### Workshop A7 - Stochastic Computation

**Organizers:** Tony Lelièvre (École Nationale des Ponts et Chaussées, France) - Arnulf Jentzen (ETH Zürich, Switzerland)

## Talks

July 10, 14:30 ~ 15:20

## Weak order analysis for SPDEs

### Arnaud Debussche

### ENS Rennes, France - arnaud.debussche@ens-rennes.fr

The numerical analysis of Stochastic Partial Differential Equations has known a lot of progress. The study of the weak order involves a lot of difficulties and started to be understood only recently. In this talk, I will present a new result of regularity for the solution of the Kolmogorov equations associated to a SPDE with a nonlinear diffusion coefficient and a Burger's type nonlinearity. This allows a complete study of the weak order of a standard semi implicit Euler scheme. I will explain why Malliavin calculus and two sided stochastic integrals are useful.

Joint work with Charles-Edouard Bréhier (Université de Lyon 1).

July 10, 15:30 ~ 15:55

## Weak convergence rates for stochastic partial differential equations with nonlinear diffusion coefficients

### Ryan Kurniawan

### ETH Zurich, Switzerland - ryan.kurniawan@sam.math.ethz.ch

Strong convergence rates for numerical approximations of semilinear stochastic partial differential equations (SPDEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for numerical approximations of such SPDEs have been investigated for about two decades and are still not yet fully understood. In particular, it has been an open problem to establish essentially sharp weak convergence rates for numerical approximations of space-time white noise driven SPDEs with nonlinear multiplication operators in the diffusion coefficients. In this talk we overcome this weak convergence problem. In particular, we establish essentially sharp weak convergence rates for numerical approximations of the continuous version of the parabolic Anderson model. Key ingredients of our approach are applications of the mild Ito type formula in UMD Banach spaces with type 2.

Joint work with Daniel Conus (Lehigh University, USA), Mario Hefter (TU Kaiserslautern, Germany) and Arnulf Jentzen (ETH Zurich, Switzerland).

July 10, 16:00 ~ 16:25

## Weak error analysis via functional Itô calculus

### Felix Lindner

### University of Kaiserslautern, Germany - lindner@mathematik.uni-kl.de

Weak errors of numerical approximations of SDEs are relatively well understood in the case where the considered functional of the solution process depends only on an evaluation of the solution at a given time $T$. In contrast, the number of available results in the literature on weak convergence rates for approximations of path-dependent functionals of SDEs is quite limited. In this talk, I present a new approach to analyzing weak approximation errors for path-dependent functionals of SDEs, based on tools from functional Ito calculus such as the functional Itô formula and functional Kolmogorov equation. It leads to a general representation formula for weak errors of the form $\mathbb{E}(f(X)-f(\tilde{X}))$, where $X$ and $\tilde{X}$ are the solution process and its approximation and the functional $f:C([0,T],\mathbb{R}^d)\to\mathbb{R}$ is assumed to be sufficiently regular. The representation formula can be used to derive explicit convergence rates, such as rate $1$ for the linearly time-interpolated explicit Euler method. Finally, I will outline how the presented approach can be extended to numerical methods for mild solutions of SPDEs.

Joint work with Mihály Kovács (Chalmers University of Technology, Sweden) and Saeed Hadjizadeh (University of Kaiserslautern, Germany).

July 10, 17:00 ~ 17:25

## On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

### Diyora Salimova

### ETH Zurich, Switzerland - diyora.salimova@sam.math.ethz.ch

In this talk we show that for every arbitrarily slow convergence speed and every natural number $d \in \{ 2, 3, ... \}$ there exist stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence.

Joint work with Máté Gerencsér (Institute of Science and Technology, Austria) and Arnulf Jentzen (ETH Zurich, Switzerland).

July 10, 17:30 ~ 17:55

## Approximation of BSDEs using random walk

### Christel Geiss

### University of Jyväskylä, Finland - christel.geiss@jyu.fi

For the FBSDE \[ X_t = x + \int_0^t b(r,X_r)dr + \int_0^t \sigma (r,X_r)dB_r \] \[ Y_t = g(X_T) + \int_t^T f(s, X_s,Y_s,Z_s)ds - \int_t^T Z_s dB_s, \,\,\,\, 0\le t \le T \]

Briand, Delyon and Memin have shown in [1] a Donsker-type theorem: If one approximates the Brownian motion $B$ by a random walk $B^n$, the according solutions $(X^n, Y^n,Z^n)$ converge weakly to $(X, Y, Z).$ \\ We investigate under which conditions $(Y^n_t,Z^n_t)$ converges to $(Y_t,Z_t)$ in $L_2$ and compute the rate of convergence in dependence of the H\"older continuity of the terminal condition function $g.$ \\

[1] P. Briand, B. Delyon, J. Memin, {\it Donsker-Type theorem for BSDEs.} Electron. Comm. Probab. 6, 1 -- 14 (2001).

Joint work with C\'eline Labart (Universit\'e de Savoie, France) and Antti Luoto (University of Jyv\"askyl\"a, Finland).

July 10, 18:00 ~ 18:25

## Lower Error Bounds for Strong Approximation of Scalar SDEs with Non-Lipschitzian Coefficients

### Thomas Mueller-Gronbach

### University of Passau, Germany - thomas.mueller-gronbach@uni-passau.de

We study the problem of pathwise approximation of the solution of a scalar SDE, either at the final time or globally in time, based on $n$ sequential evaluations of the driving Brownian motion on average. We present lower error bounds in terms of $n$ under mild local smoothness assumptions on the coefficients of the SDE. This includes SDEs with superlinearly growing or piecewise Lipschitz continuous coefficients and also certain types of CIR-processes.

Joint work with Mario Hefter (Technical University of Kaiserslautern, Germany) and Andre Herzwurm (Technical University of Kaiserslautern, Germany).

July 11, 14:30 ~ 14:55 - Room B2

## Beyond Well-Tempered Metadynamics algorithms for sampling multimodal target densities

### Gersende Fort

### CNRS, IMT (Univ. Toulouse), France - gersende.fort@math.univ-toulouse.fr

In many situations, sampling methods are considered in order to compute expectations of given observables with respect to a distribution $\pi \, \mathrm{d} \lambda$ on $\mathsf{X} \subseteq \mathbb{R}^D$, when $\pi$ is highly multimodal. Free-energy based adaptive importance sampling techniques have been developed in the physics and chemistry literature to efficiently sample from such a target distribution: the auxiliary distribution $\pi_\star \, \mathrm{d} \lambda$ from which the samples are drawn, is defined, given a partition $\{\mathsf{X}_i, i \leq d \}$ of $\mathsf{X}$, as a local biasing of the target $\pi$ such that each element $\mathsf{X}_i$ has the same weight under $\pi_\star \, \mathrm{d} \lambda$. These methods are casted in the class of adaptive Markov chain Monte Carlo (MCMC) samplers since the local biasing is unknown: it is therefore learnt on the fly and the importance function evolves along the run of the sampler. As usual with importance sampling, expectations with respect to $\pi$ are obtained from a weighted mean of the samples returned by the sampler.

Examples of such approaches are Wang-Landau algorithms, the Self-Healing Umbrella Sampling, adaptive biasing forces methods, the metadynamic algorithm or the well-tempered metadynamics algorithm. Nevertheless, the main drawback of mots of these methods is that two antagonistic phenomena are in competition: on one hand, to overcome the multimodality issue, the sampler is forced to visit all the strata $\{\mathsf{X}_i, i \leq d \}$ equally; on the other hand, the algorithm spends the same time in strata with high and low weight under $\pi \, \mathrm{d} \lambda$ which makes the Monte Carlo approximation of expectations under $\pi \mathrm{d} \lambda$ quite inefficient.

We present a new algorithm, which generalizes all the examples mentioned above: this novel algorithm is designed to reduce the two antagonistic effects. We will show that the estimation of the local bias can be seen as a Stochastic Approximation algorithm with random step-size sequence; and the sampler as an adaptive MCMC method. We will analyze its asymptotic behavior and discuss numerically the role of some design parameters.

Joint work with Benjamin Jourdain (ENPC, France), Tony Lelièvre (ENPC, France) and Gabriel Stöltz (ENPC, France).

July 11, 15:00 ~ 15:25 - Room B2

## Self-repelling processes and metadynamics

### Pierre-André Zitt

### Université Paris Est Marne la Vallée, France - Pierre-Andre.Zitt@u-pem.fr

A usual drawback of Markov Chain Monte Carlo algorithms is their inherent difficulty to overcome potential barriers, which may lead to a poor exploration of the sampling space, and large sampling errors. The "metadynamics" algorithm introduced by Bussi, Laio and Parrinello in the 00s exemplifies one of the ideas to tackle this difficulty: by keeping track of the past trajectory of the sampling process, one can use it to bias the process so that it avoids the regions it has already visited, leading to a better sampling. The processes describing the evolution of the algorithm turn out to be quite difficult to analyze rigorously. We present two toy models that are amenable to such an analysis, using results from the self-interacting processes literature.

Joint work with Benjamin Jourdain (École des Ponts ParisTech) and Tony Lelièvre (École des Ponts ParisTech).

July 11, 15:30 ~ 16:20 - Room B2

## Competing sources of variance reduction in parallel replica Monte Carlo, and optimization in the low temperature limit

### Paul Dupuis

### Brown University, USA - dupuis@dam.brown.edu

Computational methods such as parallel tempering and replica exchange are designed to speed convergence of more slowly converging Markov processes (corresponding to lower temperatures for models from the physical sciences), by coupling them through a Metropolis type swap mechanism with higher temperature processes that explore the state space more quickly. It has been shown that the sampling properties are in a certain sense optimized by letting the swap rate tend to infinity. This ``infinite swapping limit'' can be realized in terms of a process which evolves using a symmetrized version of the original dynamics, and then one produces approximations to the original problem by using a weighted empirical measure. The weights are needed to transform samples obtained under the symmetrized dynamics into distributionally correct samples for the original problem.

After reviewing the construction of the infinite swapping limit, we focus on the sources of variance reduction which follow from this construction. As will be discussed, some variance reduction follows from a lowering of energy barriers and consequent improved communication properties. A second and less obvious source of variance reduction is due to the weights used in the weighted empirical measure that appropriately transform the samples of the symmetrized process. These weights are analogous to the likelihood ratios that appear in importance sampling, and play much the same role in reducing the overall variance. A key question in the design of the algorithms is how to choose the ratios of the higher temperatures to the lowest one. As we will discuss, the two variance reduction mechanisms respond in opposite ways to changes in these ratios. One can characterize in precise terms the tradeoff and explicitly identify the optimal temperature selection for certain models when the lowest temperature is sent to zero, i.e., when sampling is most difficult.

Joint work with Jim Doll, Guo-Jhen Wu and Michael Snarski (Brown University, USA).

July 11, 17:00 ~ 17:25 - Room B2

## Modeling aggregation processes of Lennard-Jones particles via stochastic networks

### Maria Cameron

### University of Maryland, USA - cameron@math.umd.edu

An isothermal aggregation process of particles/atoms interacting according to the Lennard-Jones pair potential is modeled by mapping the energy landscapes of each cluster size N onto stochastic networks, computing transition probabilities for the network for an N-particle cluster to the one for N + 1, and connecting these networks into a single joint network. The attachment rate is a control parameter. The resulting network representing the aggregation and dynamics of up to 14 Lennard-Jones particles contains 6417 vertices. It is not only time-irreversible but also reducible. To analyze its transient dynamics, we introduce the sequence of the expected initial and pre- attachment distributions and compute them for a wide range of attachment rates and three values of temperature. As a result, we find the most likely to observe configurations in the process of aggregation for each cluster size. We examine the attachment process and conduct a structural analysis of the sets of local energy minima for every cluster size. We show that both processes taking place in the network, attachment and relaxation, lead to the dominance of icosahedral packing in small (up to 14 atom) clusters.

Joint work with Yakir Forman (Yeshiva University, USA).

July 11, 17:30 ~ 17:55 - Room B2

## A coupling approach to the kinetic Langevin equation

### Andreas Eberle

### University of Bonn, Germany - eberle@uni-bonn.de

The (kinetic) Langevin equation is an SDE with degenerate noise that describes the motion of a particle in a force field subject to damping and random collisions. It is also closely related to Hamiltonian Monte Carlo methods. An important question is, why in certain cases kinetic Langevin diffusions seem to approach equilibrium faster than overdamped Langevin diffusions.

So far, convergence to equilibrium for kinetic Langevin diffusions has almost exclusively been studied by analytic techniques. In this talk, I present a new probabilistic approach that is based on a specific combination of reflection and synchronuous coupling of two solutions of the Langevin equation. The approach yields rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime, and it may help to shed some light on the open question mentioned above.

Joint work with Arnaud Guillin (Université Blaise Pascal, Clermont-Ferrand, France) and Raphael Zimmer (Universität Bonn, Germany).

July 11, 18:00 ~ 18:25 - Room B2

## The Complexity of Best-Arm Identification

### Aurélien Garivier

### Université de Toulouse, France - aurelien.garivier@math.univ-toulouse.fr

We consider the problem of finding the highest mean among a set of probability distributions that can be sampled sequentially. We provide a complete characterization of the complexity of this task in simple parametric settings: we give a tight lower bound on the sample complexity, and we propose the 'Track-and-Stop' strategy, which we prove to be asymptotically optimal. This algorithm consists in a new sampling rule (which tracks the optimal proportions of arm draws highlighted by the lower bound) and in a stopping rule named after Chernoff, for which we give a new analysis.

Joint work with Emilie Kaufmann (CNRS, team CRIStAL, France).

July 11, 18:30 ~ 18:55 - Room B2

## Fluctuation Analysis of Fleming-Viot Particle Systems

### Arnaud Guyader

### UPMC - Paris VI, France - arnaud.guyader@upmc.fr

The distribution of a Markov process with killing, conditioned to be still alive at a given time, can be approximated by a Fleming-Viot type particle system. In such a system, each particle is simulated independently according to the law of the underlying Markov process, and branches onto another particle at each killing time. The consistency of this method in the large population limit was the subject of several recent works. The purpose of this talk is to present a central limit theorem for the law of the Fleming-Viot particle system at a given time. We will illustrate this result on an application in molecular dynamics.

Joint work with Frederic Cerou (INRIA, France), Bernard Delyon (University of Rennes, France) and Mathias Rousset (INRIA, France).

July 12, 14:30 ~ 14:55 - Room B2

## Multilevel Monte Carlo for SDEs with Random Bits

### Klaus Ritter

### TU Kaiserslautern, Germany - ritter@mathematik.uni-kl.de

Let $X$ be the solution to a stochastic differential equation (SDE), and let $\varphi$ be a real-valued functional on the path space. We study the approximation of the expectation of $\varphi(X)$ by means of randomized algorithms that may only use random bits. We provide upper and lower bounds on the complexity of the problem, as well as a multilevel algorithm that achieves the upper bound.

Joint work with M. Giles (Oxford, UK), M. Hefter (Kaiserslautern, Germany) and L. Mayer (Kaisers\-lautern, Germany).

July 12, 15:00 ~ 15:25 - Room B2

## Strong convergence properties of the Ninomiya Victoir scheme and applications to multilevel Monte Carlo

### Benjamin Jourdain

### University Paris-Est, Ecole des Ponts, CERMICS, France - benjamin.jourdain@enpc.fr

We prove that the strong convergence rate of the Ninomiya-Victoir scheme is 1/2. The normalized error converges stably to the solution of an affine SDE with a source term involving the commutators between the Brownian vector fields. When the Brownian vector fields commute, this limit vanishes and we show that the strong convergence rate improves to 1. We also show that averaging the order of integration of the Brownian fields leads to a coupling with strong order 1 with the scheme proposed by Giles and Szpruch (2014) in order to achieve the optimal complexity in the multilevel Monte Carlo method. Last, we are interested in the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme. We prove that this error converges with strong order 2 so that the convergence properties of our multilevel estimators are preserved when an explicit Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order 5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the same order of strong convergence.

Joint work with Anis Al Gerbi (Ecole des Ponts, CERMICS) and Emmanuelle Clément (Ecole Centrale Paris, MICS).

July 12, 15:30 ~ 15:55 - Room B2

## On the approximation of pathdependent BSDEs driven by the Brownian motion

### Stefan Geiss

### Department of Mathematics and Statistics, University of Jyväskylä, Finland, Finland - stefan.geiss@jyu.fi

In this talk we present recent results about the $L_p$-approximation of the $Y$-process of path-dependent quadratic and sub-quadratic backwards stochastic differential equations driven by the Brownian motion of the form \[ Y_t = \xi + \int_t^T f(s,Y_s,Z_s) ds - \int_t^T Z_s dB_s. \] In this approximation we use adapted time-nets where the adaptation is based on certain quantitative properties of the initial data $(\xi,f)$ (the terminal condition and the generator) that are related to differential properties (in the Malliavin sense) of $(\xi,f)$. The talk is mainly based on [1].

[1] S. Geiss and J. Ylinen: Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs. ArXiv:1409.5322v3

Joint work with Juha Ylinen (University of Jyväskylä).

July 12, 16:00 ~ 16:25 - Room B2

## Existence, uniqueness, and numerical approximation for stochastic Burgers equations

### Sara Mazzonetto

### Université de Lille 1, France - sara.mazzonetto@polytech-lille.fr

The talk is about a recently introduced explicit full-discrete numerical approximation scheme for some stochastic partial differential equations with additive noise and non-globally Lipschitz continuous nonlinearities. The scheme allows to prove simultaneously existence and uniqueness of the mild solution and strong convergence of the numerical approximation for some classes of equations, e.g. the stochastic Burgers equation with additive white noise.

Joint work with Arnulf Jentzen and Diyora Salimova (ETH Zürich).

July 12, 17:00 ~ 17:25 - Room B2

## Recent advances on stochastic methods in data science

### Sotirios Sabanis

### University of Edinburgh, UK - S.Sabanis@ed.ac.uk

Some recent advances on stochastic algorithms in data science will be discussed.

July 12, 17:30 ~ 17:55 - Room B2

## The stability of stochastic gradient descent

### Benjamin Recht

### University of California, Berkeley, USA - brecht@berkeley.edu

The most widely used optimization method in machine learning practice is the Stochastic Gradient Method (SGM). This method has been used since the fifties to build statistical estimators, iteratively improving models by correcting errors observed on single data points. SGM is not only scalable, robust, and simple to implement, but achieves the state-of-the-art performance in many different domains. In contemporary systems, SGM powers enterprise analytics systems and is the workhorse tool used to train complex pattern-recognition systems in speech and vision.

In this talk, I will explore why SGM has had such staying power, focusing on the notion of generalization. I will show that any model trained with a few SGM iterations has vanishing generalization error and performs as well on unseen data as on the training data. The analysis will solely employ elementary tools from convex and continuous optimization. Applying the results to the convex case provides new explanations for why multiple epochs of stochastic gradient descent generalize well in practice, and give new insights into minibatch sizes in SGM. In the nonconvex case, I will describe a new interpretation of common practices in neural networks, and provide a formal rationale for stability-promoting mechanisms in training large, deep models.

Joint work with Moritz Hardt (Google Brain/UC Berkeley), Yoram Singer (Google Brain).

July 12, 18:00 ~ 18:25 - Room B2

## Stochastic Composite Least-Squares Regression with convergence rate $O(1/n)$

### Francis Bach

### INRIA - Ecole Normale Supérieure, France - francis.bach@inria.fr

We consider the minimization of composite objective functions composed of the expectation of quadratic functions and an arbitrary convex function. We study the stochastic dual averaging algorithm with a constant step-size, showing that it leads to a convergence rate of $O(1/n)$ without strong convexity assumptions. This thus extends earlier results on least-squares regression with the Euclidean geometry to (a) all convex regularizers and constraints, and (b) all geometries represented by a Bregman divergence. This is achieved by a new proof technique that relates stochastic and deterministic recursions.

Joint work with Nicolas Flammarion.

## Posters

## Numeric for Stochastic State-Dependent Delay Differential Equations

### Bahareh Akhtari

### Assistant Professor, Institute for Advanced studies in Basic Sciences (IASBS), Iran - b.akhtari@iasbs.ac.ir

The stochastic equations of delay-type, including system memory, describe the model in a more accurate manner. The numerical solution for stochastic delay differential equations (SDDEs) have been relatively adequate discussed in the recent years. But all works deal with the case in which delay term is constant or time-dependent. In this poster, we consider a new one: ${\bf{State-dependent}}$. Under the sufficiently smooth conditions on drift and diffusion coefficients, a new interpolation based on split-step scheme for approximating history in the presence of state-dependent delay developed and then mean-square convergence of the scheme investigated.

## Strong Stability in Phase-Type Queueing Systems

### Yasmina Djabali

### Research Unit LaMOS, University of Bejaia, Algeria - dj.mina06@yahoo.fr

Abstract. Phase-type queueing systems are used to approximate queues with general distributions. In this work, we provide the justification of this approximation method by employing the strong stability method. We prove the robustness of the underlying Markov chain and estimate an upper bound of the deviation of the stationary vector, resulting from the perturbation of the inter-arrival distribution of GI/M/1 queueing system. We provide numerical examples and compare the perturbation bounds obtained in this paper with the estimates of the real deviation of the stationary vector obtained by simulation. Keywords: Queueing systems, Markov chain, Phase-type distributions, Perturbation, Strong stability.

Joint work with Rabta Boualem (Research Unit LaMOS, University of Bejaia, Algeria) and Aïssani Djamil (Research Unit LaMOS, University of Bejaia, Algeria).

## Performance Modeling of Finite-Source priority queue with vacations via Generalized Stochastic Petri Nets

### Sedda Hakmi

### Research Unit LaMOS (Modeling and Optimization of Systems),University of Bejaia., Algeria - sed.hakmi@gmail.com

Abstract. Our study deals with performance modeling and analysis of finite-source non-preemptive priority queue with vacations via Generalized Stochastic Petri Nets (GSPN). Indeed, we considered two classes of non-preemptive priority: high priority and low priority request. The service times and the inter-arrival times are assumed to be exponentially distributed. The corresponding GSPN model of the studied system provided us a formal method for generating Markov Chain (MC) which allowed us to develop the formulas of the main stationary performance measures. Therefore, we provide an exact analysis for the performance indices of both requests classes. Through numerical examples, we discuss the impact of vacation and arrival rates on the network performances.

Keywords: Generalized Stochastic Petri Nets, Modeling, Performance Evaluation, Priority Queue.

Joint work with Ouiza Lekadir (Research Unit LaMOS (Modeling and Optimization of Systems),University of Bejaia, Algeria) and Djamil Aïssani (Research Unit LaMOS (Modeling and Optimization of Systems),University of Bejaia, Algeria).

## Comparison study for random PDE optimization problems based on different matching functionals

### Hyung-Chun Lee

### Ajou University, South Korea - hclee@ajou.ac.kr

In this poster, we consider an optimal control problem for a partial differential equation with random inputs. To determine an applicable deterministic control $\hat{f}(x)$, we consider the four cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility.

Joint work with Max D. Gunzburger (Florida State University, USA).

## Coupling sample paths to the partial thermodynamic limit in stochastic chemical reaction networks

### Ethan Levien

### University of Utah, United States - levien@math.utah.edu

Many stochastic biochemical systems have a multiscale structure so that they converge to piecewise deterministic Markov processes in a thermodynamic limit. While information about stochastic fluctuations is lost in the thermodynamic limit, studying the exact process is computationally expensive for most systems of interest. We present a new technique for accelerating the convergence of Monte Carlo estimators of the exact process that makes use of a probabilistic coupling between the exact process and the thermodynamic limit. In addition to rigorous results concerning the asymptotic computational complexity of our method, we apply our method to study various models of gene expression.

Joint work with Paul C. Bressloff (University of Utah).

## Comparison of the Euler-Maruyama and backward Euler methods for neutral stochastic differential equations with time-dependent delay

### Marija Milosevic

### University of Nis, Faculty of Science and Mathematics, Serbia - 27marija.milosevic@gmail.com

Numerical solutions to a class of neutral stochastic differential equations with time-dependent delay are considered. Comparison of the explicit Euler-Maruyama and implicit backward Euler methods is presented with particular emphasis on the degree to which the numerical solutions inherit certain properties of the exact solution. The influences of the neutral term and time-dependent delay on the analysis of numerical solutions are stressed.

## Stochastic Image Processing with an Application

### Gamze OZEL KADILAR

### Hacettepe University, Turkey - gamzeozl@hacettepe.edu.tr

Stochastic computation (SC) is important for the statistical nature of application-level performance metrics. The main benefit of SC is that complicated arithmetic computation can be performed by simple logic circuits since numbers are represented in SC by random bit-streams that are interpreted as probabilities. SC is categorized as an approximate computing technique. In this study, first we give a detailed discussion of the basic concepts of SC including accuracy and correlation. It also surveys the history of the field and highlights SC’s applications. Then, in this study, we provide the application of SC to image processing. We demonstrate the design of many representative image processing circuits and compare them to conventional binary counterparts.

Joint work with Cem Kadilar (Hacettepe University, Turkey).

## Minimisation of relative entropy to efficiently capture the macro-scale behaviour of stochastic systems

### Przemyslaw Zielinski

### KU Leuven, Belgium - przemyslaw.zielinski@kuleuven.be

The purpose of this research is to develop, study and implement a micro-macro method to simulate observables of stiff SDEs. The technique exploits the separation between the fast time scale on which trajectories advance and the slow evolution of observables, by combining short bursts of paths simulation with extrapolation of a number of macroscopic states forward in time.

In the crucial step of the algorithm, we aim to obtain, after extrapolation, a new ensemble of particles/replicas compatible with given macroscopic states. To address and regularise this inference problem, we introduce the matching operator based on the minimisation of suitable distance between probability distributions — logarithmic relative entropy.

Particularly, the ongoing study focuses on establishing the convergence in the numerically weak sense, inquiring about the stability for appropriately chosen test models, and providing a convenient numerical approach. It also deals with the generic properties of the matching operator that allow to adopt other information theoretic distances besides the logarithmic entropy.

Joint work with Kristian Debrabant (University of Southern Denmark, Denmark), Tony Lelievre (Ecole des Ponts ParisTech, France) and Giovanni Samaey (KU Leuven, Belgium).

## Inverse Gamma Kernel Estimators Using Two Multiplicative Bias Correction Methods

### Nabil Zougab

### Research Unit LaMOS, University of Bejaia, Ageria - nabilzougab@yahoo.fr

Abstract. Two multiplicative bias correction (MBC) approaches for nonparametric kernel estimators based on inverse gamma (IG) distribution for probability density function in the context of nonnegative supported data are proposed. We show some properties of the MBC-IG kernel estimators like bias, variance and mean integrated square error. A simulation study and an application on a real data set illustrate the performance of the MBC estimators based on the IG kernel in terms of the integrated squared error and integrated squared bias.

Keywords: Inverse gamma kernel, Integrated square error, Multiplicative bias correction, Squared bias.

Joint work with Harfouche Lynda (Research Unit LaMOS, University of Bejaia, Ageria).