Pham Quy Muoi, Vo Quang Duy, Chau Vinh Khanh, Nguyen Trung Thành
In this paper, we propose a fast algorithm for smooth convex minimization problems in a real Hilbert space whose objective functionals have Lipschitz continuous Fréchet derivatives. The main advantage of the proposed algorithm is that it has the optimal-order convergence rate and faster than Nesterov’s algorithm with the best setting. To demonstrate the efficiency of the proposed algorithm, we compare it with Nesterov’s algorithm in several examples, including inverse source problems for elliptic and hyperbolic PDEs. The numerical tests show that the proposed algorithm converges faster than Nesterov’s algorithm.
{"title":"A fast algorithm for smooth convex minimization problems and its application to inverse source problems","authors":"Pham Quy Muoi, Vo Quang Duy, Chau Vinh Khanh, Nguyen Trung Thành","doi":"10.1093/imanum/drae091","DOIUrl":"https://doi.org/10.1093/imanum/drae091","url":null,"abstract":"In this paper, we propose a fast algorithm for smooth convex minimization problems in a real Hilbert space whose objective functionals have Lipschitz continuous Fréchet derivatives. The main advantage of the proposed algorithm is that it has the optimal-order convergence rate and faster than Nesterov’s algorithm with the best setting. To demonstrate the efficiency of the proposed algorithm, we compare it with Nesterov’s algorithm in several examples, including inverse source problems for elliptic and hyperbolic PDEs. The numerical tests show that the proposed algorithm converges faster than Nesterov’s algorithm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991118","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}
Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker
Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.
{"title":"Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems","authors":"Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker","doi":"10.1093/imanum/drae094","DOIUrl":"https://doi.org/10.1093/imanum/drae094","url":null,"abstract":"Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990075","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}
Paola F Antonietti, Michele Botti, Ilario Mazzieri
This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is coupled with a dG time integration scheme, resulting in a full space–time dG discretization. We present the stability analysis for both semidiscrete and fully discrete formulations, and derive error estimates in suitable energy norms. The method is applied to various numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.
{"title":"Discontinuous Galerkin discretization of coupled poroelasticity–elasticity problems","authors":"Paola F Antonietti, Michele Botti, Ilario Mazzieri","doi":"10.1093/imanum/drae093","DOIUrl":"https://doi.org/10.1093/imanum/drae093","url":null,"abstract":"This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is coupled with a dG time integration scheme, resulting in a full space–time dG discretization. We present the stability analysis for both semidiscrete and fully discrete formulations, and derive error estimates in suitable energy norms. The method is applied to various numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888925","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 dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which was first proposed by Alouges and Jaisson and later, applied for solving various practical problems. Since the classical energy approach fails to be applied directly to the analysis of this method, previous studies only focused on the convergence and until now, no any error estimate was established for such an NTP-FEM. This paper presents a rigorous error analysis and establishes the optimal $H^{1}$ error estimate. Numerical results are provided to confirm our theoretical analysis.
{"title":"Optimal error analysis of the normalized tangent plane FEM for Landau–Lifshitz–Gilbert equation","authors":"Rong An, Yonglin Li, Weiwei Sun","doi":"10.1093/imanum/drae084","DOIUrl":"https://doi.org/10.1093/imanum/drae084","url":null,"abstract":"The dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which was first proposed by Alouges and Jaisson and later, applied for solving various practical problems. Since the classical energy approach fails to be applied directly to the analysis of this method, previous studies only focused on the convergence and until now, no any error estimate was established for such an NTP-FEM. This paper presents a rigorous error analysis and establishes the optimal $H^{1}$ error estimate. Numerical results are provided to confirm our theoretical analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888311","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 study a higher-order surface finite-element penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated, which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyse the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite-element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.
{"title":"Parametric finite-element discretization of the surface Stokes equations: inf-sup stability and discretization error analysis","authors":"Hanne Hardering, Simon Praetorius","doi":"10.1093/imanum/drae080","DOIUrl":"https://doi.org/10.1093/imanum/drae080","url":null,"abstract":"We study a higher-order surface finite-element penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated, which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyse the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite-element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"64 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888926","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 article, convergence and quasi-optimal rate of convergence of an Adaptive Finite Element Method is shown for the Dirichlet boundary control problem that was proposed by Chowdhury et al. (2017, Error bounds for a Dirichlet boundary control problem based on energy spaces, Math. Comp., 86, 1103–1126). The theoretical results are illustrated by numerical experiments.
{"title":"Convergence and quasi-optimality of an AFEM for the Dirichlet boundary control problem","authors":"Arnab Pal, Thirupathi Gudi","doi":"10.1093/imanum/drae092","DOIUrl":"https://doi.org/10.1093/imanum/drae092","url":null,"abstract":"In this article, convergence and quasi-optimal rate of convergence of an Adaptive Finite Element Method is shown for the Dirichlet boundary control problem that was proposed by Chowdhury et al. (2017, Error bounds for a Dirichlet boundary control problem based on energy spaces, Math. Comp., 86, 1103–1126). The theoretical results are illustrated by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"139 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888312","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 study the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.
{"title":"The Milstein scheme for singular SDEs with Hölder continuous drift","authors":"Máté Gerencsér, Gerald Lampl, Chengcheng Ling","doi":"10.1093/imanum/drae083","DOIUrl":"https://doi.org/10.1093/imanum/drae083","url":null,"abstract":"We study the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823191","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 paper, we propose a conforming multi-domain spectral method that combines mapping techniques to solve the diffusive-viscous wave equation in the exterior domain of two complex obstacles. First, we confine the exterior domain within a relatively large rectangular computational domain. Then, we decompose the rectangular domain into two sub-domains, each containing one obstacle. By applying coordinate transformations along radial direction to each sub-domain, we map them into eight regular sub-blocks. Subsequently, we perform numerical simulations using classical spectral methods on these regular sub-blocks. Our analysis focuses on the optimal convergence of this approach. The numerical results demonstrate the high-order accuracy of the proposed method.
{"title":"A conforming multi-domain Legendre spectral method for solving diffusive-viscous wave equations in the exterior domain with separated star-shaped obstacles","authors":"Guoqing Yao, Zicheng Wang, Zhongqing Wang","doi":"10.1093/imanum/drae085","DOIUrl":"https://doi.org/10.1093/imanum/drae085","url":null,"abstract":"In this paper, we propose a conforming multi-domain spectral method that combines mapping techniques to solve the diffusive-viscous wave equation in the exterior domain of two complex obstacles. First, we confine the exterior domain within a relatively large rectangular computational domain. Then, we decompose the rectangular domain into two sub-domains, each containing one obstacle. By applying coordinate transformations along radial direction to each sub-domain, we map them into eight regular sub-blocks. Subsequently, we perform numerical simulations using classical spectral methods on these regular sub-blocks. Our analysis focuses on the optimal convergence of this approach. The numerical results demonstrate the high-order accuracy of the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823307","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 study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $sigma _{c}$. We provide explicit bounds on $sigma _{c}$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.
{"title":"Asymptotic consistency of the WSINDy algorithm in the limit of continuum data","authors":"Daniel A Messenger, David M Bortz","doi":"10.1093/imanum/drae086","DOIUrl":"https://doi.org/10.1093/imanum/drae086","url":null,"abstract":"In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $sigma _{c}$. We provide explicit bounds on $sigma _{c}$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142820569","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}
A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous and state-dependent delay. The framework is a natural extension of standard spectral collocation methods and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.
{"title":"A spectral collocation method for functional and delay differential equations","authors":"Nicholas Hale","doi":"10.1093/imanum/drae079","DOIUrl":"https://doi.org/10.1093/imanum/drae079","url":null,"abstract":"A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous and state-dependent delay. The framework is a natural extension of standard spectral collocation methods and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142752867","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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