SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 593-621, April 2024. Abstract. The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807–2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN.
{"title":"Numerical Analysis for Convergence of a Sample-Wise Backpropagation Method for Training Stochastic Neural Networks","authors":"Richard Archibald, Feng Bao, Yanzhao Cao, Hui Sun","doi":"10.1137/22m1523765","DOIUrl":"https://doi.org/10.1137/22m1523765","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 593-621, April 2024. <br/> Abstract. The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807–2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001008","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}
Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024. Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)[math] formulation of the implicit [math]-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.
{"title":"A Numerical Framework for Nonlinear Peridynamics on Two-Dimensional Manifolds Based on Implicit P-(EC)[math] Schemes","authors":"Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi","doi":"10.1137/22m1498942","DOIUrl":"https://doi.org/10.1137/22m1498942","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024. <br/> Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)[math] formulation of the implicit [math]-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140015424","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 646-666, April 2024. Abstract. We study the homogenization of the equation [math] posed in a bounded convex domain [math] subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic, and symmetric diffusion matrix [math] is merely assumed to be essentially bounded and (if [math]) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax–Milgram-type problem, we obtain [math]-bounds for periodic problems in double-divergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of Hölder continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments.
{"title":"Homogenization of Nondivergence-Form Elliptic Equations with Discontinuous Coefficients and Finite Element Approximation of the Homogenized Problem","authors":"Timo Sprekeler","doi":"10.1137/23m1580279","DOIUrl":"https://doi.org/10.1137/23m1580279","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 646-666, April 2024. <br/> Abstract. We study the homogenization of the equation [math] posed in a bounded convex domain [math] subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic, and symmetric diffusion matrix [math] is merely assumed to be essentially bounded and (if [math]) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax–Milgram-type problem, we obtain [math]-bounds for periodic problems in double-divergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of Hölder continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140015460","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 567-591, February 2024. Abstract. Virtual element methods (VEMs) without extrinsic stabilization in an arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local [math]-conforming macro finite element spaces such that the associated [math] projection of the gradient of virtual element functions is computable, and the [math] projector has a uniform lower bound on the gradient of virtual element function spaces in the [math] norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.
{"title":"Virtual Element Methods Without Extrinsic Stabilization","authors":"Chunyu Chen, Xuehai Huang, Huayi Wei","doi":"10.1137/22m1504196","DOIUrl":"https://doi.org/10.1137/22m1504196","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 567-591, February 2024. <br/> Abstract. Virtual element methods (VEMs) without extrinsic stabilization in an arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local [math]-conforming macro finite element spaces such that the associated [math] projection of the gradient of virtual element functions is computable, and the [math] projector has a uniform lower bound on the gradient of virtual element function spaces in the [math] norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139915934","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 533-566, February 2024. Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.
{"title":"A Universal Median Quasi-Monte Carlo Integration","authors":"Takashi Goda, Kosuke Suzuki, Makoto Matsumoto","doi":"10.1137/22m1525077","DOIUrl":"https://doi.org/10.1137/22m1525077","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 533-566, February 2024. <br/> Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750213","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 500-532, February 2024. Abstract. In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal, and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely, Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs—which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox–Ingersoll–Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh–Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of [math] and outperform schemes previously proposed in the literature.
{"title":"High Order Splitting Methods for SDEs Satisfying a Commutativity Condition","authors":"James M. Foster, Gonçalo dos Reis, Calum Strange","doi":"10.1137/23m155147x","DOIUrl":"https://doi.org/10.1137/23m155147x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 500-532, February 2024. <br/> Abstract. In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal, and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely, Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs—which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox–Ingersoll–Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh–Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of [math] and outperform schemes previously proposed in the literature.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139739343","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024. Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math], this does not define a solution isomorphism. Hence, we use an appropriate state space [math] such that the wave operator becomes an isomorphism from [math] onto [math]. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error [math] between the computed space-time finite element solution [math] and the target function [math] with respect to the regularization parameter [math], and the space-time finite element mesh size [math], depending on the regularity of the desired state [math]. These estimates lead to the optimal choice [math] in order to define the regularization parameter [math] for a given space-time finite element mesh size [math] or to determine the required mesh size [math] when [math] is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.
{"title":"Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation","authors":"Richard Löscher, Olaf Steinbach","doi":"10.1137/22m1532962","DOIUrl":"https://doi.org/10.1137/22m1532962","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024. <br/> Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math], this does not define a solution isomorphism. Hence, we use an appropriate state space [math] such that the wave operator becomes an isomorphism from [math] onto [math]. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error [math] between the computed space-time finite element solution [math] and the target function [math] with respect to the regularization parameter [math], and the space-time finite element mesh size [math], depending on the regularity of the desired state [math]. These estimates lead to the optimal choice [math] in order to define the regularization parameter [math] for a given space-time finite element mesh size [math] or to determine the required mesh size [math] when [math] is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139700968","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 422-451, February 2024. Abstract. We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fréchet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fréchet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.
{"title":"On Uncertainty Quantification of Eigenvalues and Eigenspaces with Higher Multiplicity","authors":"Jürgen Dölz, David Ebert","doi":"10.1137/22m1529324","DOIUrl":"https://doi.org/10.1137/22m1529324","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 422-451, February 2024. <br/> Abstract. We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fréchet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fréchet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705025","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 476-499, February 2024. Abstract. Finding stationary states of Landau free energy functionals has to solve a nonconvex infinite-dimensional optimization problem. In this paper, we develop a Bregman distance based optimization method for minimizing a class of Landau energy functionals and focus on its convergence analysis in the function space. We first analyze the regularity of the stationary states and show the weakly sequential convergence results of the proposed method. Furthermore, under the Łojasiewicz–Simon property, we prove a strongly sequential convergent property and establish the local convergence rate in an appropriate Hilbert space. In particular, we analyze the Łojasiewicz exponent of three well-known Landau models, the Landau–Brazovskii, Lifshitz–Petrich, and Ohta–Kawasaki free energy functionals. Finally, numerical results support our theoretical analysis.
{"title":"Convergence Analysis for Bregman Iterations in Minimizing a Class of Landau Free Energy Functionals","authors":"Chenglong Bao, Chang Chen, Kai Jiang, Lingyun Qiu","doi":"10.1137/22m1517664","DOIUrl":"https://doi.org/10.1137/22m1517664","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 476-499, February 2024. <br/> Abstract. Finding stationary states of Landau free energy functionals has to solve a nonconvex infinite-dimensional optimization problem. In this paper, we develop a Bregman distance based optimization method for minimizing a class of Landau energy functionals and focus on its convergence analysis in the function space. We first analyze the regularity of the stationary states and show the weakly sequential convergence results of the proposed method. Furthermore, under the Łojasiewicz–Simon property, we prove a strongly sequential convergent property and establish the local convergence rate in an appropriate Hilbert space. In particular, we analyze the Łojasiewicz exponent of three well-known Landau models, the Landau–Brazovskii, Lifshitz–Petrich, and Ohta–Kawasaki free energy functionals. Finally, numerical results support our theoretical analysis.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"39 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139700857","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 400-421, February 2024. Abstract. We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell’s equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretizations of Maxwell’s equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions.
{"title":"Frequency-Explicit A Posteriori Error Estimates for Discontinuous Galerkin Discretizations of Maxwell’s Equations","authors":"Théophile Chaumont-Frelet, Patrick Vega","doi":"10.1137/22m1516348","DOIUrl":"https://doi.org/10.1137/22m1516348","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 400-421, February 2024. <br/> Abstract. We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell’s equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretizations of Maxwell’s equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"32 2 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695656","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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