Workshop Dynamics & NumericsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

\(Numerics: Vanja Nikolic (30 Min)\)

Title: Analytical and numerical aspects of nonlinear acoustic wave propagation

Abstract: The need to analyze and accurately simulate nonlinear sound propagation has increased with the rise in the number of ultrasound applications in medicine and industry. In this talk, I will present some of my recent work on the well-posedness and numerical simulation of partial differential equations that model nonlinear sound propagation. In addition, I will briefly discuss the treatment of shape optimization problems that arise in the practical use of high-intensity focused ultrasound.

\(Dynamics: Maxime Breden (30 Min)\)

Title: An introduction to a posteriori validation techniques, illustrated on the study of minimum energy paths.

Abstract: To understand the global behavior of a nonlinear system, the first step is to study its invariant set. Indeed, specific solutions like steady states, periodic orbits and connections between them are building blocks that organize the global dynamics. While there are many deep, general and theoretical mathematical results about the existence of such solutions, it is often difficult to apply them to a specific example. Besides, when dealing with a precise application, it is not only the existence of these solutions, but also their qualitative properties that are of interest. In that case, a powerful and widely used tool is numerical simulations, which is well adapted to the study of an explicit system and can provide invaluable insight for problems where the nonlinearities hinder the use of purely analytical techniques. The aim of a posteriori validation techniques is to obtain mathematically rigorous and quantitative existence theorems, using those numerical simulations. Given an approximate solution, the general strategy is to combine a posteriori estimates with analytical ones to apply a fixed point theorem, which then yields the existence of a true solution in an explicit neighborhood of the numerical one. In the first part of the talk, I'll present the main ideas of a posteriori validation in more detail, and describe the general framework in which they are applicable. In the second part, I'll then focus on a specific example and explain how to validate minimum energy paths for stochastic differential equations.

\(Numerics: Elisabeth Ullmann (30 Min)\)

Title: Multilevel Sequential^2 Monte Carlo for Bayesian Inverse Problems

Abstract: The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the likelihood, and the resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed by P.S. Koutsourelakis (J. Comput. Phys. 228, 2009, pp. 6184-6211) where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive, and can also be generalized to time-dependent problems.

\(Dynamics: Christian Kühn (30 Min)\)

Title: Numerical Continuation of Ellipsoids for Stochastic Problems

Abstract: In this talk, I shall explain a method, how to analyze certain aspects of stochastic dynamical systems from a purely deterministic, discrete, and geometric perspective. In particular, we study fluctuations around steady states in stochastic differential equations using ellipsoids calculated via Lyapunov matrix equations. The method will be embedded in a numerical continuation framework to effectively study parametrized problems. I am also going to briefly mention rigorous error estimates for the numerics and several applications.