Foundations of Computational Mathematics最新文献

{"title":"Multi-index Sequential Monte Carlo Ratio Estimators for Bayesian Inverse problems","authors":"Ajay Jasra, Kody J. H. Law, Neil Walton, Shangda Yang","doi":"10.1007/s10208-023-09612-z","DOIUrl":"https://doi.org/10.1007/s10208-023-09612-z","url":null,"abstract":"Abstract We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of $$hbox {MSE}^{-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mtext>MSE</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , while single-level methods require $$hbox {MSE}^{-xi }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mtext>MSE</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>ξ</mml:mi> </mml:mrow> </mml:msup> </mml:math> for $$xi >1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where $$xi =5/4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$xi =3/2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately $$xi =9/4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>9</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $$xi = 5/4 + omega $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , for any $$omega > 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the s","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135846075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A New Approach to Handle Curved Meshes in the Hybrid High-Order Method","authors":"Liam Yemm","doi":"10.1007/s10208-023-09615-w","DOIUrl":"https://doi.org/10.1007/s10208-023-09615-w","url":null,"abstract":"Abstract We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in $$L^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135932461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Correction to: Conormal Spaces and Whitney Stratifications","authors":"M. Helmer, Vidit Nanda","doi":"10.1007/s10208-022-09602-7","DOIUrl":"https://doi.org/10.1007/s10208-022-09602-7","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"1-8"},"PeriodicalIF":3.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Polynomial Meshes Exist on any Multivariate Convex Domain","authors":"Feng Dai, Andriy Prymak","doi":"10.1007/s10208-023-09606-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09606-x","url":null,"abstract":"We show that optimal polynomial meshes exist for every convex body in $${mathbb {R}}^d$$ , confirming a conjecture by A. Kroó.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Scattering and Uniform in Time Error Estimates for Splitting Method in NLS","authors":"R. Carles, C. Su","doi":"10.1007/s10208-022-09600-9","DOIUrl":"https://doi.org/10.1007/s10208-022-09600-9","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47665434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks","authors":"Jonathan W. Siegel, Jinchao Xu","doi":"10.1007/s10208-022-09595-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09595-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44959269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time","authors":"Jess Banks, Jorge Garza-Vargas, Archit Kulkarni, Nikhil Srivastava","doi":"10.1007/s10208-022-09577-5","DOIUrl":"https://doi.org/10.1007/s10208-022-09577-5","url":null,"abstract":"<p>We exhibit a randomized algorithm which, given a square matrix <span>(Ain mathbb {C}^{ntimes n})</span> with <span>(Vert AVert le 1)</span> and <span>(delta &gt;0)</span>, computes with high probability an invertible <i>V</i> and diagonal <i>D</i> such that <span>( Vert A-VDV^{-1}Vert le delta )</span> using <span>(O(T_mathsf {MM}(n)log ^2(n/delta )))</span> arithmetic operations, in finite arithmetic with <span>(O(log ^4(n/delta )log n))</span> bits of precision. The computed similarity <i>V</i> additionally satisfies <span>(Vert VVert Vert V^{-1}Vert le O(n^{2.5}/delta ))</span>. Here <span>(T_mathsf {MM}(n))</span> is the number of arithmetic operations required to multiply two <span>(ntimes n)</span> complex matrices numerically stably, known to satisfy <span>(T_mathsf {MM}(n)=O(n^{omega +eta }))</span> for every <span>(eta &gt;0)</span> where <span>(omega )</span> is the exponent of matrix multiplication (Demmel et al. in Numer Math 108(1):59–91, 2007). The algorithm is a variant of the spectral bisection algorithm in numerical linear algebra (Beavers Jr. and Denman in Numer Math 21(1-2):143–169, 1974) with a crucial Gaussian perturbation preprocessing step. Our result significantly improves the previously best-known provable running times of <span>(O(n^{10}/delta ^2))</span> arithmetic operations for diagonalization of general matrices (Armentano et al. in J Eur Math Soc 20(6):1375–1437, 2018) and (with regard to the dependence on <i>n</i>) <span>(O(n^3))</span> arithmetic operations for Hermitian matrices (Dekker and Traub in Linear Algebra Appl 4:137–154, 1971). It is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic), thereby matching the complexity of other dense linear algebra operations such as inversion and <i>QR</i> factorization up to polylogarithmic factors. The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to <i>any</i> matrix splits its pseudospectrum into <i>n</i> small well-separated components. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts’ Newton iteration method (Roberts in Int J Control 32(4):677–687, 1980) for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"6 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bad and Good News for Strassen’s Laser Method: Border Rank of $$mathrm{Perm}_3$$ and Strict Submultiplicativity","authors":"Austin Conner, Hang Huang, J. Landsberg","doi":"10.1007/s10208-022-09579-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09579-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43274289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kähler Geometry of Framed Quiver Moduli and Machine Learning","authors":"G. Jeffreys, Siu-Cheong Lau","doi":"10.1007/s10208-022-09587-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09587-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"23 1","pages":"1899-1957"},"PeriodicalIF":3.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48055726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Stirling-Type Formula for the Distribution of the Length of Longest Increasing Subsequences","authors":"F. Bornemann","doi":"10.1007/s10208-023-09604-z","DOIUrl":"https://doi.org/10.1007/s10208-023-09604-z","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"1-39"},"PeriodicalIF":3.0,"publicationDate":"2022-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47944057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}