We present a finite element semidiscrete error analysis for the Doyle–Fuller–Newman model, which is the most popular model for lithium-ion batteries. Central to our approach is a novel projection operator designed for the pseudo-($N$+1)-dimensional equation, offering a powerful tool for multiscale equation analysis. Our results bridge a gap in the analysis for dimensions $2 le N le 3$ and achieve optimal convergence rates of $h+(varDelta r)^{2}$. Additionally, we perform a detailed numerical verification, marking the first such validation in this context. By avoiding the change of variables our error analysis can also be extended beyond isothermal conditions.
本文对锂离子电池最常用的模型Doyle-Fuller-Newman模型进行了有限元半离散误差分析。该方法的核心是为伪（$N$+1）维方程设计的一种新的投影算子，为多尺度方程分析提供了一个强大的工具。我们的结果弥补了维度$2 le N le 3$的分析空白，并实现了$h+(varDelta r)^{2}$的最佳收敛率。此外，我们执行了详细的数值验证，标志着这种情况下的第一次验证。通过避免变量的变化，我们的误差分析也可以扩展到等温条件之外。
{"title":"Optimal convergence in finite element semidiscrete error analysis of the Doyle–Fuller–Newman model beyond one dimension with a novel projection operator","authors":"Shu Xu, Liqun Cao","doi":"10.1093/imanum/draf065","DOIUrl":"https://doi.org/10.1093/imanum/draf065","url":null,"abstract":"We present a finite element semidiscrete error analysis for the Doyle–Fuller–Newman model, which is the most popular model for lithium-ion batteries. Central to our approach is a novel projection operator designed for the pseudo-($N$+1)-dimensional equation, offering a powerful tool for multiscale equation analysis. Our results bridge a gap in the analysis for dimensions $2 le N le 3$ and achieve optimal convergence rates of $h+(varDelta r)^{2}$. Additionally, we perform a detailed numerical verification, marking the first such validation in this context. By avoiding the change of variables our error analysis can also be extended beyond isothermal conditions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144918983","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 work introduces finite-element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov–Bakelman–Pucci estimate. Under rather general structural assumptions on the operator, convergence of $C^{1}$ conforming and discontinuous Galerkin methods is proven in the $L^{^infty} $ norm. Numerical experiments on the performance of adaptive mesh refinement driven by local information of the residual in two and three space dimensions are provided.
{"title":"Minimal residual discretization of a class of fully nonlinear elliptic PDE","authors":"Dietmar Gallistl, Ngoc Tien Tran","doi":"10.1093/imanum/draf075","DOIUrl":"https://doi.org/10.1093/imanum/draf075","url":null,"abstract":"This work introduces finite-element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov–Bakelman–Pucci estimate. Under rather general structural assumptions on the operator, convergence of $C^{1}$ conforming and discontinuous Galerkin methods is proven in the $L^{^infty} $ norm. Numerical experiments on the performance of adaptive mesh refinement driven by local information of the residual in two and three space dimensions are provided.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144915513","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}
On a finite time interval $(0,T)$ we consider the multiresolution Galerkin discretization of a modified Hilbert transform ${mathscr{H}}_{T}$ that arises in the space–time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $geq 1$ with sufficiently many vanishing moments that constitute Riesz bases in the Sobolev spaces $ H^{s}_{0,}(0,T)$ and $H^{s}_{,0}(0,T)$. These bases provide stable multilevel splittings of the temporal discretization spaces into ‘increment’ or ‘detail’ spaces. Furthermore, they allow to optimally compress the nonlocal integrodifferential operators that appear in stable space–time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. We then obtain sparse space–time tensor-product spaces via algebraic tensor-products of the temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases). Hence, the construction of multiresolutions in the spatial domain is not necessary. An efficient multilevel preconditioner is proposed that solves the linear system of equations resulting from the sparse space–time Galerkin discretization with essentially linear complexity (in work and memory). A substantial reduction in the number of the degrees of freedom and CPU time (compared with time-marching discretizations) is demonstrated in numerical experiments.
{"title":"Wavelet compressed, modified Hilbert transform in the space–time discretization of the heat equation","authors":"Helmut Harbrecht, Christoph Schwab, Marco Zank","doi":"10.1093/imanum/draf061","DOIUrl":"https://doi.org/10.1093/imanum/draf061","url":null,"abstract":"On a finite time interval $(0,T)$ we consider the multiresolution Galerkin discretization of a modified Hilbert transform ${mathscr{H}}_{T}$ that arises in the space–time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $geq 1$ with sufficiently many vanishing moments that constitute Riesz bases in the Sobolev spaces $ H^{s}_{0,}(0,T)$ and $H^{s}_{,0}(0,T)$. These bases provide stable multilevel splittings of the temporal discretization spaces into ‘increment’ or ‘detail’ spaces. Furthermore, they allow to optimally compress the nonlocal integrodifferential operators that appear in stable space–time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. We then obtain sparse space–time tensor-product spaces via algebraic tensor-products of the temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases). Hence, the construction of multiresolutions in the spatial domain is not necessary. An efficient multilevel preconditioner is proposed that solves the linear system of equations resulting from the sparse space–time Galerkin discretization with essentially linear complexity (in work and memory). A substantial reduction in the number of the degrees of freedom and CPU time (compared with time-marching discretizations) is demonstrated in numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900123","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 energy dissipation law and the maximum bound principle are two fundamental physical properties of the Allen–Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law most of them apply to a modified form of energy. In this work we show that, when the nonlinear term of the Allen–Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge–Kutta schemes with the linear stabilization technique preserves the original energy dissipation property under some step-size constraint. To ensure the Lipschitz condition on the nonlinear term we introduce a rescaling post-processing technique, which guarantees that the numerical solution unconditionally satisfies the maximum bound principle. As a result, our proposed schemes simultaneously maintain both the original energy dissipation law and the maximum bound principle while achieving arbitrarily high-order accuracy. We also establish the optimal error estimate for the proposed schemes. Numerical experiments fully confirm the convergence rates, the preservation of the maximum bound principle and the original energy dissipation property, as well as the high efficiency of the high-order schemes for long-time simulations.
{"title":"Maximum bound principle and original energy dissipation of arbitrarily high-order ETD Runge–Kutta schemes for Allen–Cahn equations","authors":"Chaoyu Quan, Xiaoming Wang, Pinzhong Zheng, Zhi Zhou","doi":"10.1093/imanum/draf069","DOIUrl":"https://doi.org/10.1093/imanum/draf069","url":null,"abstract":"The energy dissipation law and the maximum bound principle are two fundamental physical properties of the Allen–Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law most of them apply to a modified form of energy. In this work we show that, when the nonlinear term of the Allen–Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge–Kutta schemes with the linear stabilization technique preserves the original energy dissipation property under some step-size constraint. To ensure the Lipschitz condition on the nonlinear term we introduce a rescaling post-processing technique, which guarantees that the numerical solution unconditionally satisfies the maximum bound principle. As a result, our proposed schemes simultaneously maintain both the original energy dissipation law and the maximum bound principle while achieving arbitrarily high-order accuracy. We also establish the optimal error estimate for the proposed schemes. Numerical experiments fully confirm the convergence rates, the preservation of the maximum bound principle and the original energy dissipation property, as well as the high efficiency of the high-order schemes for long-time simulations.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144850646","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 gradient bounds of generalized barycentric coordinates (GBCs) play an essential role in the $H^{1}$ norm error estimate of generalized barycentric interpolations (Gillette, Rand & Bajaj (2012) Error estimates for generalized barycentric interpolation. Adv. Comput. Math., 37, 417–439.). Similarly, an $H^{k}$ norm error estimate, $k>1$, requires upper bounds of higher-order derivatives. Due to the nonpolynomial nature of GBCs, existing techniques for proving the gradient bounds do not easily extend to higher-order cases. In this paper, we propose a new method for deriving upper bounds of higher-order derivatives for the Wachspress GBCs on simple convex $d$-dimensional polytopes, $dge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of fourth- or higher-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.
{"title":"Upper bounds of higher-order derivatives for Wachspress coordinates on polytopes","authors":"Pengjie Tian, Yanqiu Wang","doi":"10.1093/imanum/draf063","DOIUrl":"https://doi.org/10.1093/imanum/draf063","url":null,"abstract":"The gradient bounds of generalized barycentric coordinates (GBCs) play an essential role in the $H^{1}$ norm error estimate of generalized barycentric interpolations (Gillette, Rand & Bajaj (2012) Error estimates for generalized barycentric interpolation. Adv. Comput. Math., 37, 417–439.). Similarly, an $H^{k}$ norm error estimate, $k>1$, requires upper bounds of higher-order derivatives. Due to the nonpolynomial nature of GBCs, existing techniques for proving the gradient bounds do not easily extend to higher-order cases. In this paper, we propose a new method for deriving upper bounds of higher-order derivatives for the Wachspress GBCs on simple convex $d$-dimensional polytopes, $dge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of fourth- or higher-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"723 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144763224","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 develops and analyses new low-cost second-order algorithms for natural convection problems with variable density based on time filter, including constant timestep, variable timestep and adaptive timestep methods. The technique of time filter can improve the time accuracy and enhance the efficiency of the Backward Euler (BE) algorithm. The stability of constant timestep and variable timestep implicit BE schemes are proved and the schemes are unconditionally stable. This is to our knowledge the first provable, while the existing literature focuses on the proof of linear-implicit schemes. We also prove the stability of second-order constant stepsize and variable timestep algorithms and construct adaptive algorithms by extending the approach to variable time stepsize. Finally, numerical examples are implemented to verify the stability and high efficiency of these algorithms.
{"title":"Low-cost second-order time-stepping methods for natural convection problem with variable density","authors":"Jilian Wu, Ning Li, Xinlong Feng, Leilei Wei","doi":"10.1093/imanum/draf057","DOIUrl":"https://doi.org/10.1093/imanum/draf057","url":null,"abstract":"This paper develops and analyses new low-cost second-order algorithms for natural convection problems with variable density based on time filter, including constant timestep, variable timestep and adaptive timestep methods. The technique of time filter can improve the time accuracy and enhance the efficiency of the Backward Euler (BE) algorithm. The stability of constant timestep and variable timestep implicit BE schemes are proved and the schemes are unconditionally stable. This is to our knowledge the first provable, while the existing literature focuses on the proof of linear-implicit schemes. We also prove the stability of second-order constant stepsize and variable timestep algorithms and construct adaptive algorithms by extending the approach to variable time stepsize. Finally, numerical examples are implemented to verify the stability and high efficiency of these algorithms.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144763350","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 an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and conservation of invariants. In particular, symmetric partitioned linear multistep methods with no common roots in their first characteristic polynomials, except unity, appear as efficient methods to approximate nonseparable Hamiltonian systems since they can be explicit and show good long term behaviour at the same time. As a case study, a thorough analysis is given for small oscillations of the double pendulum problem, which is illustrated by numerical experiments.
{"title":"Long-term behaviour of symmetric partitioned linear multistep methods. I: global error and conservation of invariants","authors":"Begoña Cano, Ángel Durán, Melquiades Rodríguez","doi":"10.1093/imanum/draf062","DOIUrl":"https://doi.org/10.1093/imanum/draf062","url":null,"abstract":"In this paper an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and conservation of invariants. In particular, symmetric partitioned linear multistep methods with no common roots in their first characteristic polynomials, except unity, appear as efficient methods to approximate nonseparable Hamiltonian systems since they can be explicit and show good long term behaviour at the same time. As a case study, a thorough analysis is given for small oscillations of the double pendulum problem, which is illustrated by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702019","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}
Vandermonde matrices are usually exponentially ill-conditioned and often result in unstable approximations. In this paper we introduce and analyse the multivariate Vandermonde with Arnoldi (V+A) method, which is based on least-squares approximation together with a Stieltjes orthogonalization process, for approximating continuous, multivariate functions on $d$-dimensional irregular domains. The V+A method addresses the ill-conditioning of the Vandermonde approximation by creating a set of discrete orthogonal bases with respect to a discrete measure. The V+A method is simple and general, relying only on the domain’s sample points. This paper analyses the sample complexity of the least-squares approximation that uses the V+A method. We show that, for a large class of domains, this approximation gives a well-conditioned and near-optimal $N$-dimensional least-squares approximation using $M={cal O}(N^{2})$ equispaced sample points or $M={cal O}(N^{2}log N)$ random sample points, independently of $d$. We provide a comprehensive analysis of the error estimates and the rate of convergence of the least-squares approximation that uses the V+A method. Based on the multivariate V+A techniques we propose a new variant of the weighted V+A least-squares algorithm that uses only $M={cal O}(Nlog N)$ sample points to achieve a near-optimal approximation. Our initial numerical results validate that the V+A least-squares approximation method provides well-conditioned and near-optimal approximations for multivariate functions on (irregular) domains. Additionally, the (weighted) least-squares approximation that uses the V+A method performs competitively with state-of-the-art orthogonalization techniques and can serve as a practical tool for selecting near-optimal distributions of sample points in irregular domains.
{"title":"Convergence and near-optimal sampling for multivariate function approximations in irregular domains via Vandermonde with Arnoldi","authors":"Wenqi Zhu, Yuji Nakatsukasa","doi":"10.1093/imanum/draf055","DOIUrl":"https://doi.org/10.1093/imanum/draf055","url":null,"abstract":"Vandermonde matrices are usually exponentially ill-conditioned and often result in unstable approximations. In this paper we introduce and analyse the multivariate Vandermonde with Arnoldi (V+A) method, which is based on least-squares approximation together with a Stieltjes orthogonalization process, for approximating continuous, multivariate functions on $d$-dimensional irregular domains. The V+A method addresses the ill-conditioning of the Vandermonde approximation by creating a set of discrete orthogonal bases with respect to a discrete measure. The V+A method is simple and general, relying only on the domain’s sample points. This paper analyses the sample complexity of the least-squares approximation that uses the V+A method. We show that, for a large class of domains, this approximation gives a well-conditioned and near-optimal $N$-dimensional least-squares approximation using $M={cal O}(N^{2})$ equispaced sample points or $M={cal O}(N^{2}log N)$ random sample points, independently of $d$. We provide a comprehensive analysis of the error estimates and the rate of convergence of the least-squares approximation that uses the V+A method. Based on the multivariate V+A techniques we propose a new variant of the weighted V+A least-squares algorithm that uses only $M={cal O}(Nlog N)$ sample points to achieve a near-optimal approximation. Our initial numerical results validate that the V+A least-squares approximation method provides well-conditioned and near-optimal approximations for multivariate functions on (irregular) domains. Additionally, the (weighted) least-squares approximation that uses the V+A method performs competitively with state-of-the-art orthogonalization techniques and can serve as a practical tool for selecting near-optimal distributions of sample points in irregular domains.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144684797","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}
Three types of lowest-order finite volume element methods, i.e., the conforming, nonconforming and discontinuous schemes, are introduced and analysed for a variational inequality governed by the stationary Stokes equations. The variational inequality arises due to a nonlinear and nondifferentiable relationship in the slip boundary condition of friction type. This relationship cannot be well combined into a finite volume scheme by a standard procedure based on integration by parts on dual control volumes. Thereby we propose to enforce it pointwisely at cell centres in a dual mesh, which leads to some numerical integration formula for the boundary nonlinear term in the variational inequality. The resulting finite volume schemes can be seen to be equivalent to some finite element methods. We further show their solvability and stability, as well as the a priori error estimates with optimal approximation behaviours. Numerical results are reported to demonstrate the theoretical findings.
{"title":"Numerical analysis of lowest-order finite volume methods for a class of Stokes variational inequality problem","authors":"Feifei Jing, Takahito Kashiwabara, Wenjing Yan","doi":"10.1093/imanum/draf056","DOIUrl":"https://doi.org/10.1093/imanum/draf056","url":null,"abstract":"Three types of lowest-order finite volume element methods, i.e., the conforming, nonconforming and discontinuous schemes, are introduced and analysed for a variational inequality governed by the stationary Stokes equations. The variational inequality arises due to a nonlinear and nondifferentiable relationship in the slip boundary condition of friction type. This relationship cannot be well combined into a finite volume scheme by a standard procedure based on integration by parts on dual control volumes. Thereby we propose to enforce it pointwisely at cell centres in a dual mesh, which leads to some numerical integration formula for the boundary nonlinear term in the variational inequality. The resulting finite volume schemes can be seen to be equivalent to some finite element methods. We further show their solvability and stability, as well as the a priori error estimates with optimal approximation behaviours. Numerical results are reported to demonstrate the theoretical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144677490","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}
Pavel Dvurechensky, Gabriele Iommazzo, Shimrit Shtern, Mathias Staudigl
We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art semidefinite programming solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.
{"title":"A conditional gradient homotopy method with applications to semidefinite programming","authors":"Pavel Dvurechensky, Gabriele Iommazzo, Shimrit Shtern, Mathias Staudigl","doi":"10.1093/imanum/draf059","DOIUrl":"https://doi.org/10.1093/imanum/draf059","url":null,"abstract":"We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art semidefinite programming solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"96 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144669720","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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