Related papers: A Reduced Basis Landweber method for nonlinear inv…
We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state…
The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish…
In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse problems arising from parameter identification in elliptic partial differential equations. Such…
We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized…
Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order models of parametrized partial differential equation problems. With speedups that can reach several orders of…
We introduce and analyze a fast iterative method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. The key idea, in contrast to the standard Landweber method, is to use multiple search directions per…
We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR),…
The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the…
Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…
We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial…
The aim of this paper is to investigate the use of an entropic projection method for the iterative regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
These lecture notes summarize various summer schools that I have given on the topic of solving inverse problems (state and parameter estimation) by combining optimally measurement observations and parametrized PDE models. After defining a…
We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE…
In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear…
In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how…
In this work, we propose a model order reduction framework to deal with inverse problems in a non-intrusive setting. Inverse problems, especially in a partial differential equation context, require a huge computational load due to the…
The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE…
We present novel model reduction methods for rapid solution of parametrized nonlinear partial differential equations (PDEs) in real-time or many-query contexts. Our approach combines reduced basis (RB) space for rapidly convergent…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…