# Berlin Oberseminar: Optimization, Control and Inverse Problems

## Scope

This seminar serves as a knowledge exchange and networking platform for the broad area of mathematical optimization and related applications within Berlin.

• Place: Temporarily: Zoom platform
Place: Weierstrass Institute for Applied Analysis and Stochastics
Place: Mohrenstraße 39, 10117 Berlin
Place: Erhard Schmidt Lecture Room (ESH)
• Time: Mondays (biweekly), 15:00PM - 16:00PM
• Organizers: René Henrion (WIAS)
Organizers: Michael Hintermüller (WIAS, HU Berlin)
Organizers: Dietmar Hömberg (WIAS, TU Berlin)
Organizers: Kostas Papafitsoros (WIAS)
Organizers: Gabriele Steidl (TU Berlin)
Organizers: Andrea Walther (HU Berlin)
2021

TBA TBA

### Previous Talks

05.07.2021 Patrick Farrell (University of Oxford, UK)
15:00-16:00

Computing disconnected bifurcation diagrams of partial differential equations

Computing the distinct solutions $u$ of an equation $f(u, \lambda) = 0$ as a parameter $\lambda \in \mathbb{R}$ is varied is a central task in applied mathematics and engineering. The solutions are captured in a bifurcation diagram, plotting (some functional of) $u$ as a function of $\lambda$. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is very simple: it only requires a minor modification to an existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available; no auxiliary problems must be solved.

We will present applications to hyperelastic structures, liquid crystals, and Bose-Einstein condensates, among others.

14.06.2021 Ozan Öktem (KTH, Sweden)
15:00-16:00

Data driven large-scale convex optimisation

This joint work with Jevgenjia Rudzusika (KTH), Sebastian Banert (Lund University) and Jonas Adler (DeepMind) introduces a framework for using deep-learning to accelerate optimisation solvers with convergence guarantees. The approach builds on ideas from the analysis of accelerated forward-backward schemes, like FISTA. Instead of the classical approach of proving convergence for a choice of parameters, such as a step-size, we show convergence whenever the update is chosen in a specific set. Rather than picking a point in this set through a handcrafted method, we train a deep neural network to pick the best update. The method is applicable to several smooth and non-smooth convex optimisation problems and it outperforms established accelerated solvers.

03.05.2021 Lars Ruthotto (Emory University, USA)
This talk was also part of the SPP 1962 Priority Program 2021 Keynote Presentation series.
15:00-16:00

A Machine Learning Framework for Mean Field Games and Optimal Control

We consider the numerical solution of mean field games and optimal control problems whose state space dimension is in the tens or hundreds. In this setting, most existing numerical solvers are affected by the curse of dimensionality (CoD). To mitigate the CoD, we present a machine learning framework that combines the approximation power of neural networks with the scalability of Lagrangian PDE solvers. Specifically, we parameterize the value function with a neural network and train its weights using the objective function with additional penalties that enforce the Hamilton Jacobi Bellman equations. A key benefit of this approach is that no training data is needed, e.g., no numerical solutions to the problem need to be computed before training. We illustrate our approach and its efficacy using numerical experiments. To show the framework's generality, we consider applications such as optimal transport, deep generative modeling, mean field games for crowd motion, and multi-agent optimal control.

29.03.2021 Serge Gratton (ENSEEIHT, Toulouse, France)
15:00-16:00

On a multilevel Levenberg-Marquardt method for the training of artificial neural networks and its application to the solution of partial differential equations

We propose a new multilevel Levenberg-Marquardt optimizer for the training of artificial neural networks with quadratic loss function. When the least-squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by any geometrical constraints and the standard interpolation and restriction operators cannot be employed any longer. A heuristic, inspired by algebraic multigrid methods, is then proposed to construct the multilevel transfer operators. We test the new optimizer on an important application: the approximate solution of partial differential equations by means of artificial neural networks. The learning problem is formulated as a least squares problem, choosing the nonlinear residual of the equation as a loss function, whereas the multilevel method is employed as a training method. Numerical experiments show encouraging results related to the efficiency of the new multilevel optimization method compared to the corresponding one-level procedure in this context.

Member of the Leibniz Association
Part of the Forschungsverbund Berlin e.V.
Seat of the Secretariat of the International Mathematical Union
Seat of the office of the Deutsche Mathematiker-Vereinigung
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