Parametric level-set inverse problems with stochastic background estimation. (English) Zbl 07691092
Summary: We study parametric shape reconstruction inverse problems in which the object of interest is embedded in a heterogeneous background medium that is known only approximately. We model the background medium as a Gaussian random field and pose shape reconstruction as a stochastic programming problem in which we seek to minimize the expected value, with respect to the background field, of a stochastic objective function. We develop a computationally efficient algorithm based on the sample average approximation that reduces the effect of uncertainty in the background medium on shape recovery. We demonstrate that by using accelerated stochastic gradient descent, we can apply our method to large-scale problems. The capabilities of our method are demonstrated on a simple two-dimensional model problem and in a more demanding application to a three-dimensional inverse conductivity problem in geophysical imaging.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 90Cxx | Mathematical programming |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
Keywords:
parametric level-set methods; stochastic optimization; geophysical inversion; sample average approximation; shape reconstructionReferences:
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