09:50
Hierarchical tensors approximation for feedback optimal control and Hamilton Jacobi Bellmann equations
Reinhold Schneider | TU Berlin | Germany
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Authors:
M. Sc. Mathias Oster | Germany
Leon Sallandt | Germany
Reinhold Schneider | TU Berlin | Germany
Hierarchical (Tucker) tensor format (HT - tensors ) and Tensor Trains (TT- tensors, I.Oseledets) have been introduced recently for low rank tensor product approximation. Hierarchical tensor decompositions are based on sub space approximation by extending the Tucker decomposition into a multi-level framework. This approach extend the reduced basis model order reduction to high dimensional problems. For closed loop optimal control problems, this approach offers a novel tool to circumvent the curse of dimensionality.
We propose a (variational) Monte Carlo for approximating the solution of Hamilton Bellmann equations via the Bellmann principle and dynamical programming to approximate the value function by means of hierarchical tensor (HT) approximation. We will consider a deterministic setting as well as a stochastic setting, but focusing more on the stochastic case. Monte Carlo method using the Koopman operator is used to solve an inhomogenous backward Kolmogorov equation. In principle this approach can be carried out also with kernel approximations or multi-layer deep neural networks, and has closed link to reinforcement learning. We apply this approach to bi-linear optimal control problems, e.g considered by K. Kunisch and co-workers. In particular, the present approach approach contains the case of linear quadratic control exactly. It is our first attempt to apply tensor approximation to HJB.
10:10
Efficient solution of optimization problems with BV-functions in 1D
M. Sc. Daniel Walter | RICAM, Austrian Academy of Sciences | Austria
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Author:
M. Sc. Daniel Walter | RICAM, Austrian Academy of Sciences | Austria
In this talk, the efficient solution of minimization problems over the space of functions with bounded variation on an interval is considered. These are challenging problems due to several reasons. First, the functional which is to be minimized consists of the sum of a smooth term and the non-smooth BV-seminorm as a regularizer. Second, the space of BV-functions lacks properties such as reflexivity and smoothness which are desirable for the analysis of solution algorithms. Clearly, the latter complications may be overcome by discretization of the problem. Essentially, a solution can then be obtained by any method which can cope with finite dimensional composite minimization problems. However, this approach harbours the danger of yielding algorithms whose convergence behaviour critically depends on the the fineneness of the discretization. Therefore, we are interested in stating solution procedures for the continuous problem on the function space level. Adapting such methods to discretized problems then often leads to mesh-independent methods. We present a particular approach exploiting the identification of a BV-function on the interval with a Borel measure and a constant. Convergence statements are derived. The talk is completed by numerical examples which illustrate the theoretical results.
10:30
An inexact bundle algorithm for nonconvex nondifferentiable minimization in Hilbert space
M. Sc. Lukas Hertlein | Technische Universität München | Germany
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Authors:
M. Sc. Lukas Hertlein | Technische Universität München | Germany
Prof. Dr. Michael Ulbrich | Technische Universität München | Germany
Motivated by optimal control problems for elliptic variational inequalities we develop an inexact bundle method for nonsmooth nonconvex optimization subject to general convex constraints. The proposed method requires only approximate (i.e., inexact) evaluations of the cost function and of an element of Clarke’s subdifferential. The algorithm allows for incorporating curvature information while aggregation techniques ensure that an approximate solution of the piecewise quadratic subproblem can be obtained efficiently. A global convergence theory in a suitable infinite-dimensional Hilbert space setting is presented and a posteriori error estimates for the cost function and the solution of the subproblem are developed. We discuss the application of our framework to optimal control of the obstacle problem and present numerical results.