MATH4UQ Seminar
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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

18.05.2021, 16:00: Prof. Dr. Elisabeth Ullmann, Department of Mathematics, Technical University of Munich.
 Title: Rare event estimation with PDEbased models.

Abstract: The estimation of the probability of rare events is an important task in reliability and risk assessment of critical societal systems, for example, groundwater flow and transport, and engineering structures. In this talk we consider rare events that are expressed in terms of a limit state function which depends on the solution of a partial differential equation (PDE). We present recent progress on mathematical and computational aspects of this problem: (1) the impact of the PDE approximation error on the failure probability estimate, and (2) the use of the Ensemble Kalman Filter for the estimation of failure probabilities.
This is joint work with Fabian Wagner (TUM), Iason Papaioannou (TUM) and Jonas Latz (Cambridge).
Previous talks (2021)

11.05.2021, 16:00: Prof. Bernard Ghanem, KAUST.
 Title: On Adversarial Network Attacks, Robustness, and Certification.
 Abstract: The outstanding performance of deep neural networks (DNNs), for visual recognition tasks in particular, has been demonstrated on many largescale benchmarks. This performance has immensely strengthened the line of research that aims to understand and analyze the driving reasons behind the effectiveness of these networks. One important aspect of this analysis has gained much attention, namely the sensitivity of a DNN to perturbations. This has spawned a thrust in the research community, which focuses on developing adversarial attacks that fool a DNN, training strategies that make DNNs more robust against such attacks, as well as methods to certify the behavior of a DNN irrespective of the attack. In this talk, I will introduce this exciting research thrust and span some of its landscape from synthesizing adversarial attacks all the way to randomized smoothing for DNN certification. I will also give an update on the latest research progress in this direction from the Image and Video Understanding Lab (IVUL) at KAUST.

04.05.2021, 16:00: Prof. Dr. Carsten Hartmann, Institut für Mathematik, BTU CottbusSenftenberg.
 Title: Optimal control of the underdamped Langevin sampler.
 Abstract: The underdamped Langevin equation is a popular computational model in various fields of science (e.g. molecular dynamics, meteorology, or machine learning) that is used to sample from complicated multimodal probability distributions by a combination of (dissipative) Hamiltonian dynamics and diffusion. I will explain how control theory can help to speed up the convergence of an underdamped Langevin equation to its equilibrium probability distribution, specifically in situations in which the diffusion coefficient is small (i.e. low temperature) and the asymptotic properties of the dynamics are dominated by metastability and poor convergence. I will discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower ("target") or the higher ("simulation") temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow.

27.04.2021, 16:00: Marco Ballesio, KAUST
 Title: Unbiased Estimation of LogLikelihood Gradient for a Class of ContinuousTime StateSpace Models
 Abstract: We consider static parameter estimation for a class of continuoustime statespace models. Our goal is to obtain an unbiased estimate of the loglikelihood gradient, which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a doubly randomized scheme, that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization. Our novel estimate helps facilitate the application of gradientbased estimation algorithms, such as stochastic gradient descent (SGD). We illustrate our methodology in the context of SGD in several numerical examples and compare with the Rhee &Glynn estimator. Results show consistency of the method.

20.04.2021, 16:00: Eike Cramer , Forschungszentrum Jülich GmbH and RWTH Aachen University, Germany.
 Title: A principal component normalizing flow for manifold density estimation.

Abstract: Recent developments in machine learning have enabled 'modelfree' distribution learning using socalled generative neural networks. In particular, normalizing flows have proven effective as they are trained through direct loglikelihood maximization, in contrast to other generative networks such as generative adversarial networks and variational autoencoders. However, normalizing flows fail to fit distributions of data manifolds correctly, leading to severe overfitting and generation of data that does not follow the target distribution. To leverage the modeling power of normalizing flows for manifold data, we propose an injective mapping based on principal component analysis. The resulting principal component flow (PCF) model enables density estimation in latent space and avoids complications due to manifold data. We show that the dimensionality reduction does not interfere with the density estimation procedure. As an illustrative case study, we apply the PCF model to learn the distribution of the daily total photovoltaic power generation in Germany. The PCF model is able to generate new data by sampling from a significantly lower latent dimensionality than the full data dimensionality and trains more effectively than full space flow models. The PCF performs particularly well in identifying parts of the data with zero variance, while preserving critical features of the original time series like the probability and power spectral density.

13.04.2021, 16:00: Dr. Bruno Tuffin , INRIA Rennes BretagneAtlantique, France.
 Title: Estimating by Simulation the Mean and Distribution of Hitting Times of Rare Events.

Abstract: Rare events occur by definition with a very small probability but are important to analyze because of potential catastrophic consequences. During this talk, we will focus on rare event for socalled regenerative processes, that are basically processes such that portions of the process are statistically independent of each other. For many complex and/or large models, simulation is the only tool at hand but requires specific implementations to get an accurate answer in a reasonable time. There are two main families of rareevent simulation techniques: importance sampling (IS) and splitting.
We will (somewhat arbitrarily) devote most of the talk to IS.
We will then focus on the estimation of the mean hitting time of a rarely visited set. A natural and direct estimator consists in averaging independent and identically distributed copies of simulated hitting times, but an alternative standard estimator uses the regenerative structure allowing to represent the mean as a ratio of quantities. We will see that in the setting of crude simulation, the two estimators are actually asymptotically identical in a rareevent context, but inefficient for different, even if related, reasons: the direct estimator requires a large average computational time of a single run whereas the ratio estimator faces a small probability computation. We then explain that the ratio estimator is advised when using IS.
In the third part of the talk, we will discuss the estimation of the distribution, not just the mean, of the hitting time to a rarely visited set of states. We will exploit the property that the distribution of the hitting time divided by its expectation converges weakly to an exponential as the target set probability decreases to zero. The problem then reduces to the extensively studied estimation of the mean described previously. It leads to simple estimators of a quantile and conditional tail expectation of the hitting time. Some variants will be presented and the accuracy of the estimators illustrated on numerical examples.
This talk is mostly based on collaborative works with Peter W. Glynn and Marvin K. Nakayama.

06.04.2021, 16:00: Dr. Christian Bayer , Weierstraß Institute for Applied Analysis and Stochastics.
 Title: A pricing BSPDE for rough volatility.
 Abstract: In this talk, we study the option pricing problems for rough volatility models. As the framework is nonMarkovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solutions is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forwardbackward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep learningbased method is also investigated for the numerical approximations to such BSPDEs and associated nonMarkovian pricing problems. Finally, examples of rough Bergomi type are numerically computed for both European and American options.
 30.03.2021, 16:00: Dr. André Gustavo Carlon
, KAUST.
 Title: MultiIteration Stochastic Optimizers.
 Abstract: Stochastic optimization problems are of great importance for many fields ranging from engineering to machine learning. Stochastic gradient descent methods (SGD) are the main class of methods to solve these problems. However, standard SGD converges sublinearly and is not easily parallelizable. MultiIteration Stochastic Optimizers are a novel class of firstorder stochastic optimizers where the gradient is estimated using the MultiIteration stochastiC Estimator (MICE). The MICE estimator controls the coefficient of variation of the mean gradient approximation using successive control variates along the path of iterations. The SGDMICE optimizer converges linearly in the class of stronglyconvex and Lsmooth functions. The performances of multiiteration stochastic optimizers are evaluated in numerical examples, validating our analysis, and converging faster than wellknown stochastic optimization methods for the same number of gradient evaluations.

23.03.2021, 16:00: Marco Ballesio
, KAUST.
 Title: Multilevel Particle Filters.
 Abstract: We consider the filtering problem for partially observed diffusions, which are regularly observed at discrete times. We are concerned with the case when one must resort to timediscretization of the diffusion process if the transition density is not available in an appropriate form. In such cases, one must resort to advanced numerical algorithms such as particle filters to consistently estimate the filter. It is also well known that the particle filter can be enhanced by considering hierarchies of discretizations and the multilevel Monte Carlo (MLMC) method, in the sense of reducing the computational effort to achieve a given mean square error (MSE). A variety of multilevel particle filters (MLPF) have been suggested in the literature, e.g., in Jasra et al., SIAM J, Numer. Anal., 55, 3068–3096. Here we introduce a new alternative that involves a resampling step based on the optimal Wasserstein coupling. We prove a central limit theorem (CLT) for the new method. On considering the asymptotic variance, we establish that in some scenarios, there is a reduction, relative to the approach in the aforementioned paper by Jasra et al., in computational effort to achieve a given MSE. These findings are confirmed in numerical examples. We also consider filtering diffusions with unstable dynamics; we empirically show that in such cases a change of measure technique seems to be required to maintain our findings.

09.03.2021, 16:00: Dr. Eric Hall, University of Dundee, Scotland.
 Title: Weak Error Rates for Option Pricing under the Rough Bergomi Model.
 Abstract: Modeling the volatility structure of underlying assets is a key component in the pricing of options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887904, 2016], seek to fit observed market data based on the observation that the logrealized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. In fact, both time series data of asset prices and option derived price data indicate that H often takes values close to 0.1 or even smaller, i.e. rougher than Brownian Motion. The nonMarkovian nature of the driving fractional Brownian motion in the rough Bergomi model, however, poses a challenge for numerical options pricing. Indeed, while the explicit Euler method is known to converge to the solution of the rough Bergomi model, the strong rate of convergence is only H ([Neuenkirch and Shalaiko, arXiv:1606.03854]). We prove rate H + 1/2 for the weak convergence of the Euler method and, in the case of quadratic payoff functions, we obtain rate one. Indeed, the problem is very subtle; we provide examples demonstrating that the rate of convergence for payoff functions well approximated by secondorder polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof relies on Taylor expansions and an affine Markovian representation of the underlying.

02.03.2021, 15:00: Juan Pablo Madrigal Cianci , CSQI laboratory in EPFL.
 Title: Generalized parallel tempering for Bayesian inverse problems.
 Abstract: In the current work we present two generalizations of the Parallel Tempering algorithm, inspired by the socalled continuoustime Infinite Swapping algorithm, which found its origins in the molecular dynamics community, and can be understood as the continuoustime limit of a Parallel Tempering algorithm with statedependent swapping rates. In the current work, we extend this idea to the context of timediscrete Markov chains and present two Markov chain Monte Carlo algorithms that follow the same paradigm as the continuoustime infinite swapping procedure. We present results on the reversibility and ergodicity properties of our generalized PT algorithms. Numerical results on sampling from different target distributions originating from Bayesian inverse problems, show that the proposed methods significantly improve sampling efficiency over more traditional sampling algorithms such as Random Walk Metropolis and (standard) Parallel Tempering.

23.02.2021, 16:00: Felix Terhag.
 Title: Dropout for Uncertainty Estimation in Neural Networks
 Abstract: The recent success of neural networks has led to applications in evermore domains. While they yield good results on many problems one has to be especially careful in safety critical applications, as classical approaches do not contain accurate quantifications of uncertainty. In this talk, I want to give an introduction into the topic of uncertainty estimation in deep neural networks. To do so, I will show, along an illustrative example, that neural networks are prone to predicting with high confidence even in samples, which do not belong to the training distribution. In the following I will give an introduction into a Bayesian approach to neural network and for that purpose introduce Gal and Ghahramani’s Dropout as Bayesian Approximation, which is a good starting point into the topic due to its clarity. Going back to the motivating example I will show the improvements gained by this method.

16.02.2021, 16:00: Dr. AbdulLateef HajiAli, Assistant professor, HeriotWatt University.
 Title: Multilevel Monte Carlo for Computing Probabilities.
 Abstract: In this talk, I will discuss the challenges in computing probabilities of the form $\prob{X \in \Omega}$ where $X$ is a random variable and $\Omega$ is a ddimensional set. Computing such probabilities is important in many contexts, e.g., risk assessment and finance. A frequent challenge is encountered when only a costly approximation of the random variable $X$ can be sampled. For example, when $X$ depends on an inner expectation that has to be approximated with Monte Carlo or when $X$ depends on a nontrivial (stochastic) differential equations and a numerical discretization must be employed. A naive Monte Carlo method has a prohibitive complexity that compounds the slow convergence of Monte Carlo with the complexity of approximation. Instead, I will present a variant of Multilevel Monte Carlo (MLMC) with adaptive levels that can, under certain conditions, have a complexity that is independent of the complexity of approximation.

09.02.2021, 16:00: Dr. Chiara Piazzola, CNRIMATI, Pavia, Italy.
 Title: Comparing MultiIndex Stochastic Collocation and Radial Basis Function Surrogates for Ship Resistance Uncertainty Quantification
 Abstract: A comparison of two methods for the forward Uncertainty Quantification (UQ) of complex industrial problems is presented. Specifically, the performance of MultiIndex Stochastic Collocation (MISC) and multifidelity Stochastic Radial Basis Functions (SRBF) surrogates is assessed for the UQ of a rollon/rolloff passengers ferry advancing in calm water and subject to two operational uncertainties, namely the ship speed and draught. The estimation of the expected value, standard deviation, and probability density function of the (modelscale) resistance is presented and discussed. Both methods need to repeatedly solve the freesurface NavierStokes equations for different configurations of the operational parameters. The required CFD simulations are obtained by a multigrid Reynolds Averaged NavierStokes (RANS) equations solver. Both MISC and SRBF use as fidelity levels the intermediate grids employed by the RANS solver

02.02.2021, 16:00: Dr. Joakim Beck, Stochastic Numerics Research Group, CEMSE Division, KAUST
 Title: Multiindex stochastic collocation using isogeometric analysis for random PDEs
 Abstract: We consider the forward uncertainty quantification (UQ) problem of solving partial differential equations (PDEs) with random coefficients in domains of more challenging shape than hyperrectangles. We present an extension of multiindex stochastic collocation (MISC) that uses isogeometric analysis (IGA) instead of conventional finite element analysis. MISC uses tensorized PDE solvers, and this allows IGA solvers as they build on tensorization of univariate splines. A feature of IGA that enables its solvers to handle nonstandard domains is that the basis functions that describe the domain geometry, typically standard Bsplines or nonuniform rational Bsplines (NURBS), are used as the basis in approximating the PDE solution on the domain. We numerically demonstrate the computational efficiency of IGAbased MISC for linear elliptic PDEs with random coefficients in domains of complicated shape.

26.01.2021, 16:00: Dr. Zaid Sawlan, KAUST
 Title: Statistical and Bayesian methods for fatigue life prediction.
 Abstract: Predicting fatigue in mechanical components is extremely important for preventing hazardous situations. In this work, we calibrate several plausible probabilistic stresslifetime (SN) models using fatigue experiments on unnotched specimens. To generate accurate fatigue life predictions, competing SN models are ranked according to several classical informationbased measures. A different set of predictive information criteria is then used to compare the candidate Bayesian models. Moreover, we propose a spatial stochastic model to generalize SN models to fatigue crack initiation in general geometries. The model is based on a spatial Poisson process with an intensity function that combines the SN curves with an averaged effective stress that is computed from the solution of the linear elasticity equations. The resulting model can predict the initiation of cracks in specimens made from the same material with new geometries.

19.01.2021, 16:00: Giacomo Garegnani, EPFL
 Title: Filtering the data: An alternative to subsampling for drift estimation of multiscale diffusions
 Abstract: We present a novel technique for estimating the effective drift of twoscale diffusion processes. We set ourselves in a semiparametric framework and fit to data a singlescale equation of the overdamped Langevin type. If data is given in the form of a continuous time series, a preprocessing technique is needed for unbiasedness. Oftentimes, this is achieved by subsampling the data at an appropriate rate, which lies between the two characteristic time scales. We avoid subsampling and process the data with an appropriate lowpass filter, thus proposing maximum likelihood estimators and Bayesian techniques which are based on the filtered process. We show that our methodology is asymptotically unbiased and demonstrate numerically an enhanced robustness with respect to subsampling on several test cases.
Previous talks (2020)

15.12.2020, 16:00: Prof. Sebastian Krumscheid, RWTH Aachen University.
 Title: Adaptive Stratified Sampling for Nonsmooth Problems
 Abstract: Sampling based variance reduction techniques, such as multilevel Monte Carlo methods, have been established as a generalpurpose procedure for quantifying uncertainties in computational models. It is known however, that these techniques may not provide performance gains when there is a nonsmooth parameter dependence. Moreover, in many applications (e.g. transport problems in fractured porous media of relevance to carbon storage and wastewater injection) the key idea of multilevel Monte Carlo cannot be fully exploited since no hierarchy of computational models can be constructed. An alternative means to obtain variance reduction in these cases is offered by stratified sampling methods. In this talk we will discuss various ideas on adaptive stratified sampling methods tailored to applications with a discontinuous parameter dependence. Specifically, we will build upon ideas from adaptive PDE mesh refinement strategies applied to the stochastic instead of the physical domain. That is, the stochastic domain is adaptively stratified using local sensitivity estimates, and the samples are sequentially allocated to the strata for asymptotically optimal variance reduction. The proposed methodology is demonstrated on geomechanics in fractured reservoirs, and computational speedup compared to standard Monte Carlo is obtained. This is joint work with Per Pettersson (NORCE)

08.12.2020, 16:00: Dr. Jonas Kiessling, RWTH Aachen University, and KTH.
 Title: Deep Residual Neural Networks: Generalization Error and Parameterization.
 Abstract: In this talk I will consider the supervised learning problem of reconstructing a target function from noisy data using a deep residual neural network. I will give an overview of residual networks and present estimates of optimal generalization errors. I will touch on the ever important question of "why deep and not shallow neural networks" and show that under certain circumstances, deep residual networks have better approximation capacity than shallow networks with similar number of free parameters. I will also discuss a layerbylayer training algorithm and show some results on simulated data. This is joint work with A. Kammonen, P. Plechac, M. Sandberg, A. Szepessy and R. Tempone. The seminar is based on Smaller generalization error derived for a deep residualneural network compared to shallow networks, available on arXiv.

01.12.2020, 16:00: Dr. Marco Scavino, Instituto de Estadística, Universidad de la República, Uruguay.
 Title: A SDE model with derivative tracking for wind power forecast error: inference and application.

Abstract: Reliable wind power generation forecasting is crucial for many applications in the electricity market. We propose a datadriven modelbased on parametric Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of wind power forecast errors. Our SDE framework features timederivative tracking of the forecast, timevarying meanreversion parameter, and an improved statedependent diffusion term. The statistical inference methods we developed and applied allows the simulation of future wind power production paths and to obtain sharp empirical confidence bands. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply the model to historical Uruguayan wind power production data and forecasts between April and December 2019.
This talk is based on the work: Renzo Caballero, Ahmed Kebaier, Marco Scavino, Raúl Tempone (2020). A Derivative Tracking Model for Wind Power Forecast Error (https://arxiv.org/abs/2006.15907).
 24.11.2020, 16:00: Dr. Ben Mansour Dia, College of Petroleum Engineering & Geosciences, KFUPM, KSA
 Title: Continuation Bayesian inference
 Abstract: We present a continuation method that entails generating a sequence of transition probability density functions from the prior to the posterior in the context of Bayesian inference for parameter estimation problems. The characterization of transition distributions, by tempering the likelihood function, results in a homogeneous nonlinear partial integrodifferential equation for which existence and uniqueness of solutions are addressed. The posterior probability distribution comes as the interpretation of the final state of a path of transition distributions. A computationally stable scaling domain for the likelihood is explored for the approximation of the expected deviance, where we manage to restrict the evaluations of the forward predictive model at the prior stage. It follows the computational tractability of the posterior distribution and opens access to the posterior distribution for direct sampling. To get a solution formulation of the expected deviance, we derive a partial differential equation governing the moment generating function of the loglikelihood. We show also that a spectral formulation of the expected deviance can be obtained for lowdimensional problems under certain conditions. The effectiveness of the proposed method is demonstrated through three numerical examples that focus on analyzing the computational bias generated by the method, assessing the continuation method in the Bayesian inference with nonGaussian noise, and evaluating its ability to invert a multimodal parameter of interest.

17.11.2020, 16:00: Dr. Jonas Kiessling, RWTH Aachen and KTH.
 Title: Adaptive Random Fourier Features with Metropolis Sampling.

10.11.2020, 16:00: Dr. Alexander Litvinenko, RWTH Aachen University.
 Title: Solution of the densitydriven groundwater flow problem with uncertain porosity and permeability.
 Abstract: The pollution of groundwater, essential for supporting populations and agriculture, can have catastrophic consequences. Thus, accurate modeling of water pollution at the surface and in groundwater aquifers is vital. Here, we consider a densitydriven groundwater flow problem with uncertain porosity and permeability. Addressing this problem is relevant for geothermal reservoir simulations, natural salinedisposal basins, modeling of contaminant plumes and subsurface flow predictions. This strongly nonlinear timedependent problem describes the convection of a twophase flow, whereby a liquid flows and propagates into groundwater reservoirs under the force of gravity to form socalled ``fingers'’. To achieve an accurate numerical solution, fine spatial resolution with an unstructured mesh and, therefore, high computational resources are required. Here we run a parallelized simulation toolbox UG4 with a geometric multigrid solver on a parallel cluster, and the parallelization is carried out in physical and stochastic spaces. Additionally, we demonstrate how the UG4 toolbox can be run in a blackbox fashion for testing different scenarios in the densitydriven flow. As a benchmark, we solve the Elderlike problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities for the mass fraction. We use the solution obtained from the quasiMonte Carlo method as a reference solution.
 03.11.2020, 16:00: Dr. Lorenzo Tamellini, CNRIMATI Pavia.

Title: Uncertainty quantification and identifiability of SIRlike 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 SIRlike models, that are being commonly used when attempting to predict the trend of the COVID19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIRlike models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it nontrivial 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.

30.10.2020, 13:00: Sophia Franziska Wiechert, RWTH Aachen.
 Title: Continuous Time Markov Decision Processes with Finite Time Horizon.
 Abstract: One can derive a Markov Decision Process (MDP) by adding an input to a continuoustime 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 HamiltonJacobiBellman 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 discretetime 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.
 Title: Asymptoticpreserving multilevel Monte Carlo particle methods for diffusively scaled kinetic equations.

Abstract: In many applications it is necessary to compute the timedependent distribution of an ensemble of particles subject to transport and collision phenomena. Kinetic equations are PDEs that model such particles in a positionvelocity phase space. In the low collisional regime explicit particlebased Monte Carlo methods simulate these high dimensional equations efficiently, but, as the collision rate increases, these methods suffer from severe timestep constraints.
Asymptoticpreserving particle schemes are able to avoid these timestep 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 asymptoticpreserving 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)
 Title: Multilevel Double Loop Monte Carlo Method with Importance Sampling for Bayesian Optimal Experimental Design
 Abstract: An optimal experimental setup maximizes the value of data for statistical inferences. The efficiency of strategies for finding optimal experimental setups is particularly important for experiments that are timeconsuming 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 singlelevel 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)
 Title: Consistency analysis of datadriven 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 datadriven 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 closedform. 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 rareevent estimator based on the multilevel splitting algorithm. In its original form, the splitting algorithm cannot be applied to timeindependent problems because splitting requires an underlying continuoustime Markov process whose trajectories can be split. We embed the timeindependent problem within a continuoustime 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 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.