Linear functional strategy and the approximate inverse for nonlinear ill-posed problems. (English) Zbl 07723934
Summary: This article generalizes the results of the so-called linear functional strategy [R. S. Anderssen, Inverse Problems (Oberwolfach, 1986)], used for fast reconstruction of some particular feature of interest in the solution of a linear inverse problem. Two versions are proposed for nonlinear problems. The first one applies to differentiable forward operators and is based on the One-Step Newton method. The second one, in turn, uses a linearization of the forward operator obtained by the employment of basic Machine Learning techniques, being applicable to non-differentiable operators. As a byproduct of the proposed methods, we derive two variants of the so-called approximate inverse method [A. K. Louis, Inverse Problems, 1996] for nonlinear inverse problems. Numerical tests, using electrical impedance tomography applied to a biphasic flow problem, are presented to test the efficiency of the proposed methods.
MSC:
| 65Fxx | Numerical linear algebra |
| 65Hxx | Nonlinear algebraic or transcendental equations |
| 65Rxx | Numerical methods for integral equations, integral transforms |
Keywords:
approximate inverse; biphasic flow; electrical impedance tomography; linear functional strategy; machine learning; nonlinear inverse problemsSoftware:
KAIRUAINReferences:
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