# MATH4UQ Seminar

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

• 28.09.2021, 16:00: Dr. Nadhir Ben Rached, RWTH Aachen University.
• Title: Efficient Importance Sampling for Large Sums of IID Random Variables
• Abstract: We discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $\mathbb{P}(\sum_{i=1}^{N}{X_i} \leq \gamma)$, via importance sampling (IS).  We are particularly interested in the rare event regime when  $N$ is large and/or $\gamma$ is small.  The exponential twisting is a popular technique for similar problems that, in most cases, compares favorably to other estimators. However, it has some limitations: i) it assumes the knowledge of the moment generating function of $X_i$ and ii) sampling under the new IS PDF is not straightforward and might be expensive. The aim of this work is to propose an alternative IS PDF that approximately yields, for certain classes of distributions and in the rare event regime,  at least the same performance as the exponential twisting technique and,  at the same time, does not introduce serious limitations. The first class includes distributions whose probability density functions (PDFs) are asymptotically equivalent, as $x \rightarrow 0$, to $bx^{p}$, for $p>-1$ and $b>0$. For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the rare event regime corresponding to small values of $\gamma$ and/or large values of $N$, the same performance of the estimator based on the use of the exponential twisting technique. In the second class, we consider the Log-normal setting, whose  PDF at zero vanishes faster than any polynomial, and we show numerically that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting IS PDF. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large $N$ and/or small $\gamma$

### Previous talks (2021)

• 21.09.2021, 16:00: Prof.Serge Prudhomme, Polytechnique Montréal.
• Title: Construction of reduced-order models for structural dynamics applications
• Abstract: The talk is concerned with the construction of proper-generalized decomposition formulations that preserve the structure and stability of the original system, such as its symplectic properties. The formulations are based on the  Hamiltonian and will be shown to be more stable than classical approaches. The formalism also allows one to define an optimization problem with constraints on the error in goal functionals in order to construct reduced models capable of delivering accurate approximation of quantities of interest. Numerical examples dealing with the dynamical behavior of beam structures will be presented in order to demonstrate the efficiency of the proposed approach.
• 27.07.2021, 15:00: Prof. Youssef Marzouk, MIT.
• Title: Density estimation and conditional simulation using triangular transport.
• Abstract: Triangular transformations of measures, such as the Knothe–Rosenblatt rearrangement, underlie many new computational approaches for density estimation and conditional simulation. This talk will discuss several aspects of such constructions. First we present new approximation results for triangular maps, focusing on analytic densities with bounded support. We show that there exist sparse polynomial approximations and deep ReLU network approximations of such maps that converge exponentially, where error is assessed on the pushforward distributions of such maps in a variety of distances and divergences. We also give an explicit a priori description of the polynomial ansatz space guaranteeing this convergence rate. Second, we discuss the problem of estimating a triangular transformation given a sample from a distribution of interest—and hence, transport-driven density estimation. We present a general functional framework for representing monotone triangular maps between distributions on unbounded domains, and analyze properties of maximum likelihood estimation in this framework. We demonstrate that the associated optimization problem is smooth and, under appropriate conditions, has no spurious local minima. This result provides a foundation for a greedy semi-parametric estimation procedure. Time permitting, we may also discuss a conditional simulation method that employs a specific composition of maps, derived from the Knothe–Rosenblatt rearrangement, to push forward a joint distribution to any desired conditional. We show that this composed-map approach reduces variability in conditional density estimates and reduces the bias associated with any approximate map representation. For context, and as a pointer to an interesting application domain, we elucidate links between conditional simulation with composed maps and the ensemble Kalman filter used in geophysical data assimilation. This is joint work with Ricardo Baptista (MIT), Olivier Zahm (INRIA), and Jakob Zech (Heidelberg).
• 20.07.2021, 16:00: Juan Pablo Madrigal Cianci,CSQI lab, EPFL.
• Title: Multi-level Markov chain Monte Carlo with maximally coupled proposals.
• Abstract:  In this work we present a novel class of Multi-Level Markov chain Monte Carlo (ML-MCMC) algorithms based on maximally coupled proposals, and apply them in the context of Bayesian inverse problems. In this context, the likelihood function involves a complex differential model, which is then approximated on a sequence of increasingly accurate discretizations. The key point of this algorithm is to construct highly coupled Markov chains together with the standard Multi-level Monte Carlo argument to obtain a better cost-tolerance complexity than a single level MCMC algorithm. Our approach generates these highly coupled chains by sampling from a maximal coupling of the proposals for each marginal Markov chain. By doing this, we are allowed to create novel ML-MCMC methods for which, contrary to previously used models, the proposals at each iteration can depend on the current state of this chain, while at the same time, creating chains that are highly correlated. The presented method is tested on an array of academic examples which evidence how our extended ML-MCMC method is robust when targeting some pathological posteriors, for which some of the previously proposed ML-MCMC algorithms fail.
• 13.07.2021, 16:00:  Prof. Dr. Helmut Harbrecht, Universität Basel.
• Title: Multilevel approximation of Gaussian random fields.
• Abstract: Centered Gaussian random fields are determined by their covariance operators. We consider such fields to be given sample-wise as variational  solutions to colouring operator equations driven by spatial white  noise, with pseudodifferential colouring operator being elliptic, self-adjoint and positive from the H\"ormander class. Especially, this  includes the Mat\'ern class of covariance operators. By using wavelet matrix compression, we prove that the precision and covariance operators,  respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number  independent of the dimension $N$ of this section. This results in several powerful algorithms like fast sampling, multilevel Monte Carlo oracles  for covariance estimation, and kriging. Besides the plain theory and related numerical experiments, we also present application driven  algorithm which extend the theoretical findings.
• 06.07.2021, 16:00:  Md Nurtaj Hossain, Indian Institute of Science, Bangalore.
• Title: Uncertainty quantification of dynamical systems using adaptively trained reduced order models.
• Abstract: Uncertainty quantification (UQ) of dynamical systems requires multiple solutions of the governing differential equation. Therefore, for a large dynamical system involving a million degrees of freedom, the task of UQ becomes computationally very expensive. This expensive task can be alleviated by replacing the large dynamical system with a computationally cheaper reduced order model (ROM), such as proper orthogonal decomposition (POD) based ROM. However, POD-based ROM suffers from a lack of robustness and accuracy in the absence of a diverse and large number of training points. To circumvent these issues, first, a posteriori error estimators are derived for the ROMs of both linear and nonlinear dynamical systems. Then these error estimators are used in conjunction with a modified greedy search algorithm to develop methods for adaptive training of POD-based ROMs. After adequate training, the adaptively trained ROMs are employed for UQ. Different numerical studies are performed to demonstrate the efficacy of the error estimators and developed algorithms. Usage of the adaptively trained ROMs in the estimation failure probability leads to a speed-up of more than two orders of magnitude.
• 29.06.2021, 16:00:   Dr.  Yoshihito Kazashi,  CSQI, École polytechnique fédérale de Lausanne.
• Title: Density estimation in RKHS with application to Korobov spaces in high dimensions.
• Abstract: In this talk, we will consider a kernel method for estimating a probability density function (pdf) from an i.i.d. sample drawn from such density. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. We will present an error analysis for the mean integrated squared error in a general reproducing kernel Hilbert space setting. We will discuss how this theory can be applied to estimate pdfs belonging to weighted Korobov spaces, and show a dimension independent convergence rate. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate.
This is joint work with Fabio Nobile (CSQI, EPFL).
• 22.06.2021, 16:00:  Prof. Pierre Del Moral, INRIA, France.
• Title: Stability and uniform fluctuation analysis of Ensemble Kalman-Bucy filters.
• Abstract: This talk is concerned with the long time behavior of particle filters and Ensemble Kalman filters. These filters can be interpreted as mean field type particle interpretation of the filtering equation and the Kalman recursion.  We present a series of old and new results on the stability properties of these filters. We initiate a comparison between these particle samplers and discuss some open research questions.
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• 15.06.2021, 16:00:  Prof. Ahmed Kebaier, University Sorbonne Paris Nord (USPN).
• Title: Approximation of Stochastic Volterra Equations.
• Abstract: In this talk, we present a multi-factor approximation for Stochastic Volterra Equations with Lipschitz coefficients and kernels of completely monotone type that may be singular.  Our approach consists in truncating and then discretizing the integral defining the kernel, which corresponds to a classical Stochastic Differential Equation. We prove strong convergence results for this approximation. For the particular rough kernel case with Hurst parameter lying in $(0,1/2)$, we propose various discretization procedures and give their precise rates of convergence. We illustrate the efficiency of our approximation schemes with numerical tests for the rough Bergomi model.
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• 08.06.2021, 16:00:  Loris Di Cairano, PhD student at RWTH Aachen University and FZ Jülich.
• Title: An Effective Continuum Model for studying Protein Diffusion in Lipid Membrane.
• Abstract:  A model for describing protein diffusion in lipid membrane starting from first principles is presented. The lipid membrane is modelled as a linear viscoelastic fluid (Stokes equation with an explicit time-dependence in the Cauchy's stress tensor), the protein by a sphere (First Euler’s law) and, finally, a constitutive equation, namely, a relation between the Cauchy’ stress tensor and strain tensor and it describes the response of the fluid to a given deformation. Thus, the common model for linear viscoelastic fluid is to consider the instantaneous and non instantaneous responses of fluid in an additive way and the latter is model by an appropriate time-dependent response function. Through an elimination procedure of the membrane degrees of freedom (one formally solves the membrane Stokes’ equation), one gets an only equation for the protein degrees of freedom which takes the form of a GLE. The kernel function entering the GLE is exactly the response function appearing in the Cauchy stress tensor. B exploiting methods used in the literature, one can solve the GLE, namely, write an analytical solution for the correlation function (for example, mean squared displacement) in terms of the kernel function in GLE. By choosing a 3-paramenter Mittag-Leffler, one can compare MD data with model obtaining a remarkable agreement.
• 01.06.2021, 16:00:  Dr.  Michael F. Herbst, RWTH Aachen University.
• Title: Uncertainty quantification in electronic-structure theory: Status and directions for future research.
• Abstract: The aim of electronic structure theory is to describe the behaviour of electrons in matter on a quantum-mechanical level. On the one hand related to a rich set of applications in chemistry, pharmacy and materials science, the field is on the other hand a fruitful source of numerically and mathematically interesting problems. One aspect that is only barely explored so far is the question of estimating error and uncertainties in simulation results. In practice this means that people select numerical parameters and physical models based on prior experience and rely on their physical intuition to verify the reliability of obtained results. However, this approach does not properly scale to the recently emerging high-throughput approaches where tens of thousands (or more) chemical systems need to be tackled in a computational study. More rigorous mathematical approaches to understand errors and uncertainties are therefore desperately needed to increase efficiency and automatisability. In my talk I will sketch the physical and numerical procedures used in the field and I will summarise the currently employed ideas for error estimation and uncertainty quantification. I will conclude my talk with some open questions and angles of research where input from uncertainty quantification would be needed.
• 25.05.2021, 16:00:  Prof. Ajay Jasra, CEMSE, KAUST.
• Title: On unbiased score estimation for partially observed diffusions.
• Abstract: We consider the problem of statistical inference for a class of partially-observed diffusion processes, with discretely-observed data and finite-dimensional parameters. We construct unbiased estimators of the score function, i.e. the gradient of the log-likelihood function with respect to parameters, with no time-discretization bias. These estimators can be straightforwardly employed within stochastic gradient methods to perform maximum likelihood estimation or Bayesian inference. As our proposed methodology only requires access to a time-discretization scheme such as the Euler-Maruyama method, it is applicable to a wide class of diffusion processes and observation models. Our approach is based on a representation of the score as a smoothing expectation using Girsanov theorem, and a novel adaptation of the randomization schemes developed in Mcleish [2011], Rhee and Glynn [2015], Jacob et al. [2020a]. This allows one to remove the time-discretization bias and burn-in bias when computing smoothing expectations using the conditional particle filter of Andrieu et al. [2010]. Central to our approach is the development of new couplings of multiple conditional particle filters. We prove under assumptions that our estimators are unbiased and have finite variance. The methodology is illustrated on several challenging applications from population ecology and neuroscience.
• 18.05.2021, 16:00:  Prof. Dr. Elisabeth Ullmann, Department of Mathematics, Technical University of Munich.
• Title: Rare event estimation with PDE-based 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).

• 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 large-scale 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 Cottbus-Senftenberg.
• 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 Log-Likelihood Gradient for a Class of Continuous-Time State-Space Models
• Abstract: We consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the log-likelihood 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 gradient-based 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 'model-free' distribution learning using so-called generative neural networks. In particular, normalizing flows have proven effective as they are trained through direct log-likelihood 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 Bretagne-Atlantique, 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 so-called 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 rare-event 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 rare-event 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 non-Markovian, 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 forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep learning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian 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: Multi-Iteration 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. Multi-Iteration Stochastic Optimizers are a novel class of first-order stochastic optimizers where the gradient is estimated using the Multi-Iteration 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 SGD-MICE optimizer converges linearly in the class of strongly-convex and L-smooth functions. The performances of multi-iteration stochastic optimizers are evaluated in numerical examples, validating our analysis, and converging faster than well-known 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 time-discretization 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), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized 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 non-Markovian 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 second-order 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 so-called continuous-time Infinite Swapping algorithm, which found  its origins in the molecular dynamics community, and can be understood as  the continuous-time limit of a Parallel Tempering algorithm with state-dependent swapping rates.  In the current work, we extend this idea to the context of time-discrete Markov chains and present two Markov chain Monte Carlo algorithms that follow the same paradigm as the   continuous-time 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.  Abdul-Lateef Haji-Ali, Assistant professor, Heriot-Watt 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 d-dimensional 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 non-trivial (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, CNR-IMATI, Pavia, Italy.
• Title: Comparing Multi-Index 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 Multi-Index Stochastic Collocation (MISC) and multi-fidelity Stochastic Radial Basis Functions (SRBF) surrogates is assessed for the UQ of a roll-on/roll-off 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 (model-scale) resistance is presented and discussed.  Both methods need to repeatedly solve the free-surface Navier-Stokes equations for different configurations of the operational parameters.  The required CFD simulations are obtained by a multi-grid Reynolds Averaged Navier-Stokes (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: Multi-index 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 multi-index 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 non-standard domains is that the basis functions that describe the domain geometry, typically standard B-splines or non-uniform rational B-splines (NURBS), are used as the basis in approximating the PDE solution on the domain. We numerically demonstrate the computational efficiency of IGA-based 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 stress-lifetime (S-N) models using fatigue experiments on unnotched specimens. To generate accurate fatigue life predictions, competing S-N models are ranked according to several classical information-based 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 S-N models to fatigue crack initiation in general geometries. The model is based on a spatial Poisson process with an intensity function that combines the S-N 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 two-scale diffusion processes. We set ourselves in a semi-parametric framework and fit to data a single-scale equation of the overdamped Langevin type. If data is given in the form of a continuous time series, a pre-processing 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 low-pass 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 Non-smooth Problems
• Abstract: Sampling based variance reduction techniques, such as multilevel Monte Carlo methods, have been established as a general-purpose procedure for quantifying uncertainties in computational models. It is known however, that these techniques may not provide performance gains when there is a non-smooth 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 layer-by-layer 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 data-driven modelbased on parametric Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of wind power forecast errors. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent 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 integro-differential 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 log-likelihood. We show also that a spectral formulation of the expected deviance can be obtained  for low-dimensional 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 non-Gaussian 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 density-driven 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 density-driven groundwater flow problem with uncertain porosity and permeability. Addressing this problem is relevant for geothermal reservoir simulations, natural saline-disposal basins, modeling of contaminant plumes and subsurface flow predictions.  This strongly nonlinear time-dependent problem describes the convection of a two-phase flow, whereby a liquid flows and propagates into groundwater reservoirs under the force of gravity to form so-called 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 black-box fashion for testing different scenarios in the density-driven flow.  As a benchmark, we solve the Elder-like 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 quasi-Monte Carlo method as a reference solution.
• 03.11.2020, 16:00: Dr. Lorenzo Tamellini, CNR-IMATI Pavia.
• Title: 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.
• 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 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.
• Title: 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)
• Title: 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)
• Title: 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 Ström, 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.