We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = exp (-tfrac{1}{2}varepsilon ^{2}(x-y)^{2})$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(varepsilon /2)^{n} (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates that we use to bound the Hilbert space norm of a weighted polynomial fooling function.
{"title":"Approximation in Hilbert spaces of the Gaussian and related analytic kernels","authors":"Toni Karvonen, Yuya Suzuki","doi":"10.1093/imanum/draf050","DOIUrl":"https://doi.org/10.1093/imanum/draf050","url":null,"abstract":"We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = exp (-tfrac{1}{2}varepsilon ^{2}(x-y)^{2})$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(varepsilon /2)^{n} (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates that we use to bound the Hilbert space norm of a weighted polynomial fooling function.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144639793","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}
We propose a new full discretization of the Biot’s equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters and the time horizon. We further construct an interpolant showing how the error decays for smooth solutions.
{"title":"Inf-sup stable discretization of the quasi-static Biot’s equations in poroelasticity","authors":"Christian Kreuzer, Pietro Zanotti","doi":"10.1093/imanum/draf032","DOIUrl":"https://doi.org/10.1093/imanum/draf032","url":null,"abstract":"We propose a new full discretization of the Biot’s equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters and the time horizon. We further construct an interpolant showing how the error decays for smooth solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"189 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144577905","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}
This study is concerned with the finite element approximation of the elastoplastic torsion problem. We focus on the case of a nonconstant source term, which cannot be easily recast into an obstacle problem as can be done in the case of a constant source term. We present a simple formulation that penalizes the constraint directly on the gradient norm of the solution. We study its well-posedness, derive error estimates and present numerical results to illustrate the theory.
{"title":"Finite element approximation of penalized elastoplastic torsion problem with nonconstant source term","authors":"Franz Chouly, Tom Gustafsson, Patrick Hild","doi":"10.1093/imanum/draf052","DOIUrl":"https://doi.org/10.1093/imanum/draf052","url":null,"abstract":"This study is concerned with the finite element approximation of the elastoplastic torsion problem. We focus on the case of a nonconstant source term, which cannot be easily recast into an obstacle problem as can be done in the case of a constant source term. We present a simple formulation that penalizes the constraint directly on the gradient norm of the solution. We study its well-posedness, derive error estimates and present numerical results to illustrate the theory.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144565936","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}
The well-known individual differences scaling (INDSCAL) model is intended for simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. In this work the problem of fitting the orthogonal INDSCAL model to the data is reformulated and studied as a matrix optimization problem on the product manifold of orthonormal and diagonal matrices. A Riemannian inexact Newton method is proposed to address the underlying problem, with the global and quadratic convergence of the proposed method established under some mild assumptions. Furthermore, the positive definiteness condition of the Riemannian Hessian of the objective function at a solution is derived. Some numerical experiments are provided to illustrate the efficiency of the proposed method.
{"title":"A Riemannian inexact Newton method for solving the orthogonal INDSCAL problem in multidimensional scaling","authors":"Xue-lin Zhou, Chao-qian Li, Jiao-fen Li, Xue-feng Duan","doi":"10.1093/imanum/draf047","DOIUrl":"https://doi.org/10.1093/imanum/draf047","url":null,"abstract":"The well-known individual differences scaling (INDSCAL) model is intended for simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. In this work the problem of fitting the orthogonal INDSCAL model to the data is reformulated and studied as a matrix optimization problem on the product manifold of orthonormal and diagonal matrices. A Riemannian inexact Newton method is proposed to address the underlying problem, with the global and quadratic convergence of the proposed method established under some mild assumptions. Furthermore, the positive definiteness condition of the Riemannian Hessian of the objective function at a solution is derived. Some numerical experiments are provided to illustrate the efficiency of the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"152 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144566072","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}
Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion, which are commonly used in molecular dynamics simulations. To accommodate the degenerate noise structure corresponding to inertia existing in the chain, a characteristically discrete-in-time coupling and contraction proof is devised. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance and total variation are also given, together with numerical bias estimates.
{"title":"Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case","authors":"Martin Chak, Pierre Monmarché","doi":"10.1093/imanum/draf045","DOIUrl":"https://doi.org/10.1093/imanum/draf045","url":null,"abstract":"Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion, which are commonly used in molecular dynamics simulations. To accommodate the degenerate noise structure corresponding to inertia existing in the chain, a characteristically discrete-in-time coupling and contraction proof is devised. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance and total variation are also given, together with numerical bias estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513176","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}
We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Assuming that the drift coefficient of the free stochastic differential equations is operator Lipschitz and the diffusion coefficients are locally operator Lipschitz we prove the strong convergence of the numerical methods. Moreover, we investigate the exponential stability in mean square of the equations and the numerical methods. In particular, the free stochastic theta methods with $theta in [1/2, 1]$ can inherit the exponential stability of original equations for any given step size. Our methods offer better stability than the free Euler–Maruyama method. Numerical results are reported to confirm these theoretical findings and show the efficiency of our methods compared with the free Euler–Maruyama method.
{"title":"Stochastic theta methods for free stochastic differential equations","authors":"Yuanling Niu, Jiaxin Wei, Zhi Yin, Dan Zeng","doi":"10.1093/imanum/draf044","DOIUrl":"https://doi.org/10.1093/imanum/draf044","url":null,"abstract":"We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Assuming that the drift coefficient of the free stochastic differential equations is operator Lipschitz and the diffusion coefficients are locally operator Lipschitz we prove the strong convergence of the numerical methods. Moreover, we investigate the exponential stability in mean square of the equations and the numerical methods. In particular, the free stochastic theta methods with $theta in [1/2, 1]$ can inherit the exponential stability of original equations for any given step size. Our methods offer better stability than the free Euler–Maruyama method. Numerical results are reported to confirm these theoretical findings and show the efficiency of our methods compared with the free Euler–Maruyama method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513174","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}
In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite-element method for the resulting constrained minimization problem that is relaxed by a phase-field approach. The a posteriori error estimators of the adaptive algorithm consist of three components, i.e., the state variable, the adjoint variable and the complementary relation. Moreover, using tools from adaptive finite-element analysis and nonlinear optimization, we establish the strong convergence for a subsequence of adaptively generated discrete solutions to a solution of the continuous optimality system. Several numerical examples are presented to illustrate the convergence and efficiency of the adaptive algorithm.
{"title":"Adaptive approximations of inclusions in a semilinear elliptic problem related to cardiac electrophysiology","authors":"Bangti Jin, Fengru Wang, Yifeng Xu","doi":"10.1093/imanum/draf041","DOIUrl":"https://doi.org/10.1093/imanum/draf041","url":null,"abstract":"In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite-element method for the resulting constrained minimization problem that is relaxed by a phase-field approach. The a posteriori error estimators of the adaptive algorithm consist of three components, i.e., the state variable, the adjoint variable and the complementary relation. Moreover, using tools from adaptive finite-element analysis and nonlinear optimization, we establish the strong convergence for a subsequence of adaptively generated discrete solutions to a solution of the continuous optimality system. Several numerical examples are presented to illustrate the convergence and efficiency of the adaptive algorithm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"87 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144503630","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}
Megala Anandan, Benjamin Boutin, Nicolas Crouseilles
This work deals with the numerical approximation of plasmas that are confined by the effect of a fast oscillating magnetic field [Bostan, M. (2012), Transport of charged particles under fast oscillating magnetic fields. SIAM J. Math. Anal., 44, 1415–1447] in the Vlasov model. The presence of this magnetic field induces oscillations (in time) to the solution of the characteristic equations. Due to its multiscale character, a standard time discretization would lead to an inefficient solver. In this work, time integrators are derived and analyzed for a class of highly oscillatory differential systems. We prove the uniform accuracy property of these time integrators, meaning that the accuracy does not depend on the small parameter $varepsilon $. Moreover, we construct an extension of the scheme, which degenerates towards an energy preserving numerical scheme for the averaged model, when $varepsilon to 0$. Several numerical results illustrate the capabilities of the method.
这项工作涉及受快速振荡磁场影响的等离子体的数值近似[Bostan， M.(2012)，快速振荡磁场下带电粒子的输运]。SIAM J. Math。分析的。[j]在Vlasov模型中的应用。磁场的存在引起特征方程的解（在时间上）振荡。由于其多尺度特性，标准时间离散化将导致求解效率低下。本文推导并分析了一类高振荡微分系统的时间积分器。我们证明了这些时间积分器的一致精度性质，即精度不依赖于小参数。此外，我们构造了该格式的推广，当$varepsilon 到0$时，该格式退化为平均模型的能量守恒数值格式。几个数值结果说明了该方法的能力。
{"title":"Uniformly higher order accurate schemes for dynamics of charged particles under fast oscillating magnetic fields","authors":"Megala Anandan, Benjamin Boutin, Nicolas Crouseilles","doi":"10.1093/imanum/draf048","DOIUrl":"https://doi.org/10.1093/imanum/draf048","url":null,"abstract":"This work deals with the numerical approximation of plasmas that are confined by the effect of a fast oscillating magnetic field [Bostan, M. (2012), Transport of charged particles under fast oscillating magnetic fields. SIAM J. Math. Anal., 44, 1415–1447] in the Vlasov model. The presence of this magnetic field induces oscillations (in time) to the solution of the characteristic equations. Due to its multiscale character, a standard time discretization would lead to an inefficient solver. In this work, time integrators are derived and analyzed for a class of highly oscillatory differential systems. We prove the uniform accuracy property of these time integrators, meaning that the accuracy does not depend on the small parameter $varepsilon $. Moreover, we construct an extension of the scheme, which degenerates towards an energy preserving numerical scheme for the averaged model, when $varepsilon to 0$. Several numerical results illustrate the capabilities of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"87 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144503359","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}
Robert Altmann, Martin Hermann, Daniel Peterseim, Tatjana Stykel
This paper addresses the computation of ground states of multicomponent Bose–Einstein condensates, defined as the global minimizer of an energy functional on an infinite-dimensional generalised oblique manifold. We establish the existence of the ground state, prove its uniqueness up to scaling and characterize it as the solution to a coupled nonlinear eigenvector problem. By equipping the manifold with several Riemannian metrics, we introduce a suite of Riemannian gradient descent and Riemannian Newton methods. Metrics that incorporate first- or second-order information about the energy are particularly advantageous, effectively preconditioning the resulting methods. For a Riemannian gradient descent method with an energy-adaptive metric, we provide a qualitative global and quantitative local convergence analysis, confirming its reliability and robustness with respect to the choice of the spatial discretization. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.
{"title":"Riemannian optimization methods for ground states of multicomponent Bose–Einstein condensates","authors":"Robert Altmann, Martin Hermann, Daniel Peterseim, Tatjana Stykel","doi":"10.1093/imanum/draf046","DOIUrl":"https://doi.org/10.1093/imanum/draf046","url":null,"abstract":"This paper addresses the computation of ground states of multicomponent Bose–Einstein condensates, defined as the global minimizer of an energy functional on an infinite-dimensional generalised oblique manifold. We establish the existence of the ground state, prove its uniqueness up to scaling and characterize it as the solution to a coupled nonlinear eigenvector problem. By equipping the manifold with several Riemannian metrics, we introduce a suite of Riemannian gradient descent and Riemannian Newton methods. Metrics that incorporate first- or second-order information about the energy are particularly advantageous, effectively preconditioning the resulting methods. For a Riemannian gradient descent method with an energy-adaptive metric, we provide a qualitative global and quantitative local convergence analysis, confirming its reliability and robustness with respect to the choice of the spatial discretization. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144503629","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}
This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. It is also shown how to approximate a Hessian-vector product with a minimal number of function evaluations. These approximations are obtained using the techniques called generalized simplex Hessian and generalized centred simplex Hessian. We show how to choose the matrices of directions involved in the computation of these two techniques, depending on the entries of the Hessian of interest. We discuss the number of function evaluations required in each case and develop a general formula to approximate all order-$P$ partial derivatives. Since only function evaluations are required to compute the methods discussed in this paper they are suitable for use in derivative-free optimization methods.
{"title":"Using generalized simplex methods to approximate derivatives","authors":"Gabriel Jarry-Bolduc, Chayne Planiden","doi":"10.1093/imanum/draf053","DOIUrl":"https://doi.org/10.1093/imanum/draf053","url":null,"abstract":"This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. It is also shown how to approximate a Hessian-vector product with a minimal number of function evaluations. These approximations are obtained using the techniques called generalized simplex Hessian and generalized centred simplex Hessian. We show how to choose the matrices of directions involved in the computation of these two techniques, depending on the entries of the Hessian of interest. We discuss the number of function evaluations required in each case and develop a general formula to approximate all order-$P$ partial derivatives. Since only function evaluations are required to compute the methods discussed in this paper they are suitable for use in derivative-free optimization methods.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"643 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144503628","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}
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