Foundations of Computational Mathematics最新文献

{"title":"Gaussian Beam Ansatz for Finite Difference Wave Equations","authors":"Umberto Biccari, Enrique Zuazua","doi":"10.1007/s10208-023-09632-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09632-9","url":null,"abstract":"<p>This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"42 24","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516805","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":"How Much Can One Learn a Partial Differential Equation from Its Solution?","authors":"Yuchen He, Hongkai Zhao, Yimin Zhong","doi":"10.1007/s10208-023-09620-z","DOIUrl":"https://doi.org/10.1007/s10208-023-09620-z","url":null,"abstract":"<p>In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data-driven and data-adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"42 23","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516806","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":"From Kernel Methods to Neural Networks: A Unifying Variational Formulation","authors":"Michael Unser","doi":"10.1007/s10208-023-09624-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09624-9","url":null,"abstract":"<p>The minimization of a data-fidelity term and an additive regularization functional gives rise to a powerful framework for supervised learning. In this paper, we present a unifying regularization functional that depends on an operator <span>(textrm{L})</span> and on a generic Radon-domain norm. We establish the existence of a minimizer and give the parametric form of the solution(s) under very mild assumptions. When the norm is Hilbertian, the proposed formulation yields a solution that involves radial-basis functions and is compatible with the classical methods of machine learning. By contrast, for the total-variation norm, the solution takes the form of a two-layer neural network with an activation function that is determined by the regularization operator. In particular, we retrieve the popular ReLU networks by letting the operator be the Laplacian. We also characterize the solution for the intermediate regularization norms <span>(Vert cdot Vert =Vert cdot Vert _{L_p})</span> with <span>(pin (1,2])</span>. Our framework offers guarantees of universal approximation for a broad family of regularization operators or, equivalently, for a wide variety of shallow neural networks, including the cases (such as ReLU) where the activation function is increasing polynomially. It also explains the favorable role of bias and skip connections in neural architectures.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"42 21","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516807","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":"Toward Computational Morse–Floer Homology: Forcing Results for Connecting Orbits by Computing Relative Indices of Critical Points","authors":"Jan Bouwe van den Berg, Marcio Gameiro, Jean-Philippe Lessard, Rob Van der Vorst","doi":"10.1007/s10208-023-09623-w","DOIUrl":"https://doi.org/10.1007/s10208-023-09623-w","url":null,"abstract":"<p>To make progress toward better computability of Morse–Floer homology and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper, we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"30 12","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509660","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":"Improved Resolution Estimate for the Two-Dimensional Super-Resolution and a New Algorithm for Direction of Arrival Estimation with Uniform Rectangular Array","authors":"Ping Liu, H. Ammari","doi":"10.1007/s10208-023-09618-7","DOIUrl":"https://doi.org/10.1007/s10208-023-09618-7","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43551054","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":"BGG Sequences with Weak Regularity and Applications","authors":"Andreas Čap, Kaibo Hu","doi":"10.1007/s10208-023-09608-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09608-9","url":null,"abstract":"<p>We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in <span>({mathbb {R}}^{n})</span>. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"33 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509772","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":"Optimality of Robust Online Learning","authors":"Zheng-Chu Guo, Andreas Christmann, Lei Shi","doi":"10.1007/s10208-023-09616-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09616-9","url":null,"abstract":"<p>In this paper, we study an online learning algorithm with a robust loss function <span>(mathcal {L}_{sigma })</span> for regression over a reproducing kernel Hilbert space (RKHS). The loss function <span>(mathcal {L}_{sigma })</span> involving a scaling parameter <span>(sigma &gt;0)</span> can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression aiming to estimate the conditional mean function. For properly chosen <span>(sigma )</span> and step size, we show that the last iterate of this online algorithm can achieve optimal capacity independent convergence in the mean square distance. Moreover, if additional information on the underlying function space is known, we also establish optimal capacity-dependent rates for strong convergence in RKHS. To the best of our knowledge, both of the two results are new to the existing literature of online learning.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"31 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534307","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":"Further $$exists {mathbb {R}}$$-Complete Problems with PSD Matrix Factorizations","authors":"Y. Shitov","doi":"10.1007/s10208-023-09610-1","DOIUrl":"https://doi.org/10.1007/s10208-023-09610-1","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42845528","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":"High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays","authors":"Robert Szczelina, Piotr Zgliczyński","doi":"10.1007/s10208-023-09614-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09614-x","url":null,"abstract":"Abstract We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"379 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135099805","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":"Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations","authors":"Nicolas Clozeau, Marc Josien, Felix Otto, Qiang Xu","doi":"10.1007/s10208-023-09613-y","DOIUrl":"https://doi.org/10.1007/s10208-023-09613-y","url":null,"abstract":"Abstract We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior $$a_{textrm{hom}}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mtext&gt;hom&lt;/mml:mtext&gt; &lt;/mml:msub&gt; &lt;/mml:math&gt; of a stationary random medium. The latter is described by a coefficient field a ( x ) generated from a given ensemble $$langle cdot rangle $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;⟨&lt;/mml:mo&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mo&gt;⟩&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and the corresponding linear elliptic operator $$-nabla cdot anabla $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mi&gt;∇&lt;/mml:mi&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mi&gt;∇&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . In line with the theory of homogenization, the method proceeds by computing $$d=3$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; correctors ( d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a ( x ) from the whole-space ensemble $$langle cdot rangle $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;⟨&lt;/mml:mo&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mo&gt;⟩&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . We make this point by investigating the bias (or systematic error), i.e., the difference between $$a_{textrm{hom}}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mtext&gt;hom&lt;/mml:mtext&gt; &lt;/mml:msub&gt; &lt;/mml:math&gt; and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a ( x ), we heuristically argue that this error is generically $$O(L^{-1})$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . In case of a suitable periodization of $$langle cdot rangle $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;⟨&lt;/mml:mo&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mo&gt;⟩&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; , we rigorously show that it is $$O(L^{-d})$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . In fact, we give a ","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135692550","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}