[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85307-en":3,"doc-seo-85307-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85307,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Adaptive Krylov Methods for Low-Rank Exponential Integrators","Adaptive Krylov methods for stiff, time-dependent ordinary differential equations are developed using low-rank tensor representations. Exponential integrators integrate the stiff linear part exactly while approximating the nonlinear part via linear combinations of φ-functions. Classical KIOPS and RK2EXPINT evaluate these combinations through an augmented stiffness matrix and Krylov subspace exponentials, but they assume vector–matrix formulations. This work builds a general TT-format framework, introducing KIOPS-TT and RK2EXPINT-TT by tensorizing explicit exponential Runge–Kutta schemes and augmenting the stiffness tensor to compute φ-actions with a single exponential evaluation, preserving the matrix theory and enabling speed-ups in low-rank regimes.","arXiv :2607 . 11293v1 [math .NA] 13 Jul 2026  \nADAPTIVE KRYLOV METHODS FOR LOW-RANK EXPONENTIAL  \nINTEGRATORS  \nRICO WEIGEL∗ , TOM-CHRISTIAN RIEMER∗ , AND MARTIN STOLL∗  \nAbstract. Differential equations arise in numerous applications, particularly within scientific and technical contexts. Systems of stiff, time-dependent ordinary differential equations constitute the focus of this work. Exponential integrators are designed to solve such equations by integrating the linear part exactly, while simultaneously approximating the nonlinear part through a linear combination of φ-functions. By utilizing an augmented stiffness matrix, state-of-the-art methods like KIOPS and RK2EXPINT solve the linear part and evaluate linear combinations of φ-functions for the nonlinear part in a single step, effectively reducing the computational effort to a single matrix exponential evaluation. However, these classical approaches assume that the system is represented by matrices and vectors, potentially not utilizing the underlying high-dimensional structure. Tensors address this limitation and offer significant storage efficiency through well-established decompositions like the Tensor Train (TT) format. This work provides a general framework for solving stiff, time-dependent systems directly within the TT format. Specifically, KIOPS-TT and RK2EXPINT-TT are developed as extensions of the original KIOPS and RK2EXPINT algorithms. This involves reformulating the scheme of explicit exponential Runge-Kutta integrators for tensors and augmenting the stiffness tensor to compute linear combinations of φ-functions acting on tensors through a single evaluation of the exponential function using Krylov subspace methods. Furthermore, it is shown that the underlying theory of the matrix methods remains valid, thereby enabling the transfer of key theorems to the tensor case. Numerical experiments confirm significant speed-ups for KIOPS-TT and RK2EXPINT-TT in low-rank scenarios compared to their classical counterparts.  \nKey words. exponential integrators, low-rank, Krylov method, ODE, Tensor Train format MSC codes. 15A69, 65F55, 65F60, 65L04  \n1. Introduction. Differential equations arise in numerous applications and computing their solutions is often far from trivial. Traditional numerical methods typically represent systems of differential equations using matrix methods, whereby their solutions are expressed as vectors. Often these vector and matrix formulations do not entirely reflect the mathematical structure, especially in the higher-dimensional case. Furthermore, as all entries of a vector are typically stored explicitly, the computational effort increases exponentially for high-dimensional data. To preserve the mathematical structure while still reducing the storage requirements, low-rank tensor methods can be used, with well-established decompositions such as the Tensor Train (TT) format [22, 41] .  \nIn the context of using tensor methods for solving differential equations, methodological efforts center on two approaches that are also used in combination. The first approach attempts to integrate or update the individual components of a tensor decomposition, such as the TT cores in the case of the TT format, as separately as possible [13, 14, 31, 32, 36] . The second approach aims at addressing specific problems or problem structures [12, 16, 32, 35, 36, 38, 43, 57] . For instance, it is frequently assumed that the linear operator is representable as a Kronecker sum.  \nIn the case of stiff systems, the differential equation often consists of a stiff linear part, represented by the stiffness tensor matrix A, and a non-stiff semilinear part, represented by the tensor function g. Such problems arise, for example, in the discretization of semilinear parabolic differential equations on continuous domains [6] .  \n∗ Chair of Scientific Computing, Department of Mathematics, Technische Universit¨at Chemnitz, 09107 Chemnitz, Germany ([rico.weigel@mathematik.tu-chemn","cbCaiaOCAofvQgCg","https://ap.wps.com/l/cbCaiaOCAofvQgCg","pdf",1404503,3,1,24,"English","en",105,"# Introduction\n## Low-rank tensor representations for differential equations\n## Exponential integrators and φ-functions\n## Augmented matrix/tensor approach and Krylov methods\n## Motivation for TT-format exponential integrators","[{\"question\":\"What problem does this work address?\",\"answer\":\"It targets stiff, time-dependent ODE systems and efficient numerical solution using exponential integrators and low-rank tensor representations.\"},{\"question\":\"How do KIOPS and RK2EXPINT evaluate the nonlinear contribution?\",\"answer\":\"They reduce the cost by converting linear combinations of φ-functions into actions obtained from an exponential of an augmented (stiffness) matrix via Krylov subspace methods.\"},{\"question\":\"What new methods are proposed for the Tensor Train (TT) format?\",\"answer\":\"KIOPS-TT and RK2EXPINT-TT extend KIOPS and RK2EXPINT to solve systems directly in TT format by reformulating exponential Runge–Kutta tensor schemes and augmenting the stiffness tensor to compute φ-functions acting on tensors through a single exponential 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problem does this work address?","Question",{"text":75,"@type":76},"It targets stiff, time-dependent ODE systems and efficient numerical solution using exponential integrators and low-rank tensor representations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do KIOPS and RK2EXPINT evaluate the nonlinear contribution?",{"text":80,"@type":76},"They reduce the cost by converting linear combinations of φ-functions into actions obtained from an exponential of an augmented (stiffness) matrix via Krylov subspace methods.",{"name":82,"@type":73,"acceptedAnswer":83},"What new methods are proposed for the Tensor Train (TT) format?",{"text":84,"@type":76},"KIOPS-TT and RK2EXPINT-TT extend KIOPS and RK2EXPINT to solve systems directly in TT format by reformulating exponential Runge–Kutta tensor schemes and augmenting the stiffness tensor to compute φ-functions acting on tensors through a single exponential 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