Mathematics of Computation最新文献

{"title":"Optimal analysis of finite element methods for the stochastic Stokes equations","authors":"Buyang Li, Shu Ma, Weiwei Sun","doi":"10.1090/mcom/3972","DOIUrl":"https://doi.org/10.1090/mcom/3972","url":null,"abstract":"<p>Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^infty (0, T; L^2(Omega ; L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis tau Superscript 1 slash 2 Baseline plus h squared right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(tau ^{1/2}+ h^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Towards a classification of isolated 𝑗-invariants","authors":"Abbey Bourdon, Sachi Hashimoto, Timo Keller, Z. Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla","doi":"10.1090/mcom/3956","DOIUrl":"https://doi.org/10.1090/mcom/3956","url":null,"abstract":"<p>We develop an algorithm to test whether a non-complex multiplication elliptic curve <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E slash bold upper Q\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">Q</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">E/mathbf {Q}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> gives rise to an isolated point of any degree on any modular curve of the form <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 1 left-parenthesis upper N right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X_1(N)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\u0000 <mml:semantics>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Running this algorithm on all elliptic curves presently in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\u0000 <mml:semantics>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> gives rise to an isolated point on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 1 left-parenthesis upper N right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X_1(N)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> if and only if <inline-formula content-type=\"math/mathm","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140656482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cellular approximations to the diagonal map","authors":"Khaled Alzobydi, Graham Ellis","doi":"10.1090/mcom/3981","DOIUrl":"https://doi.org/10.1090/mcom/3981","url":null,"abstract":"<p>We describe an elementary algorithm for recursively constructing diagonal approximations on those finite regular CW-complexes for which the closure of each cell can be explicitly collapsed to a point. The algorithm is based on the standard proof of the acyclic carrier theorem, made constructive through the use of explicit contracting homotopies. It can be used as a theoretical tool for constructing diagonal approximations on families of polytopes in situations where the diagonals are required to satisfy certain coherence conditions. We compare its output to existing diagonal approximations for the families of simplices, cubes, associahedra and permutahedra. The algorithm yields a new explanation of a <italic>magical formula</italic> for the associahedron derived by Markl and Shnider [Trans. Amer. Math. Soc. 358 (2006), pp. 2353–2372] and Masuda, Thomas, Tonks, and Vallette [J. Éc. polytech. Math. 8 (2021), pp. 121–146] and Theorem 4.1 provides a <italic>magical formula</italic> for other polytopes. We also describe a computer implementation of the algorithm and illustrate it on a range of practical examples including the computation of cohomology rings for some low-dimensional manifolds. To achieve some of these examples the paper includes two approaches to generating a regular CW-complex structure on closed compact <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds, one using an implementation of Dehn surgery on links and the other using an implementation of pairwise identifications of faces in a tessellated boundary of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ball. The latter is illustrated in Proposition 8.1 with a topological classification of all closed orientable <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds arising from pairwise identifications of faces of the cube.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low-regularity exponential-type integrators for the Zakharov system with rough data in all dimensions","authors":"Hang Li, Chunmei Su","doi":"10.1090/mcom/3973","DOIUrl":"https://doi.org/10.1090/mcom/3973","url":null,"abstract":"<p>We propose and analyze a type of low-regularity exponential-type integrators (LREIs) for the Zakharov system (ZS) with rough solutions. Our LREIs include a first-order integrator and a second-order one, and they achieve optimal convergence under weaker regularity assumptions on the exact solution compared to the existing numerical methods in literature. Specifically, the first-order integrator exhibits linear convergence in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript m plus 2 Baseline left-parenthesis double-struck upper T Superscript d Baseline right-parenthesis times upper H Superscript m plus 1 Baseline left-parenthesis double-struck upper T Superscript d Baseline right-parenthesis times upper H Superscript m Baseline left-parenthesis double-struck upper T Superscript d Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi>m</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^{m+2}(mathbb {T}^d)times H^{m+1}(mathbb {T}^d)times H^m(mathbb {T}^d)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for solutions in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript m plus 3 Baseline left-parenthesis double-struck upper T Superscript d Baseline right-parenthesis times upper H Superscript m plus 2 Baseline left-parenthesis double-struck upper T Superscript d Baseline right-parenthesis times upper H Superscript m plus 1 Baseline left-parenthesis double-struck upper T Superscript d Baseline right-","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140674446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Adaptive fast multiplication of ℋ²-matrices","authors":"Steffen Börm","doi":"10.1090/mcom/3978","DOIUrl":"https://doi.org/10.1090/mcom/3978","url":null,"abstract":"<p>Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H squared\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathcal {H}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms.</p> <p>The matrix multiplication, a key component of many more advanced numerical algorithms, has been a challenge: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance.</p> <p>In this article, a new and fairly general algorithm for multiplying <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H squared\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathcal {H}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application.</p> <p>The complexity and accuracy are analyzed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial preserving recovery for the finite volume element methods under simplex meshes","authors":"Yonghai Li, Peng Yang, Zhimin Zhang","doi":"10.1090/mcom/3980","DOIUrl":"https://doi.org/10.1090/mcom/3980","url":null,"abstract":"<p>The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting computing nodes such that the least-squares fitting procedure of the PPR admits a unique solution. Numerical experiments demonstrate that the recovered gradient by the PPR exhibits superconvergence.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux","authors":"Aekta Aggarwal, Ganesh Vaidya","doi":"10.1090/mcom/3976","DOIUrl":"https://doi.org/10.1090/mcom/3976","url":null,"abstract":"<p>We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Super-polynomial accuracy of multidimensional randomized nets using the median-of-means","authors":"Zexin Pan, Art Owen","doi":"10.1090/mcom/3880","DOIUrl":"https://doi.org/10.1090/mcom/3880","url":null,"abstract":"<p>We study approximate integration of a function <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 comma 1 right-bracket Superscript s\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[0,1]^s</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> based on taking the median of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 r minus 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">2r-1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> integral estimates derived from independently randomized <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis t comma m comma s right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(t,m,s)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-nets in base <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The nets are randomized by Matousek’s random linear scramble with a random digital shift. If <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is analytic over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 comma 1 right-bracket Superscript s\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 </mm","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140688187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discrete tensor product BGG sequences: Splines and finite elements","authors":"F. Bonizzoni, Kaibo Hu, Guido Kanschat, Duygu Sap","doi":"10.1090/mcom/3969","DOIUrl":"https://doi.org/10.1090/mcom/3969","url":null,"abstract":"In this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand diagrams and complexes over cubical meshes in arbitrary dimension via the use of tensor product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and \u0000\u0000 \u0000 \u0000 div\u0000 ⁡\u0000 div\u0000 \u0000 operatorname {div}operatorname {div}\u0000 \u0000\u0000 complexes as examples for our construction.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140690610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wavenumber-explicit stability and convergence analysis of ℎ𝑝 finite element discretizations of Helmholtz problems in piecewise smooth media","authors":"M. Bernkopf, T. Chaumont-Frelet, J. Melenk","doi":"10.1090/mcom/3958","DOIUrl":"https://doi.org/10.1090/mcom/3958","url":null,"abstract":"<p>We present a wavenumber-explicit convergence analysis of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h p\"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}