MATH4UQ Seminar
The MATH4UQ seminar features talks by internal and external colleagues and collaborators as well as guests visiting the chair. The talks will takes place at Kackertstraße 9, seminar room C301, if not stated otherwise. Everybody interested is welcome to attend.
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Upcoming and past talks
2020

09.06.2020, 14:00: Prof. Benjamin Berkels (RWTH)
 Webinar: zoom meeting link will be circulated through the MATH4UQ seminar mailing list verbreitet.
 Title: Image registration and segmentation using variational methods

Abstract: Image segmentation and registration are two of the fundamental image processing problems arising in many different application areas.
Registration is the task of transforming two or more images into a common coordinate system. After a short introduction to variational image registration, we demonstrate that nonrigid registration techniques can be used to achieve subpicometer precision measurements of atom positions from a series of scanning transmission electron microscopy images at atomic scale. Particular challenges here are input data with low signaltonoise ratio and periodic structures, as well as initialization bias of the resulting iterative optimization strategies for the nonconvex objective.
Segmentation is to decompose an image into disjoint regions that are roughly homogeneous in a suitable sense. If three or more regions are sought, one speaks of multiphase segmentation. We first review how to find global minimizers of the nonconvex binary MumfordShah model to solve the classical twophase segmentation problem and show segmentation problems from different application areas. Then, we propose a flexible framework for multiphase segmentation based on the MumfordShah model and highdimensional local feature vectors.

13.03.2020, 14:00: Prof. Fabio Nobile (EPFL)
 Title: A multilevel stochastic gradient algorithm for PDEconstrained optimal control problems under uncertainty
 Abstract: We consider an optimal control problem for an elliptic PDE with random coefficients. The control function is a deterministic, distributed forcing term that minimizes an expected quadratic regularized loss functional. For its numerical treatment we propose and analyze a multilevel stochastic gradient (MLSG) algorithm which uses at each iteration a full, or randomized, multilevel Monte Carlo estimator of the expected gradient, build on a hierarchy of finite element approximations of the underlying PDE. The algorithm requires choosing proper rates at which the finite element discretization is refined and the Monte Carlo sample size increased over the iterations. We present complexity bounds for such algorithm. In particular, we show that if the refinement rates are properly chosen, in certain cases the asymptotic complexity of the full MLSG algorithm in computing the optimal control is the same as the complexity of computing the expected loss functional for one given control by a standard multilevel Monte Carlo estimator. This is joint work with Matthieu Martin (CRITEO, Grenoble), Panagiotis Tsilifis (General Electric), Sebastian Krumscheid (RWTH Aachen).

05.03.2020, 14:00: Prof. Fabio Nobile (EPFL)
 Title: Dynamical Low Rank approximation of random time dependent PDEs
 Abstract: In this talk we consider time dependent PDEs with random parameters and seek for an approximate solution in separable form that can be written at each time instant as a linear combination of a fixed number of linearly independent spatial functions multiplied by linearly independent random variables (low rank approximation). Since the optimal deterministic and stochastic modes can significantly change over time, we consider a dynamical approach where those modes are computed on the fly as solutions of suitable evolution equations. We discuss the construction of the method, present an existence result for the low rank approximate solution of a random semilinear evolutionary equation of diffusion type, and introduce an operator splitting numerical discretization of the low rank equations for which we can prove (conditional) stability. This is joint work with Yoshihito Kazashi, Eleonora Musharbash, Eva Vidlicková.

27.02.2020, 14:15: Chiheb Ben Hammouda (KAUST)
 Title: Adaptive sparse grids and quasiMonte Carlo for option pricing under the rough Bergomi model
 Abstract: The rough Bergomi (rBergomi) model, introduced recently in (Bayer, Friz, Gatheral, 2016), is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits with empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasiMonte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

25.02.2020, 14:00: Chiheb Ben Hammouda (KAUST)
 Title: Numerical Smoothing and Hierarchical Approximations for Efficient Option Pricing and Density Estimation
 Abstract: When approximating expectations of certain quantity of interest (QoI), the efficiency and performance of deterministic quadrature methods, such as sparse grids, and hierarchical variance reduction methods such as multilevel Monte Carlo (MLMC), may be highly deteriorated, in different ways, by the low regularity of the QoI with respect to the input parameters. To overcome this issue, a smoothing procedure is needed to uncover the available regularity and improve the performance of the aforementioned numerical methods. In this work, we consider cases where we can not perform an analytic smoothing and introduce a novel numerical smoothing technique, based on root finding combined with a one dimensional integration with respect to a single wellchosen variable. We prove that under appropriate conditions the resulting function of the remaining variables is a highly smooth function, so potentially allowing a higher efficiency of adaptive sparse grids quadrature (ASGQ), in particular when it is combined with hierarchical transformations (Brownian bridge and Richardson extrapolation on the weak error) to treat effectively the high dimensionality. Our study is motivated by option pricing problems and our main focus is on dynamics where a discretization of the asset price is needed. Through our analysis and numerical experiments, we illustrate the advantage of combining numerical smoothing with ASGQ, over the Monte Carlo (MC) approach. Furthermore, we demonstrate how the numerical smoothing significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate, when using Euler scheme from 1/2 for the standard case (without smoothing), to 1. Due to the complexity theorem of MLMC and given a preselected tolerance, TOL, this results in an improvement of the complexity from O(TOL^{2.5}) in the standard case to O(TOL^{2}\log(TOL)^2). Finally, we show how our numerical smoothing combined with MLMC enables us also to estimate density functions, a situation where standard MLMC fails.

18.02.2020, 14:45: Chiheb Ben Hammouda (KAUST)
 Title: Importance Sampling for a Robust and Efficient Multilevel Monte Carlo Estimator for Stochastic Reaction Networks
 Abstract: The multilevel Monte Carlo (MLMC) method for continuous time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), and in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be deteriorated due to the phenomenon of high kurtosis, observed at the deep levels of MLMC, which leads to inaccurate estimates for the sample variance. In this work, we address cases where the high kurtosis phenomenon is due to catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a very small proportion), and introduce a pathwise dependent importance sampling technique that improves the robustness and efficiency of the multilevel method. Our analysis, along with the conducted numerical experiments, demonstrates that our proposed method significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate. Due to the complexity theorem of MLMC and given a preselected tolerance, TOL, this results in an improvement of the complexity from O(TOL^{2} \log(TOL)^2) in the standard case to O(TOL^{2}).

17.02.2020, 14:00: TruongVinh Hoang (Technische Universität Braunschweig)
 Title: Neural networkbased filtering technique for highdimensional and nonlinear data assimilation
 Abstract: The talk aims at an introduction about the author's research, with a particular focus on the neural networkbased filtering technique for data assimilation. We propose a novel ensemble filter for high dimensional and nonlinear data assimilation problem. The method trains an artificial neural network (ANN) to approximate the conditional expectation using as training data the prior ensemble and their predicted observations. To avoid overfitting when training the ANN on ensembles of relatively small sizes, different techniques are employed such as L2 regularisation, dataset augmentation and multilevel method. The trained ANN is then used in computing the assimilated ensemble. Our approach can be understood as a natural generalisation of the ensemble Kalman filter (EnKF) to nonlinear updates. Using ANN to approximate the conditional expectation can reduce the intrinsic bias of the EnKF and improve its predictions. To illustrate the approach, we implement our framework for tracking the states in the Lorenz64 and Lorenz93 systems.

14.02.2020, 14:00: Chiheb Ben Hammouda (KAUST)
 Title: Multilevel Hybrid Split Step Implicit TauLeap for Stochastic Reaction Networks
 Abstract: In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics are dominated by stochastic effects. To approximate those systems, discrete statespace and stochastic simulation approaches have been shown to be more relevant than continuous statespace and deterministic ones. These stochastic models constitute the theory of Stochastic Reaction Networks (SRNs). In systems characterized by having simultaneously fast and slow timescales, existing discrete spacestate stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tauleap (explicitTL) method, can be very slow. In this talk, we propose a novel implicit scheme, splitstep implicit tauleap (SSITL), to improve numerical stability and provide efficient simulation algorithms for those systems. Furthermore, to estimate statistical quantities related to SRNs, we propose a novel hybrid Multilevel Monte Carlo (MLMC) estimator in the spirit of the work by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012). This estimator uses the SSITL scheme at levels where the explicitTL method is not applicable due to numerical stability issues, and then, starting from a certain interface level, it switches to the explicit scheme. We present numerical examples that illustrate the achieved gains of our proposed approach in this context.