{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T13:19:29Z","timestamp":1789996769751,"version":"4.0.1"},"reference-count":39,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01-CA 154774"],"award-info":[{"award-number":["R01-CA 154774"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["0645347"],"award-info":[{"award-number":["0645347"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1025327"],"award-info":[{"award-number":["1025327"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1217156"],"award-info":[{"award-number":["1217156"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1217161"],"award-info":[{"award-number":["1217161"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In nonlinear imaging problems whose forward model is described by a partial differential equation (PDE), the main computational bottleneck in solving the inverse problem is the need to solve many large-scale discretized PDEs at each step of the optimization process. In the context of absorption imaging in diffuse optical tomography, de Sturler et al., [SIAM J. Sci. Compute., 37 (2015), pp. B495--B517] addres this bottleneck by using parametrized reduced order models, since the forward problem, typically posed in the frequency domain, is essentially the transfer function of a high-dimensional differential algebraic system. Although this approach drastically reduces the number of large linear systems to be solved, the construction of a candidate global basis for parametrized reduced models still requires the solution of many full order problems, the discretized PDE for multiple right-hand sides, and multiple parameter vectors. This step is followed by a rank-revealing factorization to compress the candidate global basis, in order to produce a much smaller number of vectors suitable for efficient construction of reduced transfer functions. These two steps are still expensive. First, a reduced model must be computed for every reconstruction of a distinct anomaly (say, for every patient), in contrast to applications where a reduced model is computed once and used forever. Moreover, the computation of the reduced model must be efficient on commonly used hardware. The present paper addresses the costs associated with the global basis approximation in two ways. First, we use the structure of the matrix to rewrite the full order transfer function and corresponding derivatives, such that the full order systems to be solved are symmetric, and positive definite in the zero frequency case. We apply model order reduction to the new formulation of the problem. Second, we give an approach to computing the global basis approximation directly as the full order systems are solved. This new approach (1) avoids the expensive rank-revealing factorization and (2) substantially reduces both the number of large linear systems to be solved and the number of iterations for each solve, leading to huge savings. Only the incrementally new, relevant information is solved for and added to the existing global basis, avoiding the computation of redundant information. This new approach is achieved by an inner-outer Krylov recycling approach, which has wider potential use in other applications and in contexts beyond reduced order modeling. We show the value of the new approach to approximate global basis computation on two diffuse optical tomography absorption image reconstruction problems.<\/jats:p>","DOI":"10.1137\/16m1062880","type":"journal-article","created":{"date-parts":[[2017,3,16]],"date-time":"2017-03-16T11:28:53Z","timestamp":1489663733000},"page":"B272-B297","source":"Crossref","is-referenced-by-count":10,"title":["Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography"],"prefix":"10.1137","volume":"39","author":[{"given":"Meghan","family":"O'Connell","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Misha E.","family":"Kilmer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"de Sturler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Serkan","family":"Gugercin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,3,16]]},"reference":[{"key":"atypb1","volume-title":"Proc. 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O'Connell,\n                      Advanced Techniques in the Computation of Reduced Order Models and Krylov Recycling for Diffuse Optical Tomography\n                      , Ph.D. thesis, Tufts University, 2016."},{"key":"atypb31","unstructured":"M. L. Parks, K. M. Soodhalter, and D. B. Szyld,\n                      A Block Recycled GMRES Method With Investigations into Aspects of Solver Performance\n                      , Technical Report, Center for Computing Research, Sandia National Laboratories, Albuquerque, NM;arXiv:1604.01713v1."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/120902690"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/140990073"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/180\/1\/012073"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971408"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717570"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1798"},{"key":"atypb38","unstructured":"R. Yu, E. de Sturler, and D. D. 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