IRTG Modern Inverse Problems (MIP)

EU Regional School Videos 2021 Part 1

Course 1 – Prof. Irene Gamba, Ph.D. – Non-linear Boltzmann type models in Collisional Theory

These introductory lectures will focus on a unified approach to analyze the main properties of particle interactions of binary nature and their connections to statistical physics and thermodynamics.

We will focus on the model structure and common features, whether they corresponds to the classical elastic or inelastic monoatomic gases, or the more complex dynamics of polyatomic gases as well as gas mixtures with disparate masses.

We focus in the framework that naturally arises when modeling particle density functions subject to gain and loss rates when such particle interact in a binary law  as well as the transition probability rates (or scattering mechanisms) as a natural stabilization mechanism that enable us to construct solutions, to study their uniqueness properties and their long time behavior in suitable functional (Banach) spaces associated to the description of the evolution of  particle probability densities, to the shape of the probability density tails referred by polynomial or exponential moments and Fourier transform structure.

Of special significance is the understanding of the role of scattering mechanisms in the analytical properties of such solutions to be able to raise from the  Banach space  structure into the Sobolev (or Hilbert) structures. This observation plays a fundamental role in their approximations by finitely generated functions supported in bounded domains, the numerical approximations as well as error estimates.  Towards the end, we will focus of the special task to understand the grazing collision limit that gives raise to the Landau model.

The lecture will be supported by recent numerical implementation by  the presentation of hybrid computational solvers of two kinds: conservative spectral based and Finite Element Method (FEM) based for the Boltzmann equation binary interactions.

The functional analysis tools enable  proofs of consistency schemes,  construction of error estimates and convergence to statistical equilibrium results, both, in the  conservative spectral method as well as  on  Petrov-Galerkin FEM approach.

EU Regional School Video


Course 2 – Prof. Dr. Andreas Fichtner – Probabilistic Full-Waveform Inversion

In the course of the past decade, full-waveform inversion has matured from a largely idealistic dream into a commonly applied method to image the internal structure of inaccessible bodies. Despite undeniable success, a major problem remains: The quantification of uncertainties in this often strongly nonlinear inverse problem.

In this lecture, I will present a series of computational approaches that brings probabilistic full-waveform inversion with complete uncertainty quantification within reach:

1) Hamiltonian Monte Carlo sampling of the posterior probability density treats model parameters as particles that orbit through model space, obeying Hamilton’s equations from classical mechanics. The scaling properties of Hamiltonian Monte Carlo allow us to consider high-dimensional model spaces that often cannot be considered with more traditional, derivative-free sampling methods.

2) Autotuning based on limited-memory quasi-Newton methods provides nearly optimal mass matrices for Hamiltonian Monte Carlo, thereby largely removing laborious manual tuning. A factorised version of the L-BFGS algorithm, in particular, can increase the effective sample size by more than an order of magnitude.

3) Wavefield-adapted spectral-element meshes exploit prior knowledge on the geometry of wavefields. Such prior knowledge is frequently available for media that are smooth relative to the minimum wavelength. Wavefield-adapted meshes have the potential to drastically reduce the number of elements, leading to a computational forward modelling cost that makes Monte Carlo sampling possible.

EU Regional School Video


Course 3 – Steve Lionel – Modern Fortran: Features for High-Performance Computing

Fortran may be more than sixty years old, but it has grown and changed a lot over that time. Steve Lionel, former Fortran compiler developer and current head of the ISO Fortran standards committee (ISO/IEC JTC1/SC22/WG5) will give an overview of features added to the Fortran language since FORTRAN-77, with an emphasis on those that promote programmer productivity and high-performance computing. There will be time available for questions.

EU Regional School Video

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