MATH4UQ Seminar

 

YouTube Channel

We are now on YouTube. You can find recordings of previous seminar talks on the MATH4UQ channel.

 

The MATH4UQ seminar features talks by internal and external colleagues and collaborators as well as guests visiting the chair. Everybody interested is welcome to attend.

Please subscribe to our MATH4UQ seminar mailing list to receive notifications about upcoming seminars. Recordings of several previous talks can also be found on our MATH4UQ YouTube channel.

 

Upcoming and past talks

Upcoming talks

  • 03.11.2020, 16:00: Dr. Lorenzo Tamellini, CNR-IMATI Pavia.
    • Titel: Uncertainty quantification and identifiability of SIR-like dynamical systems for epidemiology.
    • Abstract: In this talk, we provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models, that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of the points that we touch upon are actually generally valid for inverse problems in more general setups.

Previous talks (2020)

  • 30.10.2020, 13:00: Sophia Franziska Wiechert, RWTH Aachen.
    • Titel: Continuous Time Markov Decision Processes with Finite Time Horizon.
    • Abstract: One can derive a Markov Decision Process (MDP) by adding an input to a continuous-time Markov Process. These inputs, also called actions, allow us to change the states' transition rates, hence, to "control" the Markov Process. By adding a reward dependent on the current state and action, one can formulate the MDP's optimal control problem. In this talk, we restrict ourselves to finite horizon problems. The aim is to find the optimal actions, which maximize the reward over a finite time horizon. The Hamilton-Jacobi-Bellman equation gives an analytic solution of the optimal control problem. Solving this system of ordinary differential equations is difficult in general. Therefore, the problem is discretized in time and solved as a discrete-time Markov decision chain by a simple algorithm that iterates backward in time. We illustrate this approach through the example of salmon farming.
  • 27.10.2020, 16:00: Emil Loevbak, KU Leuven, Belgium.
    • Titel: Asymptotic-preserving multilevel Monte Carlo particle methods for diffusively scaled kinetic equations.
    • Abstract: In many applications it is necessary to compute the time-dependent distribution of an ensemble of particles subject to transport and collision phenomena. Kinetic equations are PDEs that model such particles in a position-velocity phase space. In the low collisional regime explicit particle-based Monte Carlo methods simulate these high dimensional equations efficiently, but, as the collision rate increases, these methods suffer from severe time-step constraints.

      Asymptotic-preserving particle schemes are able to avoid these time-step constraints by explicitly including information from models describing the infinite collision rate case. However, these schemes produce biased results when used with large simulation time steps. In recent years, we have shown that the multilevel Monte Carlo method can be used to reduce this bias by combining simulations with large and small time steps, computing accurate results with greatly reduced simulation cost. In this talk, I will present the current state of the art for this newly developed asymptotic-preserving multilevel Monte Carlo approach. This includes an overview of existing methods and numerical results. I will then conclude with a view on future prospects for these methods.

  • 20.10.2020, 16:00: Dr. Luis Espath (RWTH Aachen)
    • Titel: Multilevel Double Loop Monte Carlo Method with Importance Sampling for Bayesian Optimal Experimental Design
    • Abstract: An optimal experimental set-up maximizes the value of data for statistical inferences. The efficiency of strategies for finding optimal experimental set-ups is particularly important for experiments that are time-consuming or expensive to perform. When the experiments are modeled by Partial Differential Equations (PDEs), multilevel methods have been proven to reduce the computational complexity of their single-level counterparts when estimating expected values. For a setting where PDEs can model experiments, we propose a multilevel method for estimating the widespread criterion known as the Expected Information Gain (EIG) in Bayesian optimal experimental design. We propose a Multilevel Double Loop Monte Carlo (MLDLMC), where the Laplace approximation is used for importance sampling in the inner expectation. The method's efficiency is demonstrated by estimating EIG for inference of the fiber orientation in composite laminate materials from an electrical impedance tomography experiment.
  • 13.10.2020, 16:00: Dr. Neil Chada (KAUST)
    • Titel: Consistency analysis of data-driven bilevel learning in inverse problems
    • Abstract: One fundamental problem when solving inverse problems is how to find regularization parameters. This talk considers solving this problem using data-driven bilevel optimization. This approach can be interpreted as solving an empirical risk minimization problem, and its performance with large data sample size can be studied in general nonlinear settings. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient method. The convergence of these numerical schemes can also be analyzed under suitable assumptions. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.
  • 06.10.2020, 16:00: Arved Bartuska (RWTH)
    • Title: Laplace approximation for Bayesian experimental design
    • Abstract: The problem of finding the optimal design of an experiment in a Bayesian setting via the expected information gain (EIG) leads to the computation of two nested integrals that are usually not given in closed-form. The standard approach uses a double loop Monte Carlo estimator, which can still be very costly in many cases. Two alternative estimators based on the Laplace approximation will be presented in this talk, followed by a numerical example from the field of electrical impedance tomography (EIT).
  • 29.09.2020, 16:00: Jonas Kiessling, Emanuel Ström, Magnus Tronstad
    • Title: Wind Field Reconstruction from Historical Weather Data
    • Abstract: In this talk we will present ongoing work in wind field reconstruction from historical weather measurements. We draw on techniques from Machine Learning (ML) and Fourier Analysis, and show how standard ML models can be improved by including physically motivated penalty terms. Our model is tested on public historical weather data from Sweden, and benchmarked against a range of other published models, including Kriging, Nearest Neighbour and Average Inverse Distance. This is a joint work with Andreas Enblom, Luis Espath, Dmitry Kabanov and Raul Tempone.
  • 22.09.2020, 16:00: Dr. Nadhir Ben Rached (RWTH)
    • Title: Dynamic splitting method for rare events simulation
    • Abstract: We propose a unified rare-event estimator based on the multilevel splitting algorithm. In its original form, the splitting algorithm cannot be applied to time-independent problems because splitting requires an underlying continuous-time Markov process whose trajectories can be split. We embed the time-independent problem within a continuous-time Markov process so that the target static distribution corresponds to the distribution of the Markov process at a given time instant. To illustrate the large scope of applicability of the proposed approach, we apply it to the problem of estimating the cumulative distribution function (CDF) of sums of random variables (RVs), the CDF of partial sums of ordered RVs, the CDF of ratios of RVs, and the CDF of weighted sums of  Poisson RVs. We investigate the computational efficiency of the proposed estimator via a number of simulation studies and find that it compares favorably with existing estimators.
  • 09.06.2020, 14:00: Prof. Benjamin Berkels (RWTH)
    • Webinar: zoom meeting link will be circulated through the MATH4UQ seminar mailing list.
    • 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 non-rigid registration techniques can be used to achieve sub-picometer 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 signal-to-noise ratio and periodic structures, as well as initialization bias of the resulting iterative optimization strategies for the non-convex 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 multi-phase segmentation. We first review how to find global minimizers of the non-convex binary Mumford-Shah model to solve the classical two-phase segmentation problem and show segmentation problems from different application areas. Then, we propose a flexible framework for multi-phase segmentation based on the Mumford-Shah model and high-dimensional local feature vectors.
  • 13.03.2020, 14:00: Prof. Fabio Nobile (EPFL)
    • Title: A multilevel stochastic gradient algorithm for PDE-constrained 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 semi-linear 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 quasi-Monte 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 time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi-Monte 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 well-chosen 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 pre-selected 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 pre-selected 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: Truong-Vinh Hoang (Technische Universität Braunschweig)
    • Title: Neural network-based filtering technique for high-dimensional and non-linear data assimilation
    • Abstract: The talk aims at an introduction about the author's research, with a particular focus on the neural network-based filtering technique for data assimilation. We propose a novel ensemble filter for high dimensional and non-linear 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 over-fitting 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 non-linear 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 Lorenz-64 and Lorenz-93 systems.
    • 14.02.2020, 14:00: Chiheb Ben Hammouda (KAUST)
      • Title: Multilevel Hybrid Split Step Implicit Tau-Leap 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 state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space 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 space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (explicit-TL) method, can be very slow. In this talk, we propose a novel implicit scheme, split-step implicit tau-leap (SSI-TL),  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 SSI-TL scheme at levels where the explicit-TL 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.

     

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