We introduce a family of implicit integration methods for the dynamical low-rank approximation: the alternating-implicit dynamically orthogonal Runge-Kutta (ai-DORK) schemes. Explicit integration often requires restrictively small time steps and has stability issues; our implicit schemes eliminate these concerns in the low-rank setting. We incorporate our alternating iterative low-rank linear solver into high-order Runge-Kutta methods, creating accurate and stable schemes for a variety of previously intractable problems including stiff systems. Fully implicit and implicit-explicit (IMEX) ai-DORK are derived, and we perform a stability analysis on both. The schemes may be made rank-adaptative and can handle ill-conditioned systems. To evaluate nonlinearities effectively, we propose a local\/piecewise polynomial approximation with adaptive clustering, and on-the-fly reclustering may be performed efficiently in the coefficient space. We demonstrate the ai-DORK schemes and our local nonlinear approximation technique on an ill-conditioned matrix differential equation, a stiff, two-dimensional viscous Burgers’ equation, the nonlinear, stochastic ray equations, the nonlinear, stochastic Hamilton-Jacobi-Bellman PDE for time-optimal path planning, and the parabolic wave equation with low-rank domain decomposition in Massachusetts Bay.<\/p>\n","protected":false},"excerpt":{"rendered":"
We introduce a family of implicit integration methods for the dynamical low-rank approximation: the alternating-implicit dynamically orthogonal Runge-Kutta (ai-DORK) schemes. Explicit integration often requires restrictively small time steps and has stability issues; our implicit schemes eliminate these concerns in the low-rank setting. We incorporate our alternating iterative low-rank linear solver into high-order Runge-Kutta methods, creating […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[243],"tags":[],"class_list":["post-7047","post","type-post","status-publish","format-standard","hentry","category-presentations"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/7047","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7047"}],"version-history":[{"count":1,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/7047\/revisions"}],"predecessor-version":[{"id":7048,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/7047\/revisions\/7048"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7047"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7047"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7047"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}