Partial differential equations are a versatile tool to model and
eventually simulate physical phenomena. An important aspect in view
of the reliability and relevance of such simulations are uncertainties
arising from unknown parameters and measurement errors. In particular,
the modelling and discretization of uncertainties of the computational
domain requires special care. Such uncertainties emerge in a natural
fashion when considering products fabricated by line production which
are subject to manufacturing tolerances or shapes which are obtained by
remote sensing techniques, like e.g. ultrasound or magnetic resonance imaging.
This minisymposium is dedicated to recent developments in the numerical
treatment of shape uncertainties in partial differential equations
and welcomes contributions addressing analytical aspects,
forward modelling, assimilation of measurement data,
optimization, and applications.
08:30
Linear parabolic PDEs in uncertain non-cylindrical domains
Ana Djurdjevac | TU Berlin | Germany
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Authors:
Ana Djurdjevac | TU Berlin | Germany
Lewis Church | University of Warwick | United Kingdom
Charles M. Elliott | University of Warwick | United Kingdom
The uncertainty in the mathematical modeling appears for various reasons, in particular it comes from the incomplete knowledge of the given data. We are especially interested when the uncertainty comes from the geometric aspect, i.e., the domain. We study linear parabolic PDEs that are posed on a domain which evolves by a given random velocity. As a result, we have a PDE on a random non-cylindrical domain. As concrete examples we consider a heat equation and a parabolic Stokes equation on a random moving domain. Utilizing the domain mapping method, we transfer the problem into a PDE with random coefficients on a fixed domain. We analyze the case of uniformly bounded and log-normal type of the transformation of the domain. From the application point of view (biological modeling) it is of interest to formulate these PDEs on curved domains. We comment the extension of the presented results in this direction. For numerical analysis, we consider surface FEM coupled with Monte Carlo sampling. Our theoretical convergence rates are confirmed by numerical experiments.
This is a joint work with L. Church, C. Elliott, C. Gräser, P. Herbert. This work is supported by DFG through project AA1-3 of MATH + .
09:00
- CANCELED - Stokes flow in a channel with randomly rough walls
Daniel Tartakovsky | Stanford University | United States
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Author:
Daniel Tartakovsky | Stanford University | United States
Surface roughness is a key property affecting fluid flow in bounded domains. Its effect on bulk flow is usually quantified by means of an empirical roughness coefficient which is introduced into models that treat bounding surfaces as smooth. We present a new approach, which treats the irregular geometry of rough walls as a random field, whose statistical properties (mean, standard deviation, and spatial correlation) are inferred from measurements. The subsequent stochastic mapping of a random flow domain onto its deterministic counterpart and stochastic homogenization of the transformed Stokes equations yield an expression for the roughness coefficient in terms of the wall's statistical parameters. The analytical nature of our solutions allows us to handle random surfaces with short correlations lengths, which cannot be treated by numerical stochastic simulations.
09:30
Higher Order Quasi-Monte Carlo for the Computation of Far Field Statistics in Acoustic Wave Scattering by Uncertain Penetrable Domains
Fernando Henriquez | ETH Zurich | Switzerland
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Authors:
Fernando Henriquez | ETH Zurich | Switzerland
Christoph Schwab | ETH Zurich | Switzerland
We consider the Helmholtz transmission problem with a collection of connected, disjoint and penetrable obstacles of uncertain geometry in two spatial dimensions. Our objective is to compute the expected value of the far-field pattern produced by the previously described configuration over all possible shapes. To this end, we firstly recast the aforementioned problem by means of boundary integral operators and obtain an equivalent well-posed boundary integral equation (BIE). After introducing a suitable parametric (possibly high-dimensional) description of the involved boundaries, we obtain a parametric family of BIEs in a reference domain. Due to the intrinsic high-dimensional nature of this problem, the numerical approximation in the parameter space of the solution to these BIEs, and that of the far-field pattern, becomes computationally challenging and may suffer from the the so-called curse of dimensionality in the parametric dimension. To circumvent this issue, in our numerical scheme, we use recently developed higher-order Quasi-Monte Carlo integration (HoQMC) techniques for the numerical approximation of the far-field pattern's expected value. In this talk, we establish parametric regularity of the far-field pattern and provide a rigorous analysis of its numerical approximation, including spatial discretization of the BIEs, dimension truncation effects, and approximation of the arising high-dimensional integral by means of HoQMC techniques.
10:00
Rapid computation of far-field statistics for random obstacle scattering
Helmut Harbrecht | University of Basel | Switzerland
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Author:
Helmut Harbrecht | University of Basel | Switzerland
In this talk, we consider the numerical approximation of far-field statistics for acoustic scattering problems in the case of random obstacles. In particular, we consider the computation of the expected far-field pattern and the expected scattered wave away from the scatterer as well as the computation of the corresponding variances. To that end, we introduce an artificial interface, which almost surely contains all realizations of the random scatterer. At this interface, we directly approximate the second order statistics, i.e., the expectation and the variance, of the Cauchy data by means of boundary integral equations. From these quantities, we are able to rapidly evaluate statistics of the scattered wave everywhere in the exterior domain, including the expectation and the variance of the far-field. By employing a low-rank approximation of the Cauchy data's two-point correlation function, we drastically reduce the cost of the computation of the scattered wave's variance. Numerical results are provided in order to demonstrate the feasibility of the proposed approach.