Optimization with learning-informed differential equation constraints and its applications

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided

A combined shape Newton and topology optimization technique in real time image segmentation

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Fast level-set based algorithms using shape and topological sensitivity information

A framework for descent algorithms using shape as well as topological sensitivity information is introduced. The concept of gradient-related descent velocities in shape optimization is defined, a corresponding algorithmic approach is developed, and a convergence analysis is provided. It is shown that for a particular choice of the bilinear form involved in the definition of gradient-related directions a shape Newton method can be obtain. The level set methodology is used for representing and updating the geometry during the iterations. In order to include topological changes in addition to merging and splitting of existing geometries, a descent algorithm based on topological sensitivity is proposed. The overall method utilizes the shape sensitivity and topological sensitivity based methods in a serial fashion. Finally, numerical results are presented

A primal-dual active set algorithm for bilaterally control constrained optimal control problems

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On a globalized augmented Lagrangian SQP-algorithm for nonlinear optimal control problems with box constraints

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A proximal bundle method based on approximate subgradients

In this paper a proximal bundle method is introduced that is capable to deal with approximate subgradients. No further knowledge of the approximation quality (like explicit knowledge or controllability of error bounds) is required for proving convergence. It is shown that every accumulation point of the sequence of iterates generated by the proposed algorithm is a well-defined approximate solution of the exact minimization problem. In the case of exact subgradients the algorithm behaves like well-established proximal bundle methods. Numerical tests emphasize the theoretical findings

Special issue to honour Guenter Leugering on His 65th birthday, guest edited by Michael Hintermueller, Michael Hinze, Jan Sokołowski and Stefan Ulbrich

International audienceSpecial issue, containing papers by leading specialists in control theory and optimisation, dedicated to Guenter Leugering. It is continued as the next, also special, issue (2/2019

A Class of Second-Order Geometric Quasilinear Hyperbolic PDEs and Their Application in Imaging

Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems associated to gradient flows for energy decaying. In numerical computations, it turns out that the second-order methods are superior to their first-order counter-parts. We concentrate on (i) a damped second-order total variation flow for, e.g., image denoising and (ii) a damped second-order mean curvature flow for level sets of scalar functions. The latter is connected to a nonconvex variational model capable of correcting displacement errors in image data (e.g., dejittering). For the former equation, we prove the existence and uniqueness of the solution and its long time behavior and provide an analytical solution given some simple initial datum. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces and show the existence and uniqueness of the solution for a regularized version of the equation. Finally, some numerical comparisons of the solution behavior for the new equations with first-order flows are presented

Special issue to honour Guenter Leugering on His 65th birthday, guest edited by Michael Hintermueller, Michael Hinze, Jan Sokołowski and Stefan Ulbrich

International audienceSpecial issue, containing papers by leading specialists in control theory and optimisation, dedicated to Guenter Leugering. It is continued as the next, also special, issue (2/2019