SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 391-429, April 2026. Abstract. The filtering distribution captures the statistics of the state of a possibly stochastic dynamical system from partial and noisy observations. Classical particle filters provably approximate this distribution in quite general settings; however, they behave poorly for high dimensional problems, suffering weight collapse. This issue is circumvented by the ensemble Kalman filter, which is an equal-weights interacting particle system. However, this finite particle system is only proven to approximate the true filter in the linear Gaussian case. In practice, however, it is applied in much broader settings; as a result, establishing its approximation properties more generally is important. There has been recent progress in the theoretical analysis of the algorithm in discrete time, establishing stability and error estimates, in relation to the true filter, in non-Gaussian settings; but the assumptions on the dynamics and observation models rule out the unbounded vector fields that arise in practice, and the analysis applies only to the mean field limit of the discrete time ensemble Kalman filter. The present work establishes error bounds between the filtering distribution and the finite particle discrete time ensemble Kalman filter when the dynamics and observation vector fields may be unbounded, allowing linear growth.
{"title":"Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting","authors":"E. Calvello, P. Monmarché, A. M. Stuart, U. Vaes","doi":"10.1137/25m1732544","DOIUrl":"https://doi.org/10.1137/25m1732544","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 391-429, April 2026. <br/> Abstract. The filtering distribution captures the statistics of the state of a possibly stochastic dynamical system from partial and noisy observations. Classical particle filters provably approximate this distribution in quite general settings; however, they behave poorly for high dimensional problems, suffering weight collapse. This issue is circumvented by the ensemble Kalman filter, which is an equal-weights interacting particle system. However, this finite particle system is only proven to approximate the true filter in the linear Gaussian case. In practice, however, it is applied in much broader settings; as a result, establishing its approximation properties more generally is important. There has been recent progress in the theoretical analysis of the algorithm in discrete time, establishing stability and error estimates, in relation to the true filter, in non-Gaussian settings; but the assumptions on the dynamics and observation models rule out the unbounded vector fields that arise in practice, and the analysis applies only to the mean field limit of the discrete time ensemble Kalman filter. The present work establishes error bounds between the filtering distribution and the finite particle discrete time ensemble Kalman filter when the dynamics and observation vector fields may be unbounded, allowing linear growth.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147478698","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 64, Issue 2, Page 370-390, April 2026. Abstract. In this paper, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form; cf. [W. Dörfler et al., Wave Phenomena: Mathematical Analysis and Numerical Approximation, Springer, Cham, 2023]. We focus on a discontinuous Galerkin space discretization applied to a locally refined mesh or a small region with high wave speed. This results in a stiff system of ordinary differential equations, where the stiffness is mainly caused by a small region of the spatial mesh. When using explicit time-integration schemes, the time stepsize is severely restricted by a few spatial elements, leading to a loss of efficiency. As a remedy, we propose and analyze a general leapfrog-based scheme which is motivated by [C. Carle and M. Hochbruck, SIAM J. Numer. Anal., 60 (2022), pp. 2897–2924]. The new, fully explicit, local time-integration method filters the stiff part of the system in such a way that its CFL condition is significantly weaker than that of the leapfrog scheme while its computational cost is only slightly larger. For this scheme, the filter function is a suitably scaled and shifted Chebyshev polynomial. While our main interest is in explicit local time-stepping schemes, the filter functions can be much more general, for instance, a certain rational function leads to the locally implicit method, proposed and analyzed in [M. Hochbruch and A. Sturm, SIAM J. Numer. Anal., 54 (2016), pp. 3167–3191]. Our analysis provides sufficient conditions on the filter function to ensure full order of convergence in space and second order in time for the whole class of local time-integration schemes.
SIAM数值分析杂志，64卷，第2期，370-390页，2026年4月。摘要。在本文中，我们讨论了具有双场结构的Friedrichs系统的完全离散化问题，如麦克斯韦方程组或垂阶形式的声波方程；cf。W。Dörfler et al.，波浪现象：数学分析和数值近似，[j].中国科学：自然科学版，2009。重点研究了局部精细网格或高波速小区域的不连续伽辽金空间离散化方法。这导致了一个刚性的常微分方程组，其中的刚度主要是由空间网格的一个小区域引起的。当使用显式时间积分方案时，时间步长受到少数空间元素的严重限制，导致效率损失。作为补救措施，我们提出并分析了一个通用的基于跨越式的方案，该方案的动机是[C]。卡尔和M. Hochbruck, SIAM J. number。分析的。， 60 (2022), pp. 2897-2924。新的、完全显式的局部时间积分方法过滤了系统的刚性部分，使其CFL条件明显弱于跳越方案，而其计算成本仅略大。对于该方案，滤波函数是一个适当缩放和移位的切比雪夫多项式。虽然我们的主要兴趣是在显式局部时间步进方案，但过滤函数可以更一般，例如，一个特定的有理函数导致局部隐式方法，在[M]中提出和分析。霍赫布鲁赫和A. Sturm, SIAM J. number。分析的。， 54 (2016), pp. 3167-3191]。我们的分析提供了滤波函数在空间上是满阶收敛，在时间上是二阶收敛的充分条件。
{"title":"Local Time Integration for Friedrichs’ Systems","authors":"Marlis Hochbruck, Malik Scheifinger","doi":"10.1137/25m1735627","DOIUrl":"https://doi.org/10.1137/25m1735627","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 370-390, April 2026. <br/> Abstract. In this paper, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form; cf. [W. Dörfler et al., Wave Phenomena: Mathematical Analysis and Numerical Approximation, Springer, Cham, 2023]. We focus on a discontinuous Galerkin space discretization applied to a locally refined mesh or a small region with high wave speed. This results in a stiff system of ordinary differential equations, where the stiffness is mainly caused by a small region of the spatial mesh. When using explicit time-integration schemes, the time stepsize is severely restricted by a few spatial elements, leading to a loss of efficiency. As a remedy, we propose and analyze a general leapfrog-based scheme which is motivated by [C. Carle and M. Hochbruck, SIAM J. Numer. Anal., 60 (2022), pp. 2897–2924]. The new, fully explicit, local time-integration method filters the stiff part of the system in such a way that its CFL condition is significantly weaker than that of the leapfrog scheme while its computational cost is only slightly larger. For this scheme, the filter function is a suitably scaled and shifted Chebyshev polynomial. While our main interest is in explicit local time-stepping schemes, the filter functions can be much more general, for instance, a certain rational function leads to the locally implicit method, proposed and analyzed in [M. Hochbruch and A. Sturm, SIAM J. Numer. Anal., 54 (2016), pp. 3167–3191]. Our analysis provides sufficient conditions on the filter function to ensure full order of convergence in space and second order in time for the whole class of local time-integration schemes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"89 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147380646","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}
Benjamin Berkels, Alexander Effland, Martin Rumpf, Jan Verhülsdonk
SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 350-369, April 2026. Abstract. In this paper, a posteriori error estimates are derived for the approximation error of minimizers of functionals on the space of functions with bounded variation with a nonconvex lower-order term. To this end, the calibration method by Alberti, Bouchitté, and Dal Maso [Calc. Var. Partial Differential Equations, 16 (2003), pp. 299–333] allows the problem to be reformulated as a uniformly convex variational problem over characteristic functions of subgraphs in one dimension higher. A primal-dual approach is formulated where the duality of divergence and gradient properly incorporates boundary conditions for the primal variable. Based on this, a posteriori error estimates can be derived first for the relaxed problem in the [math]-norm. A cut-out argument allows converting this into an [math]-error estimate for the characteristic subgraph functions apart from the jump interface, whereas the area of the interfacial region is estimated separately. To apply the estimate, we consider as one possible discretization a conforming finite element space for the primal variable and a nonconforming space for the dual variable. Finally, we validate the a posteriori error estimates in numerical experiments for a prototypical nonconvex functional in one and two dimensions as well as depth estimation in stereo imaging, a classical computer vision problem.
{"title":"A Posteriori Error Control for Nonconvex Problems via Calibration","authors":"Benjamin Berkels, Alexander Effland, Martin Rumpf, Jan Verhülsdonk","doi":"10.1137/25m1782959","DOIUrl":"https://doi.org/10.1137/25m1782959","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 350-369, April 2026. <br/> Abstract. In this paper, a posteriori error estimates are derived for the approximation error of minimizers of functionals on the space of functions with bounded variation with a nonconvex lower-order term. To this end, the calibration method by Alberti, Bouchitté, and Dal Maso [Calc. Var. Partial Differential Equations, 16 (2003), pp. 299–333] allows the problem to be reformulated as a uniformly convex variational problem over characteristic functions of subgraphs in one dimension higher. A primal-dual approach is formulated where the duality of divergence and gradient properly incorporates boundary conditions for the primal variable. Based on this, a posteriori error estimates can be derived first for the relaxed problem in the [math]-norm. A cut-out argument allows converting this into an [math]-error estimate for the characteristic subgraph functions apart from the jump interface, whereas the area of the interfacial region is estimated separately. To apply the estimate, we consider as one possible discretization a conforming finite element space for the primal variable and a nonconforming space for the dual variable. Finally, we validate the a posteriori error estimates in numerical experiments for a prototypical nonconvex functional in one and two dimensions as well as depth estimation in stereo imaging, a classical computer vision problem.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147358813","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 64, Issue 2, Page 303-325, April 2026. Abstract. The modified electromagnetic transmission eigenvalue problem (METEP) arises from the inverse scattering theory and can be used to detect changes of the material properties in nondestructive testing. This paper proposes and analyzes a conforming edge element method for the METEP. We establish a rigorous error analysis of the numerical eigenpairs by proving the uniform convergence of the discrete operator. In particular, as the problem contains two second order equations and is indefinite, we introduce auxiliary problems and show that they satisfy [math]-coercivity, based on which we prove the existence of both the continuous and discrete solution operators to the source problem. We then prove the uniform convergence of the discrete solution operator by reformulating the continuous and discrete solution operators. Optimal error estimates are obtained by investigating the adjoint problems and using the spectral approximation theory for compact operators. The theory is validated by numerical examples with various coefficients for different domains in both two and three dimensions.
{"title":"Error Analysis of a Conforming Finite Element Method for the Modified Electromagnetic Transmission Eigenvalue Problem","authors":"Jiayu Han, Jiguang Sun, Qian Zhang","doi":"10.1137/25m1723608","DOIUrl":"https://doi.org/10.1137/25m1723608","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 303-325, April 2026. <br/> Abstract. The modified electromagnetic transmission eigenvalue problem (METEP) arises from the inverse scattering theory and can be used to detect changes of the material properties in nondestructive testing. This paper proposes and analyzes a conforming edge element method for the METEP. We establish a rigorous error analysis of the numerical eigenpairs by proving the uniform convergence of the discrete operator. In particular, as the problem contains two second order equations and is indefinite, we introduce auxiliary problems and show that they satisfy [math]-coercivity, based on which we prove the existence of both the continuous and discrete solution operators to the source problem. We then prove the uniform convergence of the discrete solution operator by reformulating the continuous and discrete solution operators. Optimal error estimates are obtained by investigating the adjoint problems and using the spectral approximation theory for compact operators. The theory is validated by numerical examples with various coefficients for different domains in both two and three dimensions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147329709","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 64, Issue 2, Page 326-349, April 2026. Abstract. Preasymptotic error estimates are derived for the second-type Nédélec linear edge element method and the linear [math]-conforming interior penalty edge element method (CIP-EEM) for the time-harmonic Maxwell equations with large wave number. It is shown that under the mesh condition that [math] is sufficiently small, the errors of the solutions to both methods are bounded by [math] in the energy norm and [math] in the [math]-scaled [math] norm, where [math] is the wave number and [math] is the mesh size. Numerical tests are provided to illustrate our theoretical results and the potential of CIP-EEM in significantly reducing the pollution effect.
{"title":"Preasymptotic Error Estimates of Linear EEM and CIP-EEM for the Time-Harmonic Maxwell Equations with Large Wave Number","authors":"Shuaishuai Lu, Haijun Wu","doi":"10.1137/24m1680362","DOIUrl":"https://doi.org/10.1137/24m1680362","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 326-349, April 2026. <br/> Abstract. Preasymptotic error estimates are derived for the second-type Nédélec linear edge element method and the linear [math]-conforming interior penalty edge element method (CIP-EEM) for the time-harmonic Maxwell equations with large wave number. It is shown that under the mesh condition that [math] is sufficiently small, the errors of the solutions to both methods are bounded by [math] in the energy norm and [math] in the [math]-scaled [math] norm, where [math] is the wave number and [math] is the mesh size. Numerical tests are provided to illustrate our theoretical results and the potential of CIP-EEM in significantly reducing the pollution effect.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147329708","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}
Markus Bachmayr, Riccardo Bardin, Matthias Schlottbom
SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 277-302, February 2026. Abstract. The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption, and scattering in many relevant societal applications and requires numerical approximations. However, classical numerical algorithms scale unfavorably with respect to the dimensionality of such radiative transfer problems, where solutions depend on physical as well as angular variables. In this paper, we address this dimensionality issue by developing a low-rank tensor product framework for the RTE in plane-parallel geometry. We exploit the tensor product nature of the phase space to recover an operator equation where the operator is given by a short sum of Kronecker products. This equation is solved by a preconditioned and rank-controlled Richardson iteration in Hilbert spaces. Using exponential sums approximations, we construct a preconditioner that is compatible with the low-rank tensor product framework. The use of suitable preconditioning techniques yields a transformation of the operator equation in Hilbert space into a sequence space with a Euclidean inner product, enabling rigorous error and rank control in the Euclidean metric.
{"title":"Low-Rank Tensor Product Richardson Iteration for Radiative Transfer in Plane-Parallel Geometry","authors":"Markus Bachmayr, Riccardo Bardin, Matthias Schlottbom","doi":"10.1137/24m1648065","DOIUrl":"https://doi.org/10.1137/24m1648065","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 277-302, February 2026. <br/> Abstract. The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption, and scattering in many relevant societal applications and requires numerical approximations. However, classical numerical algorithms scale unfavorably with respect to the dimensionality of such radiative transfer problems, where solutions depend on physical as well as angular variables. In this paper, we address this dimensionality issue by developing a low-rank tensor product framework for the RTE in plane-parallel geometry. We exploit the tensor product nature of the phase space to recover an operator equation where the operator is given by a short sum of Kronecker products. This equation is solved by a preconditioned and rank-controlled Richardson iteration in Hilbert spaces. Using exponential sums approximations, we construct a preconditioner that is compatible with the low-rank tensor product framework. The use of suitable preconditioning techniques yields a transformation of the operator equation in Hilbert space into a sequence space with a Euclidean inner product, enabling rigorous error and rank control in the Euclidean metric.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"87 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146210246","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 64, Issue 1, Page 251-276, February 2026. Abstract. We address the convergence analysis of lattice Boltzmann methods for scalar nonlinear conservation laws, focusing on two-relaxation-times (TRT) schemes. Unlike Finite Difference/Finite Volume methods, lattice Boltzmann schemes offer exceptional computational efficiency and parallelization capabilities. However, their monotonicity and [math]-stability remain underexplored. Extending existing results on simpler BGK schemes, we derive conditions ensuring that TRT schemes are monotone and stable by leveraging their unique relaxation structure. Our analysis culminates in proving convergence of the numerical solution to the weak entropy solution of the conservation law. Compared to BGK schemes, TRT schemes achieve reduced numerical diffusion while retaining provable convergence. Numerical experiments validate and illustrate the theoretical findings.
{"title":"Monotonicity and Convergence of Two-Relaxation-Times Lattice Boltzmann Schemes for a Nonlinear Conservation Law","authors":"Denise Aregba-Driollet, Thomas Bellotti","doi":"10.1137/25m1725218","DOIUrl":"https://doi.org/10.1137/25m1725218","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 251-276, February 2026. <br/> Abstract. We address the convergence analysis of lattice Boltzmann methods for scalar nonlinear conservation laws, focusing on two-relaxation-times (TRT) schemes. Unlike Finite Difference/Finite Volume methods, lattice Boltzmann schemes offer exceptional computational efficiency and parallelization capabilities. However, their monotonicity and [math]-stability remain underexplored. Extending existing results on simpler BGK schemes, we derive conditions ensuring that TRT schemes are monotone and stable by leveraging their unique relaxation structure. Our analysis culminates in proving convergence of the numerical solution to the weak entropy solution of the conservation law. Compared to BGK schemes, TRT schemes achieve reduced numerical diffusion while retaining provable convergence. Numerical experiments validate and illustrate the theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146134778","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 64, Issue 1, Page 224-250, February 2026. Abstract. The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.
{"title":"A Primal-Dual Level Set Method for Computing Geodesic Distances","authors":"Hailiang Liu, Laura Zinnel","doi":"10.1137/24m1721086","DOIUrl":"https://doi.org/10.1137/24m1721086","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 224-250, February 2026. <br/> Abstract. The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"89 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146121993","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}
Ziqian Li, Kang Liu, Lorenzo Liverani, Enrique Zuazua
SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 193-223, February 2026. Abstract. In this paper, we introduce semiautonomous neural ODEs (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses, we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.
{"title":"Universal Approximation of Dynamical Systems by Semiautonomous Neural ODEs and Applications","authors":"Ziqian Li, Kang Liu, Lorenzo Liverani, Enrique Zuazua","doi":"10.1137/24m1679690","DOIUrl":"https://doi.org/10.1137/24m1679690","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 193-223, February 2026. <br/> Abstract. In this paper, we introduce semiautonomous neural ODEs (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses, we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"94 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101983","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 64, Issue 1, Page 170-192, February 2026. Abstract. Off-lattice agent-based models (or cell-based models) of multicellular systems are increasingly used to create in-silico models of in-vitro and in-vivo experimental setups of cells and tissues, such as cancer spheroids, neural crest cell migration, and liver lobules. These applications, which simulate thousands to millions of cells, require robust and efficient numerical methods. At their core, these models necessitate the solution of a large friction-dominated equation of motion, resulting in a sparse, symmetric, and positive definite matrix equation. The conjugate gradient method is employed to solve this problem, but this requires a good preconditioner for optimal performance. In this study, we develop a graph-based preconditioning strategy that can be easily implemented in such agent-based models. Our approach centers on extending support graph preconditioners to block-structured matrices. We prove asymptotic bounds on the condition number of these preconditioned friction matrices. We then benchmark the conjugate gradient method with our support graph preconditioners and compare its performance to other common preconditioning strategies.
{"title":"Support Graph Preconditioners for Off-Lattice Cell-Based Models","authors":"Justin Steinman, Andreas Buttenschön","doi":"10.1137/25m1727904","DOIUrl":"https://doi.org/10.1137/25m1727904","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 170-192, February 2026. <br/> Abstract. Off-lattice agent-based models (or cell-based models) of multicellular systems are increasingly used to create in-silico models of in-vitro and in-vivo experimental setups of cells and tissues, such as cancer spheroids, neural crest cell migration, and liver lobules. These applications, which simulate thousands to millions of cells, require robust and efficient numerical methods. At their core, these models necessitate the solution of a large friction-dominated equation of motion, resulting in a sparse, symmetric, and positive definite matrix equation. The conjugate gradient method is employed to solve this problem, but this requires a good preconditioner for optimal performance. In this study, we develop a graph-based preconditioning strategy that can be easily implemented in such agent-based models. Our approach centers on extending support graph preconditioners to block-structured matrices. We prove asymptotic bounds on the condition number of these preconditioned friction matrices. We then benchmark the conjugate gradient method with our support graph preconditioners and compare its performance to other common preconditioning strategies.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"289 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101992","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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