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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 376-399, February 2024. Abstract. We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On the one hand, the essential properties of the solution, including positivity, global bounds, mass conservation, and energy dissipation, are all guaranteed by construction. On the other hand, our approach enjoys sufficient flexibility when applied to a large variety of problems including different free energy functionals, general wetting boundary conditions, and degenerate mobilities. The performance of our methods is demonstrated through a suite of examples.
{"title":"Structure Preserving Primal Dual Methods for Gradient Flows with Nonlinear Mobility Transport Distances","authors":"José A. Carrillo, Li Wang, Chaozhen Wei","doi":"10.1137/23m1562068","DOIUrl":"https://doi.org/10.1137/23m1562068","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 376-399, February 2024. <br/> Abstract. We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On the one hand, the essential properties of the solution, including positivity, global bounds, mass conservation, and energy dissipation, are all guaranteed by construction. On the other hand, our approach enjoys sufficient flexibility when applied to a large variety of problems including different free energy functionals, general wetting boundary conditions, and degenerate mobilities. The performance of our methods is demonstrated through a suite of examples.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695658","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 353-375, February 2024. Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.
{"title":"Numerical Methods and Analysis of Computing Quasiperiodic Systems","authors":"Kai Jiang, Shifeng Li, Pingwen Zhang","doi":"10.1137/22m1524783","DOIUrl":"https://doi.org/10.1137/22m1524783","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 353-375, February 2024. <br/> Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660060","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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