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IRTG Modern Inverse Problems (MIP)

Kategorie: ‘Videos’

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

Charlemagne Distinguished Lecture Series Video 2020 Part 2

12. November 2020 | von

Prof. George Em Karniadakis, Ph.D. – Physics-Informed Neural Networks (PINNs): An Alphabet of Algorithms for Diverse Applications

Twice a year, the AICES & IRTG-2379 fellows are in charge of the organization of the prestigious Charlemagne Lecture, whose objective is to invite persons, who have achieved impressive accomplishments throughout their career and, in this sense, to get inspired by their scientific achievements as well as experience. As the 19th speaker of the Charlemagne Distinguished Lecture Series, the IRTG & AICES fellows invited Prof. George Em Karniadakis, Ph.D.,Professor of Applied Mathematics, of the The Charles Pitts Robinson and John Palmer Barstow, Brown University, USA. Professor Karniadakis gave a talk on „Physics-Informed Neural Networks (PINNs): An Alphabet of Algorithms for Diverse Applications“ on October 19, 2020 at 3:00 pm.

EU Regional School Videos 2020 Part 2

04. August 2020 | von

Course 6 – Prof. Leszek Demkowicz, Ph.D. – The Discontinuous Petrov-Galerkin (DPG) Method (with Optimal Test Functions)

The 3h short course will cover the fundamentals of the DPG method:
1. Three hats of the ideal and practical Petrov-Galerkin method with
    optimal test functions.
2. Breaking forms and test spaces.
3. Implementation aspects and examples of computations.
The presentation is based on the chapter in [1]
[1] L. Demkowicz, „Lecture Notes on Mathematical Theory of Finite Elements“,
    Oden Institute Report 2020/11, May 2020.

Course 8 – Prof. Dr. Felix Krahmer – Structure and Randomness in Data Science

The goal of this short course is to convince you that a smart combination of structure and randomness can be very useful for many data science applications, including sensing and data acquisition, dimension reduction, and computational tasks. Randomness can help, as random parameter choices often lead to good conditioning with high probability due to concentration phenomena, while structure arises either as imposed by applications, or to make the computations more feasible. Here the role of randomness can be twofold. On the one hand, randomness can help establish deterministic properties which are too complex in nature to be understood for any deterministic constructions. For example, a sufficient condition for guaranteed recovery of sparse signals in compressive sensing is the so-called Restricted Isometry Property, that holds when the sensing matrix acts as an approximate isometry on sparse vectors. This property is known to hold for random matrices of embedding dimension  near-linear in the sparsity level with high probability, while no comparable deterministic construction is known to date. This way of thinking can also help in the analysis of stochastic partial differential equations.
In many applications related to computing, in contrast, randomness plays another essential role: A random preprocessing of the data, such as a random projection or a random subsampling, can allow to reduce high dimensional problems to lower dimensional problems that can be solved more efficiently. In most of these scenarios, comparable deterministic constructions are not feasible, as for any realization of the preprocessing operation, the procedure will fail for some data sets. The role of randomness in this context is that it translates a method that works for most instances – which may be useless as the actual data set may be one of the bad instances – into a method that for any data set works with high probability. Structure is important to ensure that the preprocessing remains efficient and its computational complexity does not exceed the one of the task to be performed. Examples of problems where this strategy is of use include nearest neighbour search and principal component analysis.
In the course, we will give an overview of instances of both these paradigms from various application areas and also present some key ideas of the underlying mathematical theory.

EU Regional School Videos 2020 Part 1

03. Februar 2020 | von

Course 1 – Prof. Dr. Siddhartha Mishra – Current Topics in Numerical methods for hyperbolic systems of conservation laws: uncertainty quantification, statistical solutions and machine learning

This short course is devoted to the topic of state and parameter estimation problems where the goal is to compute a fast reconstruction of the state and associated parameters of interest of a physical system from available measurement observations and the knowledge of a physical PDE model. Due to their ill-posedness, these problems are often addressed with Bayesian approaches. However, in view of their high numerical cost, especially in a high dimensional framework, strategies involving reduced models have recently been proposed as a vehicle to reduce complexity. In this talk, we present an overview of this alternative approach and illustrate its potential for several different applications: inverse problems in biological flows, pollutant estimation in urban areas and nuclear engineering applications.

Charlemagne Distinguished Lecture Series Videos 2020 Part 1

03. Februar 2020 | von

Prof. Anthony T. Patera, Ph.D. – Parametrized Partial Differential Equations: Mathematical Models, Computational Methods, and Applications

Twice a year, the IRTG & AICES fellows organize the prestigious Charlemagne Lecture, whose objective is to invite persons, who have achieved impressive accomplishments throughout their career and, in this sense, to get inspired by their scientific achievements. 
As the 18th speaker, the fellows invited Prof. Anthony T. Patera, Ph.D., Ford Professor of Engineering and Professor of Mechanical Engineering, of the Department of Mechanical Engineering, Massachusetts Institute of Technology, USA. Professor Patera gave a talk on „Parametrized Partial Differential Equations: Mathematical Models, Computational Methods, and Applications“ on January 31, 2020 at 3:00 pm at HKW Aachen.

EU Regional School Videos 2019 Part 1

24. Juni 2019 | von

Course 1 – Prof. Dr. Torsten Hoefler – MPI Remote Memory Access Programming and Scientific Benchmarking of Parallel Codes

We will provide an overview of advanced MPI programming techniques. Specifically, we will focus on MPI-3’s new Remote Memory Access (RMA) programming and an implementation thereof. We will discuss how to utilize MPI-3 RMA in modern applications. Furthermore, we will discuss issues in large-scale implementation and deployment. The lecture will then continue to a small number of other advanced MPI usage scenarios that every scientific computing researcher should know. Finally, we will discuss how to benchmark parallel applications in a scientifically rigorous way. This turns out to be surprisingly difficult and the state of the art is suboptimal. We will present twelve simple rules that can be used as guidelines for good scientific practice when it comes to measuring and reporting performance results.

Course 3 – Prof. Alessandro Reali, Ph.D. – Isogeometric Analysis: An Introduction and Some Recent Advances

Isogeometric Analysis (IGA) is a recent simulation framework, originally proposed by T.J.R. Hughes and coworkers in 2005, to bridge the gap between Computational Mechanics and Computer Aided Design (CAD). The basic IGA paradigm consists of adopting the same basis functions used for geometry representations in CAD systems – such as, e.g., Non-Uniform Rational B-Splines (NURBS) – for the approximation of field variables, in an isoparametric fashion. This leads to a cost-saving simplification of the typically expensive mesh generation and refinement processes required by standard finite element analysis. In addition, thanks to the high-regularity properties of its basis functions, IGA has shown a better accuracy per-degree-of-freedom and an enhanced robustness with respect to standard finite elements in a number of applications ranging from solids and structures to fluids, opening also the door to geometrically flexible discretizations of higher-order partial differential equations in primal form, as well as to highly efficient (strong-form) collocation methods.
The first part of this short course is devoted to the introduction of the basic concepts of IGA (including a brief primer on B-Splines and NURBS). The unique potential of IGA is then shown through some convincing applications, mainly belonging to the field of structural mechanics and of fluid-structure interaction, where the superior results that can be provided by IGA with respect to standard finite elements are clearly pointed out. 
The lecture is finally concluded by a brief presentation of further IGA works in progress and new ideas.

Charlemagne Distinguished Lecture Series Videos 2019 Part 1

10. April 2019 | von

Prof. Karen Willcox, Ph.D. – Projection-based Model Reduction: Formulations for Scientific Machine Learning

Twice a year, the AICES fellows organize the prestigious Charlemagne Lecture, whose objective is to invite persons, who have achieved impressive accomplishments throughout their career and, in this sense, to get inspired by their scientific achievements. 
Prof. Karen Willcox, Ph.D., Director of the Oden Institute for Computational Engineering and Sciences, Professor of Aerospace Engineering and Engineering Mechanics at Oden Institute for Computational Engineering and Sciences, the University of Texas at Austin, USA. Professor Willcox gave a talk on „Projection-based Model Reduction: Formulations for Scientific Machine Learning“ on April 30, 2019 at 4:30 pm.

Charlemagne Distinguished Lecture Series Videos

10. April 2019 | von

EU Regional School Videos 2019 Part 2

10. April 2019 | von

Course 5 – Dr. David Ham – Automated Simulation from Equations to Computation with Firedrake

Creating simulations by numerically solving PDEs often requires large amounts of complex low-level code which is hard to write, hard to debug, and hard to change. It doesn’t need to be like that! In this tutorial we’ll present the Firedrake automated finite element system. Firedrake users write finite element problems mathematically using the Unified Form Language (UFL) embedded in Python. High performance parallel operator and residual assembly is automatically generated using advanced compiler technology. Firedrake integrates with the PETSc framework to provide a full suite of sophisticated linear and nonlinear solvers. In this hands-on Jupyter-based tutorial, you will have the chance to solve linear and nonlinear PDEs using Firedrake and try out some of its advanced features.

Course 7 – Prof. Dariusz Uciński Ph.D. – Optimum Experimental Design for Distributed Parameter System Identification

The impossibility of observing the states of distributed parameter systems over the entire spatial domain raises the question of where to locate measurement sensors so as to estimate the unknown system parameters as accurately as possible. Both researchers and practitioners do not doubt that making use of sensors placed in an ‘intelligent’ manner may lead to dramatic gains in the achievable accuracy of the parameter estimates, so efficient sensor location strategies are highly desirable. In turn, the complexity of the sensor location problem implies that there are few sensor placement methods which are readily applicable to practical situations. What is more, they are not well known among researchers. The aim of the minicourse is to give account of both classical and recent original work on optimal sensor placement strategies for parameter identification in dynamic distributed systems modeled by partial differential equations. The reported work constitutes an attempt to meet the needs created by practical applications, especially regarding environmental processes, through the development of new techniques and algorithms or adopting methods which have been successful in akin fields of optimal control and optimum experimental design. While planning, real-valued functions of the Fisher information matrix of parameters are primarily employed as the performance indices to be minimized with respect to the sensor positions. Particular emphasis is placed on determining the ‘best’ way to guide moving and scanning sensors, and making the solutions independent of the parameters to be identified. A couple of case studies regarding the design of air quality monitoring networks are adopted as an illustration aiming at showing the strength of the proposed approach in studying practical problems. The course will be complemented by a discussion of more advanced topics including the related problem of optimum input design and the Bayesian approach to deal with the ill-posedness of parameter estimation.

EU Regional School Videos

10. April 2019 | von