Solving Implicit Inverse Problems with Homotopy-Based Regularization Path
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912394967515136 |
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| author | Parodi, Davide Benvenuto, Federico Garbarino, Sara Piana, Michele |
| author_facet | Parodi, Davide Benvenuto, Federico Garbarino, Sara Piana, Michele |
| contents | Implicit inverse problems, in which noisy observations of a physical quantity are used to infer a nonlinear functional applied to an associated function, are inherently ill posed and often exhibit non uniqueness of solutions. Such problems arise in a range of domains, including the identification of systems governed by Ordinary and Partial Differential Equations (ODEs/PDEs), optimal control, and data assimilation. Their solution is complicated by the nonlinear nature of the underlying constraints and the instability introduced by noise. In this paper, we propose a homotopy based optimization method for solving such problems. Beginning with a regularized constrained formulation that includes a sparsity promoting regularization term, we employ a gradient based algorithm in which gradients with respect to the model parameters are efficiently computed using the adjoint state method. Nonlinear constraints are handled through a Newton Raphson procedure. By solving a sequence of problems with decreasing regularization, we trace a solution path that improves stability and enables the exploration of multiple candidate solutions. The method is applied to the latent dynamics discovery problem in simulation, highlighting performance as a function of ground truth sparsity and semi convergence behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solving Implicit Inverse Problems with Homotopy-Based Regularization Path Parodi, Davide Benvenuto, Federico Garbarino, Sara Piana, Michele Numerical Analysis Optimization and Control Implicit inverse problems, in which noisy observations of a physical quantity are used to infer a nonlinear functional applied to an associated function, are inherently ill posed and often exhibit non uniqueness of solutions. Such problems arise in a range of domains, including the identification of systems governed by Ordinary and Partial Differential Equations (ODEs/PDEs), optimal control, and data assimilation. Their solution is complicated by the nonlinear nature of the underlying constraints and the instability introduced by noise. In this paper, we propose a homotopy based optimization method for solving such problems. Beginning with a regularized constrained formulation that includes a sparsity promoting regularization term, we employ a gradient based algorithm in which gradients with respect to the model parameters are efficiently computed using the adjoint state method. Nonlinear constraints are handled through a Newton Raphson procedure. By solving a sequence of problems with decreasing regularization, we trace a solution path that improves stability and enables the exploration of multiple candidate solutions. The method is applied to the latent dynamics discovery problem in simulation, highlighting performance as a function of ground truth sparsity and semi convergence behavior. |
| title | Solving Implicit Inverse Problems with Homotopy-Based Regularization Path |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2505.19608 |