SIAM Journal on Numerical Analysis最新文献

{"title":"A Posteriori Lipschitz Bounds for Deep Neural Networks","authors":"Moreno Pintore, Bruno Després","doi":"10.1137/25m1742503","DOIUrl":"https://doi.org/10.1137/25m1742503","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 779-803, June 2026. <br/> Abstract. Deriving sharp a posteriori upper bounds of the Lipschitz constant of deep neural networks is crucial to formally guarantee the robustness of neural network based models. We analyze three existing upper bounds written for the [math] norm. We highlight the interest of working with the [math] and [math] norms and we propose two novel bounds for feed-forward fully connected neural networks, which can be extended to convolutional neural networks. Several numerical tests confirm the theoretical results, help to quantify the relationship between the presented bounds, and establish the better accuracy of the new bounds. Four numerical tests are studied: one with random matrices; two where the output has an analytical closed form; and one for convolutional neural networks trained on the MNIST dataset. The last numerical test is an application to a reference convolutional neural network. One of our bounds is optimal in the sense that it is exact for the second test uniformly with respect to the depth of the neural network and it is always better than the other ones.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"46 1","pages":"779-803"},"PeriodicalIF":2.9,"publicationDate":"2026-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288352","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":"Numerical Analysis for Saddle Dynamics of Some Semilinear Elliptic Problems","authors":"Lei Zhang, Xiangcheng Zheng, Shangqin Zhu","doi":"10.1137/25m1761185","DOIUrl":"https://doi.org/10.1137/25m1761185","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 755-778, June 2026. <br/> Abstract. This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via index-1 saddle dynamics or, equivalently, the gentlest ascent dynamics. To establish clear connections between saddle dynamics and numerical methods of PDEs, as well as improving their compatibility, we first propose the continuous-in-space formulation of saddle dynamics for semilinear elliptic problems. This formulation yields a parabolic system that converges to saddle points. We then analyze the well-posedness, [math] stability, and error estimates of semidiscrete and fully discrete finite element schemes. Significant efforts are devoted to addressing the coupling, gradient nonlinearity, nonlocality of the proposed parabolic system, and the impacts of retraction due to the norm constraint. The error estimate results demonstrate the accuracy and index-preservation of the discrete schemes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"14 1","pages":"755-778"},"PeriodicalIF":2.9,"publicationDate":"2026-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288203","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":"On the Convergence of Split Exponential Integrators for Semilinear Parabolic Problems","authors":"Marco Caliari, Fabio Cassini, Lukas Einkemmer, Alexander Ostermann","doi":"10.1137/25m1743454","DOIUrl":"https://doi.org/10.1137/25m1743454","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 737-754, June 2026. <br/> Abstract. Splitting the exponential-like [math] functions, which typically appear in exponential integrators, is attractive in many situations since it can dramatically reduce the computational cost of the procedure. However, depending on the employed splitting, this can result in order reduction. The aim of this paper is to analyze different such split approximations. We perform the analysis for semilinear problems in the abstract framework of commuting semigroups and derive error bounds that depend, in particular, on whether the vector (to which the [math] functions are applied) satisfies appropriate boundary conditions. We then present the convergence analysis for two split versions of a second-order exponential Runge–Kutta integrator in the context of analytic semigroups, and show that one suffers from order reduction while the other does not. Numerical results for semidiscretized parabolic PDEs confirm the theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":"737-754"},"PeriodicalIF":2.9,"publicationDate":"2026-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288278","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":"Uniformly HP-Stable Elements for the Elasticity Complex","authors":"Francis R. A. Aznaran, Kaibo Hu, Charles Parker","doi":"10.1137/24m1701605","DOIUrl":"https://doi.org/10.1137/24m1701605","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 708-736, June 2026. <br/> Abstract. For the discretization of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu–Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein–Gelfand–Gelfand framework of the finite element exterior calculus. We also construct [math]-bounded projection operators satisfying a commuting diagram property and [math]-stable Hodge decompositions. Numerical examples are provided.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"14 1","pages":"708-736"},"PeriodicalIF":2.9,"publicationDate":"2026-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288205","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":"A Structure-Preserving and Thermodynamically Compatible Cell-Centered Lagrangian Finite Volume Scheme for Continuum Mechanics","authors":"Walter Boscheri, Michael Dumbser, Raphaël Loubère, Pierre-Henri Maire","doi":"10.1137/25m1765857","DOIUrl":"https://doi.org/10.1137/25m1765857","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 685-707, June 2026. &lt;br/&gt; Abstract. In this work we present a novel structure-preserving scheme for the compatible discretization of the Godunov–Peshkov–Romenski (GPR) model of continuum mechanics written in Lagrangian form. The governing equations fall into the larger class of overdetermined hyperbolic and thermodynamically compatible systems of partial differential equations (PDE). This model admits an extra conservation law for the total energy (first principle of thermodynamics) and satisfies the entropy inequality (second principle of thermodynamics). Furthermore, in the absence of algebraic source terms, the distortion field of the continuum and the specific thermal impulse satisfy a curl-free condition, provided the initial data are curl-free. Last but not least, the determinant of the distortion field is related to the density of the medium, i.e., the system is also endowed with a nonlinear algebraic constraint. In the stiff relaxation limit, the system tends to the compressible Navier–Stokes equations, i.e., the GPR model is able to describe at the same time the dynamics of nonlinear solids as well as the one of fluids. The objective of this work is to construct and analyze a new semidiscrete thermodynamically compatible cell-centered Lagrangian finite volume scheme on moving unstructured meshes that satisfies the following structural properties of the governing PDE exactly at the discrete level: (i) compatibility with the first law of thermodynamics, i.e., discrete total energy conservation; (ii) compatibility with the second law of thermodynamics, i.e., discrete entropy inequality; (iii) exact discrete compatibility between the density and the determinant of the distortion field; (iv) exact preservation of the curl-free property of the distortion field and of the specific thermal impulse in the absence of algebraic source terms. We will show that it is indeed possible to achieve all the above properties simultaneously. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of an appropriate and thermodynamically compatible discretization of all the other equations. From this discretization, property (ii) above follows trivially by construction, while (i) leads to provable nonlinear stability, which is an important feature, in particular for the complex PDE system under consideration here. The thermodynamic compatibility and thus nonlinear stability in the sense of total energy conservation is achieved via a very simple and general approach recently introduced by Abgrall, Dumbser, and Maire. by using a scalar correction factor that is defined at the nodes of the grid. This perfectly fits into the formalism of nodal solvers which is typically adopted in cell-centered Lagrangian finite volume methods. The new scheme is run on some academic benchmark problems for computational fluid and solid","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"14 1","pages":"685-707"},"PeriodicalIF":2.9,"publicationDate":"2026-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288207","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":"Lévy Score Function and Score-Based Particle Algorithm for Nonlinear Lévy–Fokker–Planck Equations","authors":"Yuanfei Huang, Chengyu Liu, Xiang Zhou","doi":"10.1137/24m1721311","DOIUrl":"https://doi.org/10.1137/24m1721311","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 656-684, June 2026. <br/> Abstract. The score function for the diffusion process, also known as the gradient of the log-density, is a basic concept to characterize the probability flow with important applications in the score-based diffusion generative modeling and the simulation of Itô stochastic differential equations. However, neither the probability flow nor the corresponding score function for the diffusion-jump process is known. This paper delivers mathematical derivation, a numerical algorithm, and error analysis focusing on the corresponding score function in non-Gaussian systems with jumps and discontinuities represented by the nonlinear Lévy–Fokker–Planck equations. We propose the Lévy score function for such stochastic equations, which features a nonlocal double-integral term, and we develop its training algorithm by minimizing the proposed loss function from samples. Based on the equivalence of the probability flow with deterministic dynamics, we develop a self-consistent score-based transport particle algorithm to sample the interactive Lévy stochastic process at discrete-time grid points. We provide an error bound for the Kullback–Leibler divergence between the numerical and true probability density functions by overcoming the nonlocal challenges in the Lévy score. The full error analysis with the Monte Carlo error and the time discretization error is furthermore established. To show the usefulness and efficiency of our approach, numerical examples from applications in biology and finance are tested.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"10 1","pages":"656-684"},"PeriodicalIF":2.9,"publicationDate":"2026-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288208","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":"A Novel Augmented Subspace Adaptive Finite Element Method for the Eigenvalue Problem with Discontinuous Coefficients","authors":"Manting Xie, Meiling Yue","doi":"10.1137/25m1727266","DOIUrl":"https://doi.org/10.1137/25m1727266","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 631-655, June 2026. <br/> Abstract. This paper proposes an efficient adaptive finite element method (AFEM) for solving the eigenvalue problem with discontinuous coefficients. Different from the existing adaptive algorithms for eigenvalue problems, the method innovatively focuses on solving linearized boundary value problems in each adaptively refined space, complemented by solving small-scale eigenvalue problems on low-dimensional augmented subspaces which are automatically controlled by the algorithm. Notably, it does not require solving the small-scale eigenvalue problem at every iteration, which improves the computational efficiency. Moreover, a novel a posteriori error estimator, which relies on the local oscillations of coefficients near singular points, guides the adaptive refinement process. To further substantiate this method, we give a thorough and rigorous convergence analysis in this paper. Finally, two numerical examples are provided to illustrate the accuracy and efficiency of our new AFEM.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"14 1","pages":"631-655"},"PeriodicalIF":2.9,"publicationDate":"2026-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288211","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":"Unfitted Hybrid High-Order Methods Stabilized by Polynomial Extension for Elliptic Interface Problems","authors":"Erik Burman, Alexandre Ern, Romain Mottier","doi":"10.1137/25m1741790","DOIUrl":"https://doi.org/10.1137/25m1741790","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 601-630, June 2026. <br/> Abstract. In this work, we study the design and analysis of a novel hybrid high-order (HHO) method on unfitted meshes. HHO methods rely on a pair of unknowns, combining polynomials attached to the mesh faces and the mesh cells. In the unfitted framework, the interface can cut through the mesh cells in a very general fashion, and the polynomial unknowns are doubled in the cut cells and the cut faces. In order to avoid the ill-conditioning issues caused by the presence of small cut cells, the novel approach introduced herein is to use polynomial extensions in the definition of the gradient reconstruction operator. Stability and consistency results are established, leading to optimally decaying error estimates. The theory is illustrated by numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 8 1","pages":"601-630"},"PeriodicalIF":2.9,"publicationDate":"2026-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148287400","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":"A Deformation-Based Framework for Learning Solution Mappings of PDEs Defined on Varying Domains","authors":"Shanshan Xiao, Pengzhan Jin, Yifa Tang","doi":"10.1137/24m171499x","DOIUrl":"https://doi.org/10.1137/24m171499x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 537-564, April 2026. <br/> Abstract. In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation; then the solution mapping is regarded as a continuous metric-to-metric mapping, and subsequently can be represented by another continuous metric-to-Banach mapping using two different strategies, referred to as the D2D subframework and the D2E subframework, respectively. We point out that such a metric-to-Banach mapping can be learned by neural networks, and hence the solution mapping is accordingly learned. With this framework, a rigorous convergence analysis is built for the problem of learning solution mappings of PDEs on varying domains. As the theoretical framework holds based on several pivotal assumptions which need to be verified for a given specific problem, we study the star domains as a typical example, and other situations could be similarly verified. There are four important features of this framework: (1) The domains under consideration are not required to be diffeomorphic, and therefore a wide range of regions can be covered by one model, provided they are homeomorphic. (2) The deformation mapping is unnecessary to be continuous, and thus it can be flexibly established via combining a primary identity mapping and a local deformation mapping. This capability facilitates the resolution of large systems where only local parts of the geometry undergo change. (3) In fact, the recent methods (Geo-FNO, DIMON, etc.) belong to the D2D subframework. We point out that the D2D subframework introduces regularity issues, whereas the proposed D2E subframework remains free from such problems. From a comprehensive perspective, the D2E subframework is better. (4) If a linearity-preserving neural operator such as MIONet is adopted, this framework still preserves the linearity of the surrogate solution mapping on its source term for linear PDEs, and thus it can be applied to the hybrid iterative method. We finally present several numerical experiments to validate our theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"103 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148287427","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":"Complex Scaling for the Helmholtz Equation with Dirichlet Boundary Conditions in a Perturbed Half-Space","authors":"Charles Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh","doi":"10.1137/24m1707523","DOIUrl":"https://doi.org/10.1137/24m1707523","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 510-536, April 2026. <br/> Abstract. We present a new analysis of complex scaling applied to the classical double layer potential for the solution of the Helmholtz equation with Dirichlet boundary conditions in compactly perturbed half-spaces in two and three dimensions. The kernel for the double layer potential is the normal derivative of the free-space Green’s function, which has a well-known analytic continuation into the complex plane as a function of both target and source locations. Here, we prove that—when the incident data are analytic and satisfy a precise asymptotic estimate—the solution to the boundary integral equation itself admits an analytic continuation into specific regions of the complex plane and satisfies a related asymptotic estimate (this class of data includes both plane waves and fields induced by point sources). We then show that, with a carefully chosen contour deformation, the oscillatory integrals are converted to exponentially decaying integrals, effectively reducing the infinite domain to a domain of finite size. No other modifications of the domain or the governing equations are introduced. We illustrate the performance of the scheme with two and three dimensional examples and discuss its extension to other boundary conditions and open waveguides.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"28 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148287428","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}