l_shaped_domain}.In our experiments with h- and hp-adaptive mesh refinement, we used a standard isotropic h-subdivision rule.
Namely, at each refinemen... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
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2d526f85b66023a5e19d18b1ff0c14216ab97dec | subsection | 118 | 126 | Adaptive mesh refinement | Alternatively, in the case of hp-adaptive mesh refinement, a common flagging strategy \cite {ainsworth1997aspects} was used to decide whether to h or p refine; see {sub:l_shaped_domain}. | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
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93c500bb9174fd77db91535d500951cce6eb0968 | subsection | 119 | 126 | Pure Dirichlet boundary conditions on a square domain | Recall eq:ModelProblem.
In this first example, = _\square and _{\protect \scalebox {0.6}{\mathrm {D}}}= \partial .
We considered two seemingly benign cases for the loads: (i) f=2\pi ^2\sin (\pi x)\sin (\pi y) and v_0 = 0; and (ii) f=0 and v_0 = 1.
In both cases, the exact solution is infinitely smooth.
Indeed, in case ... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
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7568e893330ce3dec48b6115f9e120611e957f13 | subsection | 120 | 126 | Pure Dirichlet boundary conditions on a square domain | Although both figures correspond only to a test space enrichment of \operatorname{d}\!p=1, similar results were observed for each choice \operatorname{d}\!p\in \lbrace 0,1,2\rbrace .
[Figure: Convergence under h-uniform mesh refinements with the manufactured solution v(x,y) = \sin (\pi x)\sin (\pi y). (Here, \operatorn... | {
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7147ba7e071e7f2c4a7dbb14f0946005cac2ec3b | subsection | 121 | 126 | Pure Dirichlet boundary conditions on a square domain | Notably, this artifact also agrees with previous results seen with a DPG* method for acoustics which can be found in the original technical report on the method (which portions of this text are based off of) and clearly warrants further analysis.Recall eq:ModelProblem.
In this first example, = _\square and _{\protect \... | {
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8385d5303f1fe3ead3db2ac38400c12a95f81905 | subsection | 122 | 126 | Pure Dirichlet boundary conditions on a square domain | Indeed, fig:hUnifSinA demonstrates the convergence of the corresponding discrete solution \vec{v}_h = (\vec{p}_h,v_h) to the exact solution, \vec{v}= (\operatorname{grad}v,v), measured in the full test norm above, starting with an single-element mesh with (isotropic) polynomial order p_K=q_K = p\in \lbrace 1,2,3,4\rbra... | {
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f626e597b3291f30ed3175da2a202060e974fecc | subsection | 123 | 126 | Pure Dirichlet boundary conditions on a square domain | In testing more complicated manufactured solutions (not documented here) which also feature a singular Lagrange multiplier \lambda , we discovered superconvergence effects from this choice of enrichment parameter.
Indeed, in our numerous additional verification experiments with \operatorname{d}\!p set to zero, the meth... | {
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cdf067097d5e5c408d865e3b6d990b37b1b706a7 | subsection | 124 | 126 | Mixed boundary conditions on an L-shaped domain | Again, recall eq:ModelProblem.
In this final example, set = _{\begin{}[](0,0) -- (0ex,1ex);(0.5ex,0) -- (0.5ex,1ex);(1ex,0.5ex) -- (1ex,1ex);(0,1ex) -- (1ex,1ex);(0,0.5ex) -- (1ex,0.5ex);(0,0) -- (0.5ex,0ex);\end{}}, _{\protect \scalebox {0.6}{\mathrm {D}}}= [0,1)\times \lbrace 0\rbrace \cup \lbrace 0\rbrace \times [0,... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
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4dd26a4d332dc8b95bfb1eb0f0d8541934b4a8b9 | subsection | 125 | 126 | Mixed boundary conditions on an L-shaped domain | For this problem, it is well known that the solution v \in H^{1+s}(), for all s<2/3.In each of our experiments, we began with a single three-element mesh composed of congruent squares and uniform order p_K = q_K =2 and \operatorname{d}\!p=1 in all three elements.
fig:hpAdaptivity demonstrates the convergence of the sol... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
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0c4181b35aee6a2d0802ec0e8518b937109fd28f | abstract | 0 | 83 | Abstract | We prove a quenched invariance principle for a class of random walks in
random environment on $\mathbb{Z}^d$, where the walker alters its own
environment. The environment consists of an outgoing edge from each vertex. The
walker updates the edge $e$ at its current location to a new random edge $e'$
(whose law depends o... | {
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} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
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39aba92cb1313fe410b77e2ceb6b2b8fde26fef1 | subsection | 1 | 83 | A random environment altered by the walker | Label each site of \mathbb {Z}^2 with either `H' or `V'. A walker starts at the origin. At each discrete time step the walker resamples the label at its current location (changing `H' to `V' and `V' to `H' with probability q, independent of the past) and then takes a mean zero horizontal step if the new label is `H' an... | {
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"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
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4fcecd7c304eee09b3ff05804edc96dc8ad4615f | subsection | 2 | 83 | A random environment altered by the walker | The wired uniform spanning forest is then the unique infinite-volume limit of \mu _n .To build a native environment for the random walk with local memory, we orient the connected component of the origin in the \mathsf {WUSF} toward the origin, orient all other components toward infinity, and add an independent outgoing... | {
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"doi": "10.1214/aop/1176990223",
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"Swee Hong Chan",
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b4718cf9cbca23c4defaae732543d7840e627df9 | subsection | 3 | 83 | A random environment altered by the walker | We illustrate the flavor with the `H,V'-walk described above (with q strictly between 0 and 1). By the martingale CLT,
the problem reduces to showing that the
walker encounters the label `V' half of the time, i.e.,\frac{1}{n} \sum _{i=0}^{n-1} \mathbb {1}\lbrace \text{the label used by the walker at the $i$-th step is ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
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93a908f47098ab3be55cb043d9ea454c2c3ef1e7 | subsection | 4 | 83 | Related work | When each vertex uses a deterministic rule to update its local memory,
the random walk with local memory is known as rotor walk, , .
That is to say, each vertex is given a prescribed cyclic ordering on its outgoing edges,
and for every update the vertex changes the current edge to the next edge in the cyclic order.
A f... | {
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5f3093ad63b0008c20549c7a89302d3f60fbb378 | subsection | 5 | 83 | Outline | Section contains the precise definition of our walks with local memory.
Certain walks with local memory admit a routine application of the martingale CLT, regardless of initial environment. We treat these first in Section .In Section we show the reduction that converts random walks with more complicated forms of loca... | {
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"source_ref... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
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7921d3184b3cdf3c61206b2cdde1864ccb6585cb | subsection | 6 | 83 | Random walks with local memory | Throughout this paper G:=(V(G),E(G)) denotes a connected, undirected graph that is locally finite (every vertex has finite degree) and simple (no loops, no multiple edges), with the exception of Section where a graph may have multiple edges.
When the graph G is evident from context, we will omit G from the notation and... | {
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} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
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75cf16e929361423064e92604c7a26fe9a077195 | subsection | 7 | 83 | Random walks with local memory | A walker-and-rotor configuration is a pair (x,\rho ), where x is a vertex of G and \rho is a rotor configuration of G.Remark 2.1
A
rotor configuration can be interpreted as either:A function \rho : V \rightarrow V such that \rho (x) \in {N}(x) for all x\in V; or
An oriented subgraph of G that has exactly one outgoing... | {
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"Swee Hong Chan",
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94cb910be6503afb0131c6d0a5782f2a175aae00 | subsection | 8 | 83 | Random walks with local memory | We refer to , for more details.
p-rotor walk on \mathbb {Z} for p\in [0,1], in which the probability transition function p_x (x \in \mathbb {Z}) is
given by
p_x(x\pm 1, x\mp 1)=1-p; \qquad p_x(x\pm 1, x\pm 1)=p.We now present three other examples of RWLMs.Example 2.3 (p-rotor walk on \mathbb {Z}^d)
Fix d\ge 2 and p\... | {
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"raw": "Alexander E. Holroyd, Lionel Levine, Karola Mészáros, Yuval Peres, James Propp, and David B. Wilson, Chip-firing and rotor-routing on dire... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
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d8846944d9c67068f66b0e6ea26183eaf67634d2 | subsection | 9 | 83 | Random walks with local memory | See Figure REF for an illustration of this mechanism on \mathbb {Z}^2.
[Figure: The mechanism for p,\!q-rotor walk on \mathbb {Z}^2, which stays at the current rotor with probability a:=\frac{q}{4}, rotates 180 degrees with probability a,rotates 90 degrees counterclockwise with probability b:=\frac{q}{4}+(1-q)p, and ro... | {
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"raw": "Ofer Zeitouni, Random walks in random environment, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189–312. MR 2071631",
"source_r... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
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"Peter Li"
] | [
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d105c57cf6aab2da85437410b87a2f6bda678fea | subsection | 10 | 83 | Random walks with local memory | We denote by {{E} the set of oriented edges of G.
}The running example for a graph in this paper is the integer lattice \mathbb {Z}^d of dimension d, i.e., the graph given byV&:=\lbrace \mathbf {x}\mid \mathbf {x}\in \mathbb {Z}^d \rbrace ; \qquad {E}:=\lbrace \lbrace \mathbf {x}, \mathbf {y}\rbrace \in \mathbb {Z}^d \... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
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1633451c4e1ca6b2c6d11c99fefbb1498c57d839 | subsection | 11 | 83 | Random walks with local memory | At time n,
the walker updates
the rotor of X_n
using the Markov chain M_{X_n} (which depends only on X_n and \rho _n(X_n)),
and then moves to the vertex to which the new rotor is pointing.
The local memory in the name refers to the fact that the walker records the last exit from each vertex that it visits via the rotor... | {
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"Swee Hong Chan",
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c82ed15a665abf0b8481ed54a611a282c53b554e | subsection | 12 | 83 | Random walks with local memory | \end{array}\right.}}See Figure REF for an illustration of this mechanism on \mathbb {Z}^2.
[Figure: (a) The mechanism for p-rotor walk on \mathbb {Z}^2, in which the rotor rotates counterclockwise with probability p, and clockwise with probability 1-p.The location of the walker and the rotor after one step of the RWLMi... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
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2a998b48cbd1e48d3398f508f75f715de91da7cd | subsection | 13 | 83 | Random walks with local memory | Such a walk is called elliptic in the literature of random walks in random environments ,
which we will explore more in Section .Example 2.5 (Triangular walk)
The triangular lattice is the graph embedded in \mathbb {R}^2 given by:V&:=\mathopen {}\mathclose {\left\lbrace a \begin{pmatrix}
1\\0
\end{pmatrix} + b\begin{p... | {
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da66170611d24650423c9d24abe752cc1609696a | subsection | 14 | 83 | Scaling limit of random walks with local memory | In this section we show that, under certain assumptions on the mechanism, the trajectory of the walker of a random walk with local memory has a scaling limit of a d-dimensional Brownian motion.Our main tool is the vector-valued martingale central limit theorem proved in .
We denote by D_{\mathbb {R}^d}[0,\infty ) the S... | {
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fcb78d3f10a54372aa12a76f613fc973fa23cbea | subsection | 15 | 83 | Scaling limit of random walks with local memory | Suppose that\sup _{ \lbrace \mathbf {x},\mathbf {y}\rbrace \in E} ||\mathbf {x}-\mathbf {y}||<\infty ;
There exists a matrix \Gamma such that
for any \mathbf {x}\in V, any neighbor \mathbf {y} of \mathbf {x}, and any random variable Y sampled from p_{\mathbf {x}}(\mathbf {y},\cdot ),
&\mathbb {E}\mathopen {}\mathclo... | {
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} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
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7f83d9544ed85154237b83cea5063a90c9672563 | subsection | 16 | 83 | Scaling limit of random walks with local memory | This implies that ||X_n||\le Cn+||X_0|| for all n\ge 0,
and it then follows that (X_n)_{n \ge 0} is square-integrable.We now check that (X_n)_{n \ge 0} is a martingale process with respect to the filtration {F}_n:=\sigma (X_0,\dots ,X_n,\rho _0,\dots ,\rho _n).
For any \mathbf {x}\in V and \mathbf {y}\in {N}(\mathbf {x... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
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"Lionel Levine",
"Peter Li"
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a412383ff53baa8ff39848846066ee8f63612869 | subsection | 17 | 83 | Scaling limit of random walks with local memory | It follows from the the transition rule of RWLM that, for any n\ge 0:\mathbb {E}\mathopen {}\mathclose {\left[V_n V_n^\top \mid {F}_n\right]} &= \sum _{\mathbf {x}\in V}\sum _{\mathbf {y}\in {N}(\mathbf {x})} \mathbb {E}\mathopen {}\mathclose {\left[ (Y_{\mathbf {x},\mathbf {y}}- \mathbf {x}) (Y_{\mathbf {x},\mathbf {y... | {
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2cf3330dc81dcf5c5a1a5f1d7cc89e428c5e4c99 | subsection | 18 | 83 | Scaling limit of random walks with local memory | This implies that\frac{1}{n}\sum _{i=0}^{n-1} \mathbb {E}\mathopen {}\mathclose {\left[\Vert V_i\Vert ^2 \mathbb {1}\lbrace \Vert V_i\Vert \ge \epsilon \sqrt{n}\rbrace \mid {F}_i\right]}= 0,which proves ().
The proof is now complete.In this section we show that, under certain assumptions on the mechanism, the trajector... | {
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{
"arxiv_id": "",
"doi": "10.1007/s00440-004-0424-1",
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"raw": "Firas Rassoul-Agha and Timo Seppäläinen, An almost sure invariance principle for random walks in a space-time random environment, Probab. The... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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6ce6a2e6e4bb3667e230f86766f8104b622c6897 | subsection | 19 | 83 | Scaling limit of random walks with local memory | Suppose that\sup _{ \lbrace \mathbf {x},\mathbf {y}\rbrace \in E} ||\mathbf {x}-\mathbf {y}||<\infty ;
There exists a matrix \Gamma such that
for any \mathbf {x}\in V, any neighbor \mathbf {y} of \mathbf {x}, and any random variable Y sampled from p_{\mathbf {x}}(\mathbf {y},\cdot ),
&\mathbb {E}\mathopen {}\mathclo... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b273f24c9c5058817cb9e20331449b9f77cb362c | subsection | 20 | 83 | Scaling limit of random walks with local memory | This implies that ||X_n||\le Cn+||X_0|| for all n\ge 0,
and it then follows that (X_n)_{n \ge 0} is square-integrable.We now check that (X_n)_{n \ge 0} is a martingale process with respect to the filtration {F}_n:=\sigma (X_0,\dots ,X_n,\rho _0,\dots ,\rho _n).
For any \mathbf {x}\in V and \mathbf {y}\in {N}(\mathbf {x... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ddd6bc6c5590431ae90e8ecd22268af5be0adf10 | subsection | 21 | 83 | Scaling limit of random walks with local memory | It follows from the the transition rule of RWLM that, for any n\ge 0:\mathbb {E}\mathopen {}\mathclose {\left[V_n V_n^\top \mid {F}_n\right]} &= \sum _{\mathbf {x}\in V}\sum _{\mathbf {y}\in {N}(\mathbf {x})} \mathbb {E}\mathopen {}\mathclose {\left[ (Y_{\mathbf {x},\mathbf {y}}- \mathbf {x}) (Y_{\mathbf {x},\mathbf {y... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
d124df8320e7c43d18d678f6619768981957769b | subsection | 22 | 83 | Scaling limit of random walks with local memory | This implies that\frac{1}{n}\sum _{i=0}^{n-1} \mathbb {E}\mathopen {}\mathclose {\left[\Vert V_i\Vert ^2 \mathbb {1}\lbrace \Vert V_i\Vert \ge \epsilon \sqrt{n}\rbrace \mid {F}_i\right]}= 0,which proves ().
The proof is now complete. | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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5ca9e9a3339ab5995103f47560da94229410f5e6 | subsection | 23 | 83 | Random walks with hidden local memory | In this section we present a more general version of random walk with local memory
inspired by hidden Markov chains (see for references on hidden Markov chains).
We remark that the content of this section is independent of the later sections.For each x \in V, a hidden mechanism at x is a Markov chain M_x with finite st... | {
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{
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"doi": "10.1093/ietisy/e89-d.3.869",
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"raw": "Jeff A. Bilmes, What hmms can do, IEICE - Trans. Inf. Syst. E89-D (2006), no. 3, 869–891.",
"source_ref_id": "35e2ee79971c1ef3fb81c601... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
fbe25a531474e2619253d0ced088b2d2a46e77e8 | subsection | 24 | 83 | Random walks with hidden local memory | Let N_1 \sqcup N_2 be the partition of the neighbors {N}(\mathbf {x}) of \mathbf {x} given by:N_1:= \mathbf {x}+\mathopen {}\mathclose {\left\lbrace \begin{pmatrix}
1\\ 0
\end{pmatrix}, \frac{1}{2}\begin{pmatrix}
-{1} \\ {\sqrt{3}}
\end{pmatrix} , \frac{1}{2}\begin{pmatrix}
-1 \\ {-\sqrt{3}}
\end{pmatrix}\right\rbrace ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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615596c45f22b3caba3e9cc4b7eb8a319db42be8 | subsection | 25 | 83 | Random walks with hidden local memory | Let (X_n,\rho _n,\kappa _n)_{n \ge 0} be an RWHLM on G.
Start an RWLM (X_n^\times , \rho _n^\times )_{n \ge 0} on G^\times with the following initial configuration:X_0^\times &:= X_0; \qquad \rho ^\times _0(x):= e({x, \rho _0(x), \kappa _0(x))} \quad (x \in V).Then (X_n, \rho _n)_{n \ge 0} is equal in distribution to
(... | {
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{
"arxiv_id": "",
"doi": "10.1093/ietisy/e89-d.3.869",
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"raw": "Jeff A. Bilmes, What hmms can do, IEICE - Trans. Inf. Syst. E89-D (2006), no. 3, 869–891.",
"source_ref_id": "35e2ee79971c1ef3fb81c60... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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f682c2001ccb45a4884a2d4bb3eb4d0fb2bb5634 | subsection | 26 | 83 | Random walks with hidden local memory | \end{array}\right.}};
\rho _{n+1}(x):={\mathopen {}\mathclose {\left\lbrace \begin{array}{ll} Y_n & \text{if } x=X_n;\\
\rho _{n}(x) & \text{if } x\ne X_n, \end{array}\right.}}
X_{n+1}:=Y_n,where K_n is a random element of S_{X_n} sampled from p_{X_n}(\kappa _{n}(X_n), \cdot )
independent of the past, and Y_n is a r... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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8753dd79906648a82a6516db5f2c8a85ffc9a398 | subsection | 27 | 83 | Random walks with hidden local memory | Let N_1 \sqcup N_2 be the partition of the neighbors {N}(\mathbf {x}) of \mathbf {x} given by:N_1:= \mathbf {x}+\mathopen {}\mathclose {\left\lbrace \begin{pmatrix}
1\\ 0
\end{pmatrix}, \frac{1}{2}\begin{pmatrix}
-{1} \\ {\sqrt{3}}
\end{pmatrix} , \frac{1}{2}\begin{pmatrix}
-1 \\ {-\sqrt{3}}
\end{pmatrix}\right\rbrace ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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605b6307a106e852a2e06f0f9d32a834a1e0212e | subsection | 28 | 83 | Random walks with hidden local memory | Let (X_n,\rho _n,\kappa _n)_{n \ge 0} be an RWHLM on G.
Start an RWLM (X_n^\times , \rho _n^\times )_{n \ge 0} on G^\times with the following initial configuration:X_0^\times &:= X_0; \qquad \rho ^\times _0(x):= e({x, \rho _0(x), \kappa _0(x))} \quad (x \in V).Then (X_n, \rho _n)_{n \ge 0} is equal in distribution to
(... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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22d40c0816bb3d5ee535fdb7fc4cfe2635ac32f2 | subsection | 29 | 83 | Wired spanning forest oriented toward a root | In this section we present two methods to generate the wired spanning forest oriented toward a chosen root vertex, which we will use to construct an initial rotor configuration for random walks with local memory in Sections and .Let (G,c) be an electrical network (we emphasize that G is always an unoriented graph, and ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
c92b81b592fb636ccb132c16c4ef50bcfa0819ba | subsection | 30 | 83 | Wired spanning forest oriented toward a root | Let $(G_n,c_n)_{n \ge 0}$ be a wired exhaustion of $G$.
We denote by ${{\mu _{r,n}}$ the probability distribution
${{\operatorname{\mathsf {{WSF}}}}_r(G_n,c_n)$ on the oriented subgraphs of $G_n$.
$}{{}{\bfseries {Definition 5.10 (Rooted oriented wired spanning forest for infinite graphs)}}
Let $(G,c)$ be an electrica... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ccfca06624f9f018093e9ec3d277ec6368d0f367 | subsection | 31 | 83 | Wired spanning forest oriented toward a root | The lemma now
follows from Definition~\ref {definition: unoriented wsf finite graph} and Definition~\ref {definition: wsf finite graphs}, and the proof is complete.
}
\end{}}As in the unoriented case, a random oriented subgraph {{F} sampled from {{\operatorname{\mathsf {{WSF}}}}_r is not necessarily an oriented spannin... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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059c1d6e8ee376e3a1ee2599a9da4b8908908103 | subsection | 32 | 83 | Wired spanning forest oriented toward a root | Let $x_1,x_2,\ldots $ be an ordering of elements of the $V(G) \setminus \lbrace r\rbrace $.
Define a growing sequence
$({{{T}}(i))_{i \ge 0} of oriented trees recursively as follows:
\begin{}
\item Set
{{{T}}(0)
to be the tree with the single vertex r and with no edges.
\item Suppose that
{{{T}}(i)
has been generated.
... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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bd1bfdae5f94fe088d75a82916385363233dc592 | subsection | 33 | 83 | Wired spanning forest oriented toward a root | Then for any finite subset {{{B}} of {{E}(G), any ordering of V(G) \setminus \lbrace r\rbrace , and any wired exhaustion of G,
\mathbb {P}[{{{B}}\subseteq {{{T}}] =\lim _{n \rightarrow 0} \ {{\mu _{r,n}}[{{{B}}\subseteq {{{T}}_n],
where {{{T}} is a random tree of G generated using Wilson^{\prime }s method,
with root ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b05c8c0054b6ff463e3238232642bd037c06c50b | subsection | 34 | 83 | Wired spanning forest oriented toward a root | Start a network random walk at x_{i+1}.
Stop the walk the first time it hits {{{F}}(i); if it never hits {{{F}}(i) then let it run indefinitely.
This walk is locally finite a.s.\ by transience.
Let \langle y_0^{\prime },y_1^{\prime },\ldots \rangle be the loop erasure of this random walk.
}\item Set {{{F}}(i+1) to be t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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dcf1ad8c887400ef253062b13fe579ecfe3d05fa | subsection | 35 | 83 | Wired spanning forest oriented toward a root | That is, if \operatorname{{\mathsf {LE}}}\langle x_i \mid i \le I\rangle =\langle y_{I,i} \mid i \le m_I\rangle and \operatorname{{\mathsf {LE}}}\langle x_i \mid i \ge 0 \rangle =\langle y_{i} \mid i \ge 0\rangle , then for every i and all sufficiently large I we have y_{I,i}=y_i.
Since G is transient,
it follows that ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0... |
281494fcd0fc845a4549c4a828806e4fd106d6d5 | subsection | 36 | 83 | Wired spanning forest oriented toward a root | It then follows from definition of oriented wired spanning forest for finite graphs (Definition~\ref {definition: wsf finite graphs}) that
{{{T}}_n has the law of {{\operatorname{\mathsf {{WSF}}}}_{r}(G_n,c_n).
}Let \tau ^j_n be the first time that \langle X_i^j \mid i \ge 0\rangle reaches the portion of the spanning t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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afc3b957f888b81527d68fda3c2038e1e3f7fcd5 | subsection | 37 | 83 | Wired spanning forest oriented toward a root | Starting an RWLM from a native environment allows us to use ergodic theory in Section~\ref {SLLN}.
The main result of this section is Theorem~\ref {stationarity theorem},
which gives an explicit distribution as a native environment for the RWLM.
}\end{equation}In this section the underlying graph of the RWLM will be a ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0... |
c1e6fbff456e62c9d9494ef072457b74504eb98c | subsection | 38 | 83 | Wired spanning forest oriented toward a root | Note that the measure \mu _{\operatorname{\mathnormal {o}}} is symmetric (i.e., \mu _{\operatorname{\mathnormal {o}}}(x)=\mu _{\operatorname{\mathnormal {o}}}(x^{-1})) as a consequence of c: \mathcal {S}\rightarrow \mathbb {R}_{>0} being symmetric.
}}{{}{\bfseries {Definition 5.16 (Transitive mechanism)}}
Let $(G,c)$ ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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2c783b26453397602bfdb958fba66662e54653f5 | subsection | 39 | 83 | Wired spanning forest oriented toward a root | \end{}}The scenery process $(\widehat{\rho }_n)_{n \ge 0}$ is a Markov chain with state space the set of rotor configurations of $G$ and with transition rule
\begin{equation}
\widehat{\rho }_{n+1}(x) :=
{\mathopen {}\mathclose {\left\lbrace \begin{array}{ll}\operatorname{\mathnormal {o}}&\text{ if } x= Y_{n}^{-1};\\
Y_... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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63ad809a54086e3d1172b6e6198ddd5b9ab60299 | subsection | 40 | 83 | Wired spanning forest oriented toward a root | The \emph {$r$-oriented wired spanning forest plus one edge}, denoted ${{\operatorname{\mathsf {{WSF}}}}_r^+:={{\operatorname{\mathsf {{WSF}}}}_r^+(G,c), is the law of the random subgraph {{{F}}\sqcup \lbrace (r, Y)\rbrace , where {{{F}} is a random r-oriented forest of G sampled from {{\operatorname{\mathsf {{WSF}}}}_... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ab75556677a02fc5867747cf099c8db7ecbaedf9 | subsection | 41 | 83 | Wired spanning forest oriented toward a root | Each unicycle {{{U}} is picked with probability proportional to
{\Xi ({{{U}})}, where \Xi ({{{U}}) is as in Definition~\ref {definition: weight of a directed tree}.
This implies that
{{F_{Y}} \, \sqcup \, \lbrace (Y, r) \rbrace is distributed as {{\operatorname{\mathsf {{WSF}}}}^+_r, as desired.
}
}}\begin{}[Proof of T... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
4b1a94b170648ee6dff80b7f68c59be34283a600 | subsection | 42 | 83 | Wired spanning forest oriented toward a root | It now follows from Lemma~\ref {lemma: wsfplus sum}
that \widehat{\rho }_1 is distributed according to {{\operatorname{\mathsf {{WSF}}}}^+_{\operatorname{\mathnormal {o}}}, and the proof is complete.
}
}An important property of {{\operatorname{\mathsf {{WSF}}}}_r^+ is that it is a tail trivial measure, defined below.
T... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.04... |
76effdc24447c6a31e7814ad2f3aa5b4903620c1 | subsection | 43 | 83 | Wired spanning forest oriented toward a root | Also note that {{\operatorname{\mathsf {{WSF}}}}_r[g({{{B}})]=\operatorname{\mathsf {{WSF}}}[f\circ g({{{B}})] by
Lemma~\ref {lemma: wsf and oriented wsf}.
Finally, note that
the set f \circ g({{{B}}) is a tail event in {F} since {{{B}} is a tail event in {{{F}}.
The conclusion of the proposition now follows from the t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
c6ab171070483077fe24a1e97b43061081a5c1ad | subsection | 44 | 83 | Wired spanning forest oriented toward a root | \end{}
We will further assume that the initial rotor configuration of the RWLM is sampled from a {tail trivial} native environment~(Definitions~\ref {definition: native environment} and \ref {definition: tail trivial}).
}}Several remarks are in order.
Condition \ref {item: ELL} is known as an \emph {ellipticity} condit... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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