Abstracts and slides of talks

• Alexey Chernov (Uni Oldenburg): Estimation of probability density functions by the Maximum Entropy method (Slides)
  Abstract: The Maximum Entropy method is a powerful tool to recover the distribution density function when only a finite series of generalised moments of a random variable are known or are estimated numerically, e.g. by inexact (multilevel) Monte Carlo simulation. We recall the two-stage simulation procedure from [1], discuss the error analysis and test the method in a set of numerical experiments.
  [1] C. Bierig and A. Chernov, Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method, J. Comput. Physics 314 (2016), 661–681 (preprint)




• Paul Dupuis (Brown University): Methods for Model Approximation and Optimization in the Presence of Model Uncertainty Using Information Divergences (Slides)
  Abstract: Constructing a model that adequately describes complex phenomena forces one to evaluate the trade-offs between a simpler model which allows for calculations and/or computer simulations, and one of sufficient generality and complexity to capture all features thought to be present (e.g., multiple scales of space and time). Even within a definite class of models the determination of the parameters using data leads to another trade-off between the availability of data or the expense in its acquisition, the cost of training the parameters and the precision in the model. In such a situation one wants to know which parameters are important and quantify errors resulting from misspecification of these parameters. Challenges of this sort occur in many fields (e.g., material science and design, complex macroeconomics models, stochastic networks, machine learning). When a system is to be controlled or optimized, related questions center on how to optimize in the presence of unresolvable model uncertainty. One can formulate problems of this sort as quantifying the errors that result when a specific probabilistic 'nominal' model P (which we consider as the design model, or the computational model) is used in place of the 'true' model Q. In this talk we describe how these problems may be addressed using information divergences between Q and P, and in particular using variational formulas that give tight bounds for the difference between performance measures under Q (the true) and under P (the design) through a divergence. We identify the scaling properties needed for the bounds to be meaningful in various limits (e.g., steady state/large time, large system size), which are typical in many problems. The main example of such a variational formula is that which links exponential integrals, ordinary integrals, and Kullback-Leibler divergence (or relative entropy). We show how it can be used to solve various problems such as sensitivity bounds for steady state or ergodic performance measures (bounds on the sensitivity indicies), obtain tight bounds on performance measures for O(1) sized model errors, and address problems of optimization under model uncertainty. While the relative entropy variational formula has the most attractive qualitative properties and is perhaps the most useful overall, it is not well suited to every situation. If time permits we will discuss alternative divergences and distances and associated variational formulas, and in particular performance measures involving tail properties or rare events and Renyi divergence.




• Andreas Frommer (Uni Wuppertal): Multigrid methods for linear systems with stochastic entries arising in lattice QCD (Slides)
  Abstract: The numerically by fast most challenging part of a simulation in lattice QCD is to solve linear systems arising from a discretization of the Dirac operator. Up to now, the only discretization scheme in use basically is to take (covariant) finite differences on the 4-dimensional lattice. Since the Dirac operator by itself is of first order, one usually needs an additional stabilization term. Our talk will focus on the discrete Wilson-Dirac operator. While it preserves some important symmetries of the continuum theory, the main numerical challenge is to cope with the stochastic background field which appears as "gauge links" in the discretization. As a consequence, there is no natural hierarchy of the discretized operators in terms of just the discretization step size. We will motivate how multigrid techniques can still be used in this context, provided they are based on an adaptive, algebraic approach which exposes the right restriction operators from information gathered computationally on the finer grid systems. We show how a bootstrap setup procedure can be employed for these purposes, ending up with multigrid methods which substantially outperform the typically used odd-even preconditioned standard Krylov subspace solvers.




• Michael Günther (Uni Wuppertal): Lattice Quantumchromodynamics - a mathematical perspective (Slides)
  Abstract: The task of lattice QCD is to approximate the expectation of operators acting on some Lie groups via sampling. One prominent approach to approximate the associated integrals is hybrid monte carlo, which uses an augmented Markov chain to construct samples of pairs of Lie group and Lie algebra elements. We show that the detailed balance equation holds, if the proposal step is both time-reversible and volume preserving. Thus geometric time integration has to be used to approxmiate numerically the Hamiltonian flow. We give an overview on current schemes and ideas used within the integration kernel of lattice QCD.




• Helmut Harbrecht (Uni Basel): Shape optimization for quadratic functionals and states with random right-hand sides (Slides) (Marc Dambrine, Charles Dapogny, Helmut Harbrecht, and Benedicte Puig)
  Abstract: In this talk, we investigate a particular class of shape optimization problems under uncertainties on the input parameters. More precisely, we are interested in the minimization of the expectation and variance of a quadratic objective in a situation where the state function depends linearly on a random input parameter. This framework covers important objectives such as the Kohn-Vogelius functional in electric impedance tomography and the compliance in linear elasticity. We show that the robust objective and its gradient are completely and explicitly determined by low-order moments of the random input. We then derive a cheap, deterministic algorithm to minimize the objective. Numerical results are given.




• Katja Ickstadt (TU Dortmund University, Faculty of Statistics): Prediction for Stochastic Growth Processes in Fatigue Experiments (Slides)
  Abstract: In this talk we propose a general Bayesian approach for stochastic versions of deterministic growth models to provide predictions for crack propagation in an early stage of a growth process. To improve the prediction, the information of other crack growth processes is used in a hierarchical model. Two stochastic versions of a deterministic growth model are considered. One is a nonlinear regression model, in which the trajectory is assumed to be the solution of an ordinary differential equation with additive errors. The other is a diffusion model defined by a stochastic differential equation. The predictive distributions will be approximated simulating from the posterior distributions resulting from a corresponding Markov chain Monte Carlo algorithm. As an example, we consider the famous data set of Virkler et al. (1979), in which the number of cycles that lead to a fixed crack length are observed.




• Jean-Christophe Mourrat (ENS Paris): Quantitative stochastic homogenization, theory and practice
  Abstract: Divergence-form operators with random coefficients homogenize over large scales. Recently, major progress has been made to make this convergence quantitative. I will outline some of the main results in this direction, and focus on some related insights concerning the computational aspects of homogenization.




• Fabio Nobile (EPFL): Multi-level and multi-index Monte Carlo methods in Uncertainty Quantification (Slides)
  Abstract: Complex mathematical models, based on partial differential equations, are widely used in many areas of physics and engineering. However, it is often the case that some of the parameters entering those models are affected by uncertainty either due to an intrinsic variability or lack of knowledge. Forward Uncertainty Quantification aims at propagating the input uncertainty, often described in probabilistic terms, through the model and quantifying the corresponding uncertainty in its solution or output quantities of interest. Monte Carlo methods, although robust and simple to implement, often lead to unaffordable computational costs in this setting as they require an excessive number of forward runs of the complex model to achieve acceptable tolerances. The multilevel Monte Carlo method has proven to be very powerful to compute expectations of output quantities of a stochastic model governed by differential equations. It exploits several discretization levels of the underlying equation to dramatically reduce the overall complexity with respect to a standard Monte Carlo method. In this talk we review the main ideas of the Multilevel Monte Carlo method and discuss practical implementation aspects as well as extensions to accommodate concurrent types of discretizations (multi-index Monte Carlo method) and compute derived quantities such as central moments, quantiles, or cdfs of output quantities. We illustrate then the power of the MLMC method on applications such as compressible aerodynamics, shape optimization under uncertainty, ensemble Kalman filter and data assimilation.




• Felix Otto (MPI MIS Leipzig): Characterization of fluctuations in stochastic homogenization (Slides)
  Abstract: We are interested in elliptic systems \(\nabla(\cdot a\nabla u+f)=0\) in \(d\)-dimensional whole space, with a stationary coefficient field \(a\) of finite correlation length but deterministic r.~h.~s. \(f\). The goal is to characterize the fluctuations $$\int g\cdot\nabla u-E(\int g\cdot\nabla u)$$ of macroscopic observables \(\int g\cdot\nabla u\) for some (deterministic) averaging function \(g\). We consider the situation where the scale \(L\) of \(f\) and \(g\) imposed by \(f(x)=\hat f(\frac{x}{L})\) and \(g(x)=\frac{1}{L^d}\hat g(\frac{x}{L})\) is much larger than the correlation length of \(a\), which is set to unity.
  We argue that these fluctuations can be characterized to leading order by injecting the two-scale expansion into what we call the homogenization corrector. On the level of the gradients, the two-scale expansion is given by $$\nabla u\approx \partial_i\bar u(e_i+\nabla\phi_i),$$ where \(\phi_i\) is the corrector in direction \(i=1,\dots,d\) (summation convention) and \(\bar u\) solves the homogenized equation $$\nabla\cdot(\bar a\nabla\bar u+f)=0 \quad (\bar ae_i=E(a(e_i+\nabla\phi_i))).$$ The homogenization commutator is, like in the general or H-theory of homogenization, $$\mbox{``flux''} - \bar a \mbox{``field''}=a\nabla u-\bar a\nabla u.$$ This is easily seen by computing the (Malliavin) derivative w.~r.~t.~\(a\).
  We argue that this approach also works to higher order (up to nearly order \(\frac{d}{2}\)): Inserting a higher-order two-scale expansion $$\nabla u \approx\partial_i\bar u(e_i+\nabla\phi_i)+\partial_{ij}\bar u(\phi_ie_j+\nabla\psi_{ij})$$ into the homogenization commutator yields the ``pathwise'' approximation $$\int g\cdot\nabla u-E(\int g\cdot\nabla u) =F+O(L^{-\frac{d}{2}+}L^{-\frac{d}{2}})$$ with a random variable \(F=O(L^{-\frac{d}{2}})\) that can be computed ``off-line'' solely on the basis of the correctors, of the solution \(\bar u\) of the homogenized problem, and of the solution \(\bar v\) of the dual homogenized problem \(\nabla\cdot\bar a^*(\nabla\bar v+g)=0\).
  This is joint work with Mitia Duerinckx and Antoine Gloria.




• Alois Pichler (TU Chemnitz): Risk measures: their role in quantifying uncertainty (Slides)
  Abstract: Risk measures originate from mathematical finance as they allow quantifying subjective risk of some random outcome. Risk measures are nowadays indispensable in mathematical problem formulations involving uncertainty or risk. This talk highlights their mathematical key properties and illustrates/ demonstrates their use in various problems formulations, mainly originating from optimization under uncertainty. A special emphasize will be given on stochastic optimization, i.e., optimization under uncertainty in various stages.




• Björn Sprungk (Uni Mannheim) Metropolis-Hastings algorithms for Bayesian inverse problems in Hilbert spaces (Slides)
  Abstract: In this talk we consider the Bayesian approach to inverse problems and infer uncertain coefficients in elliptic PDEs given noisy observations of the associated solution. After a short introduction to this approach, we focus on Metropolis-Hastings (MH) algorithms for approximate sampling of the resulting posterior distribution. These methods used to suffer from a high dimensional state space or a highly concentrated target measure, respectively. In recent years dimension-independent MH algorithms have been developed and analyzed, suitable for Bayesian inference in infinite dimensions. However, the second issue has drawn less attention yet, despite its importance for applications with large or highly informative data. We present a MH algorithm well-defined in Hilbert spaces which possesses both properties: a dimension-independent spectral gap as well as a robust behaviour w.r.t. small noise levels in the observational data. Moreover, we show a first analysis of noise-independent performance of MH algorithms in terms of the expected acceptance rate and the expected squared jump distance of the resulting Markov chains. Numerical experiments confirm the theoretical results and also indicate that they hold in more general situations than proven.




• Peter Stollmann (TU Chemnitz): Quantum dynamics for random models: what, why and how (Slides, Handout)
  Abstract: Models of uncertainty quantification and those of mathematical physics share certain features: typically, they can be expressed in terms of [partial] differential [or difference] equations with coefficients that depend on random variables. We will introduce some classes of such models, present the typical phenomena one wants to establish and how these phenomena can be treated mathematically. Moreover we collect typical techniques and mathematical tools that have been established and might be useful in other contexts as well.




• Tim Sullivan (Free University of Berlin and Zuse Institute Berlin): Bayesian Probabilistic Numerical Methods (Slides)
  Abstract: In this work, numerical computation - such as numerical solution of a PDE - is treated as a statistical inverse problem in its own right. The popular Bayesian approach to inversion is considered, wherein a posterior distribution is induced over the object of interest by conditioning a prior distribution on the same finite information that would be used in a classical numerical method. The main technical consideration is that the data in this context are non-random and thus the standard Bayes' theorem does not hold. General conditions will be presented under which such Bayesian probabilistic numerical methods are well-posed, and a sequential Monte-Carlo method will be shown to provide consistent estimation of the posterior. The paradigm is extended to computational ``pipelines'', through which a distributional quantification of numerical error can be propagated. A sufficient condition is presented for when such propagation can be endowed with a globally coherent Bayesian interpretation, based on a novel class of probabilistic graphical models designed to represent a computational work-flow. The concepts are illustrated through explicit numerical experiments involving both linear and non-linear PDE models. Further details are available in the preprint arXiv:1702.03673.

• Peter Zaspel (Basel): Scalable solvers for meshless methods on many-core clusters (Slides)
  Abstract: Our goal is to solve large-scale stochastic collocation problems in a high-order convergent and scaling fashion. To this end, we recently discussed the radial basis function (RBF) kernel-based stochastic collocation method. In this meshless method, the higher-dimensional stochastic space is sampled by (quasi-)Monte Carlo sequences, which are used as centers of radial basis functions in a collocation scheme. This non-intrusive approach combines high-order algebraic or even exponential convergence rates of spectral (sparse) tensor-product methods with good pre-asymptotic convergence of kriging, the profound stochastic framework of Gaussian process regression and parts of the simplicity of Monte Carlo methods. Preliminary applications for this uncertainty quantification framework were (elliptic) model problems and incompressible two-phase flows with applications in chemical bubble reactors and river engineering. One specific challenge of the discussed approach is the solution of a well-structured large to huge dense linear system to compute the quadrature weights. inear systems of similar type arise in Gaussian process regression and several machine learning approaches. Classical direct factorization techniques to solve the above linear system for a large to huge kernel sample count are barely tractable, even on large parallel computers. Therefore, we discuss iterative approaches to solve such linear systems on large parallel clusters with a special emphasis on many-core hardware. To keep the iteration count small, a large-scale preconditioner with excellent strong scalability properties has been developed for GPU clusters. Moreover, we work on an optimal-complexity matrix approximation by hierarchical matrices on many-core hardware. The presentation will cover the latest results with respect to numerical methods and applications. Performance and scalability results will be given based on studies on the Titan GPU cluster at Oak Ridge National Lab. This work is partly based on joint work with Michael Griebel, Helmut Harbrecht and Christian Rieger.