A Reduced Basis Landweber method for nonlinear inverse problems
For researchers solving inverse problems with expensive PDE forward solvers, this method reduces computational cost while handling high-dimensional parameters.
The paper tackles nonlinear ill-posed inverse problems in parametrized PDEs by coupling the Landweber method with reduced basis model order reduction. The new method handles high-dimensional parameter spaces and reduces computational time, demonstrated on reconstructing conductivity in the heat equation.
We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require numerous amounts of forward solutions during the solution process. In this article we consider the nonlinear Landweber method and want to couple it with the reduced basis method as a model order reduction technique in order to reduce the overall computational time. In particular, we consider PDEs with a high-dimensional parameter space, which are known to pose difficulties in the context of reduced basis methods. We present a new method that is able to handle such high-dimensional parameter spaces by combining the nonlinear Landweber method with adaptive online reduced basis updates. It is then applied to the inverse problem of reconstructing the conductivity in the stationary heat equation.