{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/lap-a-linearize-and-project-method-for","title":"LAP: a Linearize and Project Method for Solving Inverse Problems with Coupled Variables","arxiv_id":"1705.09992","date":"2017-05-28","proceeding":null,"authors":["James Herring","James Nagy","Lars Ruthotto"],"abstract":"Many inverse problems involve two or more sets of variables that represent\ndifferent physical quantities but are tightly coupled with each other. For\nexample, image super-resolution requires joint estimation of the image and\nmotion parameters from noisy measurements. Exploiting this structure is key for\nefficiently solving these large-scale optimization problems, which are often\nill-conditioned.\n  In this paper, we present a new method called Linearize And Project (LAP)\nthat offers a flexible framework for solving inverse problems with coupled\nvariables. LAP is most promising for cases when the subproblem corresponding to\none of the variables is considerably easier to solve than the other. LAP is\nbased on a Gauss-Newton method, and thus after linearizing the residual, it\neliminates one block of variables through projection. Due to the linearization,\nthis block can be chosen freely. Further, LAP supports direct, iterative, and\nhybrid regularization as well as constraints. Therefore LAP is attractive,\ne.g., for ill-posed imaging problems. These traits differentiate LAP from\ncommon alternatives for this type of problem such as variable projection\n(VarPro) and block coordinate descent (BCD). Our numerical experiments compare\nthe performance of LAP to BCD and VarPro using three coupled problems whose\nforward operators are linear with respect to one block and nonlinear for the\nother set of variables.","url_abs":"http://arxiv.org/abs/1705.09992v3","url_pdf":"http://arxiv.org/pdf/1705.09992v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"lap-a-linearize-and-project-method-for","repo_url":"https://github.com/herrinj/LAP","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"image-super-resolution","task_name":"Image Super-Resolution"},{"task_slug":"super-resolution","task_name":"Super-Resolution"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}