Session I.7 - Stochastic Computation
Talks
Monday, June 12, 14:00 ~ 14:30
Behavior of solutions to the 1D focusing stochastic $L^2$-critical and supercritical nonlinear Schrödinger equation
Annie Millet
University Paris 1, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white or space colored noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting, nor is the mass (i.e. $L^2$-norm) conserved in the additive case.
For a space-time white noise, we show that the noise may induce blow-up, thus, ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings. We obtain profiles and rates of the blow-up solutions.
For a space-correlated noise, we investigate both theoretically and numerically how the energy is affected by various types of space correlation, and its dependence on the discretization parameters and the schemes. We then perform numerical investigation of the noise influence on the global dynamics, measuring the probability of blow-up versus scattering behavior depending on parameters of correlation kernels. Finally, we study numerically the effect of the spatially correlated noise on the blow-up behavior.
We conclude that when blow-up occurs, such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location.
Joint work with Svetlana Roudenko (Florida International University, USA) and Kai Yang (Florida International University, USA).
Monday, June 12, 14:30 ~ 15:00
The robustness of the Euler scheme for scalar SDEs with non-Lipschitz diffusion coefficient
Andreas Neuenkirch
University of Mannheim, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider stochastic differential equations (SDEs) that are given by \[ dV_t = a(V_t)dt + \left(b(V_t)\right)^{1-\gamma}dW_t, \qquad t\in[0,T], \] where $V_0=v_0 \in \mathbf{R}$ is deterministic, $W=(W_t)_{t \in [0,T]}$ is a Brownian motion and $\gamma\in\left(0,\frac{1}{2}\right]$. We assume that $a:\mathbf{R}\rightarrow \mathbf{R}$ and $b:\mathbf{R}\rightarrow [0,\infty)$ are globally Lipschitz continuous. Well-known examples that fall into this class of SDEs are the CIR process, the CEV process or the Wright-Fisher diffusion. We analyze the equidistant Euler scheme for the above SDE and, among other results, we show $L^1$-convergence order $1/2-\varepsilon$ in the discretization points (for $\varepsilon \gt 0$ arbitrarily small) if \[ \int_0^T \mathbf{E} \left[ \frac{1}{b(V_t)^{2 \gamma}} \right] dt \lt \infty. \] Thus, the loss of Lipschitzness, i.e. $\gamma \gt 0$, for the diffusion coefficient can be compensated by an appropriate inverse moment condition. This result yields in particular a unifying framework for the above mentioned SDEs: for the CIR or Wright-Fisher process, the above condition corresponds to the non-attainability of the boundaries of their support, while for the CEV process this inverse moment condition is always fulfilled.
Joint work with Annalena Mickel (University of Mannheim, Germany).
Monday, June 12, 15:00 ~ 15:30
Explicit stabilized integrators for stiff systems: from deterministic to stochastic (partial) differential equations
Gilles Vilmart
University of Geneva, Switzerland - This email address is being protected from spambots. You need JavaScript enabled to view it.
Explicit stabilized integrators are an efficient and popular alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff dissipative systems of differential equations.
In this talk, we explain how the versatility of explicit stabilized methods permits to integrate efficiently in time not only deterministic dissipative systems, but also stiff SDEs. We then analyze how such explicit stabilized methods with optimally large stability domains also apply to the case of semilinear parabolic SPDEs.
Joint work with Assyr Abdulle† (EPF Lausanne), Ibrahim Almuslimani (INRIA Rennes), Charles-Edouard Bréhier (University of Pau) and Konstantinos C. Zygalakis (University of Edinburgh).
Monday, June 12, 15:30 ~ 16:00
Generative modeling for time series via Schrödinger bridge
Huyên Pham
Université Paris Cité, FRANCE - This email address is being protected from spambots. You need JavaScript enabled to view it.
We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The resulting solution is described by a stochastic differential equation over a finite horizon with a path-dependent drift function, which accurately captures the temporal dynamics of the time series distribution. We can estimate the drift function from data samples either by kernel regression methods or with LSTM neural networks, and the simulation of the SB diffusion yields new synthetic data samples of the time series.
The performance of our generative model is evaluated through a series of numerical experiments. First, we test with a toy autoregressive model, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal and temporal dependencies metrics. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. Finally, we illustrate the SB approach for generating sequence of images.
Joint work with Mohamed Hamdouche (Université Paris Cité), Pierre Henry-Labordère (Qube RT).
Monday, June 12, 16:30 ~ 17:00
Learning the random variables: Combining Monte Carlo simulations with machine learning
Philippe von Wurstemberger
Chinese University of Hong Kong, Shenzhen, China and ETH Zurich, Switzerland - This email address is being protected from spambots. You need JavaScript enabled to view it.
In recent years, there has been a tremendous amount of research activity aimed at developing new deep learning-based methods for scientific computing problems. While many of these methods are promising, they often do not fully utilize existing numerical approaches and theory. In this talk, we introduce a novel method that merges machine learning techniques with problem-specific knowledge to tackle high-dimensional parametric stochastic approximation problems. Our strategy is based on the idea of combining Monte Carlo (MC) algorithms (e.g., standard MC or multilevel MC) with stochastic gradient descent (SGD) optimization methods by treating the realizations of random variables in the MC approximation as trainable parameters for the SGD optimization method. In other words, our approach focuses on learning the random variables appearing in MC approximations rather than training standard artificial neural networks. We present numerical results for this Learning the Random Variables (LRV) strategy applied to the pricing of financial options in the Black-Scholes model. In the considered examples, the LRV strategy produces highly convincing numerical results when compared with standard numerical methods (such as MC and Quasi-MC) and other machine learning-based methods.
Joint work with Arnulf Jentzen (The Chinese University of Hong Kong, Shenzhen, China and University of Münster, Germany), Sebastian Becker (ETH Zurich, Switzerland), Marvin S. Müller (2Xideas Switzerland AG) and Adrian Riekert (University of Münster, Germany).
Monday, June 12, 17:00 ~ 17:30
Numerical analysis of Euler scheme for SDE driven by fractional Brownian motion
Ludovic Goudenège
CNRS, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In a first part, we will present how to use stochastic sewing lemma from [Le20] for building solutions of SDE driven by additive fractional Brownian motion. When the drift is regular or bounded, we can build these solutions as the limit of Euler schemes and obtain a strong rate of convergence, as in [BDG21,DGL21,DAGI19].
Moreover, using roughness of fractional Brownian motion, we can define solutions to SDEs with distributional drift as limit of the previous solutions build with bounded regular drifts. In a second part, we will present numerical simulations of these singular SDEs involving Dirac measure or indicator functions in dimension 1 or 2.
Finally, we will present how to obtain a strong rate of convergence by combining the speed of approximation of the distributional drift and the time-step size in the Euler scheme. However it will force the noise to be ``rough enough'', essentially by adding constraints on the H\"older regularity of the noise [GHR22].
\textbf{References}
[BDG21] O. Butkovsky, K. Dareiotis, and M. Gerencs\'er. Approximation of SDEs: a stochastic sewing approach. {\em Probab. Theory Related Fields}, 181(4), 975-1034, 2021.
[GHR22] L. Gouden\`ege, E. M. Haress et A. Richard. Numerical approximation of SDEs with fractional noise and distributional drift. {\em hal-03715427v1}, 2022.
[Le20] K. L\^e. A stochastic sewing lemma and applications. {\em Electronic Journal of Probability}, 25, 1-55, 2020.
[DGL21] K. Dareiotis, M. Gerencs\'er and K. L\^e. Quantifying a convergence theorem of Gyongy and Krylov. {\em arXiv preprint arXiv:2101.12185}, 2021.
[DAGI19] T. De Angelis, M. Germain and E. Issoglio. A numerical scheme for stochastic differential equations with distributional drift. {\em arXiv preprint arXiv:1906.11026}, 2019.
Joint work with El Mehdi Haress (MICS - CentraleSupélec - Paris-Saclay University, France) and Alexandre Richard (MICS - CentraleSupélec - Paris-Saclay University, France).
Monday, June 12, 17:30 ~ 18:00
Stochastic partial differential equations arising in self-organized criticality
Benjamin Gess
Universität Bielefeld & MPI MIS Leipzig, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider scaling limits for the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to an SPDE with singular-degenerate diffusion. In addition, the deterministic BTW model is proved to converge to a singular-degenerate PDE. Alternatively, the proof of convergence can be understood as a proof of convergence of a finite-difference discretization for singular-degenerate SPDE. This extends recent work on finite difference approximation of (deterministic) quasilinear diffusion equations to discontinuous diffusion coefficients and SPDE. In addition, we perform numerical simulations illustrating relevant features of the considered models and the convergence to stochastic PDEs in spatial dimension $d=1,2$.
Monday, June 12, 18:00 ~ 18:30
Beyond strong rate $1/2$ for approximations of space-time white noise driven SPDEs
Mate Gerencser
TU Wien, Austria - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider $1+1$-dimensional space-time white noise driven reaction-diffusion type equations (e.g. stochastic Allen-Cahn equation), more precisely their full discretisation. Strong rate of convergence $1/2$ has been proven before and has been considered optimal, supported by rigorous lower bounds. We show that weakening the path topology where the error is measured results in higher strong rate of convergence (which is not the case for finite dimensional SDEs). The proof leverages tools from singular SPDEs.
Joint work with Harprit Singh (Imperial College London).
Tuesday, June 13, 14:00 ~ 14:30
On the existence of optimal shallow networks
Steffen Dereich
University Münster, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we discuss existence of global minima in optimisation problems over shallow neural networks. More explicitly, the function class over which we minimise is the family of all functions that can be expressed as artificial residual or feedforward neural networks with one hidden layer featuring a specified number of neurons with ReLU (or Leaky ReLU) activation. We give existence results. Moreover, we provide counterexamples that illustrate the relevance of the assumptions imposed in the theorems.
Joint work with Arnulf Jentzen (University Münster) and Sebastian Kassing (University Bielefeld).
Tuesday, June 13, 14:30 ~ 15:00
Unbiased sampling using reversibility checks
Tony Lelievre
Ecole des Ponts ParisTech, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will present recent results concerning the importance of using a reversibility check in some Markov Chain Monte Carlo algorithms based on a Metropolis Hastings procedure. More precisely, we will discuss two situations: sampling measures supported on submanifolds and Hamiltonian Monte Carlo with non-separable Hamiltonians. In both cases, the numerical procedure requires to solve an implicit problem at some point, which induces some numerical difficulty concerning the actual reversibility of the proposed move. Special care should be taken in the rejection procedure to avoid biases. These reversibility checks can be seen as generalizations of a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds.
References:
- T. Lelièvre, M. Rousset and G. Stoltz, Langevin dynamics with constraints and computation of free energy differences, Mathematics of Computation, 81(280), 2071-2125, (2012).
- T. Lelièvre, M. Rousset and G. Stoltz, Hybrid Monte Carlo methods for sampling probability measures on submanifolds, Numerische Mathematik, 143(2), 379-421, (2019).
- T. Lelièvre, G. Stoltz and W. Zhang, Multiple projection MCMC algorithms on submanifolds, to appear in IMA Journal of Numerical Analysis.
- T. Lelièvre, R. Santet and G. Stoltz, Unbiasing Hamiltonian Monte Carlo algorithms for a general Hamiltonian function, work in progress.
Joint work with M. Rousset (Inria Rennes), R. Santet (Ecole des Ponts ParisTech), G. Stoltz (Ecole des Ponts ParisTech) and W. Zhang (FU Berlin).
Tuesday, June 13, 15:00 ~ 15:30
Computing committor function and invariant distribution for randomly perturbed dynamical systems
Weiqing Ren
National University of Singapore, Singapore - This email address is being protected from spambots. You need JavaScript enabled to view it.
The committor function is a central object in understanding transitions between metastable states in complex systems. It has a simple mathematical description – it satisfies the backward Kol- mogorov equation. However, computing the committor function for realistic systems at low temperature is a challenging task, due to the curse of dimensionality and the scarcity of transition data. In this talk, I will present a computational approach that overcomes these issues and achieves good performance on complex benchmark problems with rough energy landscapes. The new approach combines deep learning, importance sampling and feature engineering techniques. I will also discuss the computation of invariant distributions from short trajectories using deep learning.
Joint work with Qianxiao Li (National University of Singapore) and Bo Lin (National University of Singapore).
Tuesday, June 13, 15:30 ~ 16:00
Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence
Jiequn Han
Flatiron Institute, Simons Foundation, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we introduce a novel deep learning approach to solve McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs), a core challenge in mean-field control and mean-field games. Our method overcomes limitations of existing techniques by addressing full distribution dependence in mean-field interactions.
By building on fictitious play, we transform the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs' model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration's FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under suitable assumptions, we demonstrate convergence without the curse of dimensionality using integral probability metrics. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward process.
Joint work with Ruimeng Hu (University of California, Santa Barbara, USA) and Jihao Long (Princeton University, USA).
Tuesday, June 13, 16:30 ~ 17:30
Overcoming the curse of dimensionality in the numerical approximation of BSDEs
Martin Hutzenthaler
University of Duisburg Essen, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Backward stochastic differential equations (BSDEs) in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. In this talk we show how to overcome the curse of dimensionality by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.
Tuesday, June 13, 17:30 ~ 18:00
On the strong approximation of SDEs with superlinear growing coefficients: convergence and stability of the exponential Euler scheme.
Mireille Bossy
INRIA, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the problem of approximating the solution of an SDE with a non-globally Lipschitz drift, possibly discontinuous, and a diffusion coefficient with polynomial growth. By studying the strong error, we show the usual convergence rate of 1/2 for the exponential Euler scheme.
The condition for obtaining a convergence rate is mainly determined by the possible control of the moments, and the exponential moment of the exact process and the scheme. The proof relies on a time change technique.
Joint work with Kerlyns Martínez (University of Valparaíso).
Tuesday, June 13, 18:00 ~ 18:30
Sharp lower error bounds for strong approximation of SDEs with a drift coefficient of Sobolev regularity $s\in (1/2,1)$
Thomas Müller-Gronbach
University of Passau, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study strong approximation of scalar SDEs $dX_t = \mu(X_t)\, dt + dW_t$ at time $t=1$ in the case that $\mu$ is bounded and has fractional Sobolev regularity $s\in (0,1)$. Recently, it has been shown in [1] that in this case the equidistant Euler scheme achieves a root mean squared error of order $(1+s)/2$, up to an arbitrary small $\epsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In this talk we show that, for $s\in (1/2,1)$, this order can not be improved in general.
References [1] K. Dareiotis, M. Gerencsér and K. Lê. Quantifying a convergence theorem of Gy\"ongy and Krylov. arXiv:2101.12185v2 (2022).
Joint work with Simon Ellinger (University of Passau, Germany) and Larisa Yaroslavtseva (University of Graz, Austria).
Wednesday, June 14, 14:00 ~ 15:00
Sampling Via Gradient Flows In The Space of Probability Measures
Andrew Stuart
Caltech, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial distribution can be evolved to the desired minimizer (the target distribution) dynamically via gradient flows. Mean-field models, whose law is governed by the gradient flow in the space of probability measures, may also be identified; particle approximations of these mean-field models form the basis of algorithms. The gradient flow approach is also the basis of algorithms for variational inference, in which the optimization is performed over a parameterized family of probability distributions such as Gaussians, and the underlying gradient flow is restricted to the parameterized family.
By choosing different energy functionals and metrics for the gradient flow, different algorithms with different convergence properties arise. In this work, we concentrate on the Kullback–Leibler divergence as the energy functional after showing that, up to scaling, it has the unique property (among all f -divergences) that the gradient flows resulting from this choice of energy do not depend on the normalization constant of the target distribution. For the metrics, we focus on variants of the Fisher-Rao, Wasserstein, and Stein metrics; we introduce the affine invariance property for gradient flows, and their corresponding mean-field models, determine whether a given metric leads to affine invariance, and modify it to make it affine invariant if it does not.
We study the resulting gradient flows in both the space of all probability density functions and in the subset of all Gaussian densities. The flow in the Gaussian space may be understood as a Gaussian approximation of the flow in the density space. We demonstrate that, under mild assumptions, the Gaussian approximation based on the metric and through moment closure coincide; the moment closure approach is more convenient for calculations. We establish connections between these approximate gradient flows, discuss their relation to natural gradient methods in parametric variational inference, and study their long-time convergence properties showing, for some classes of problems and metrics, the advantages of affine invariance. Furthermore, numerical experiments are included which demonstrate that affine invariant gradient flows have desirable convergence properties for a wide range of highly anisotropic target distributions.
Joint work with Y. CHEN and D.Z. HUANG (Caltech), J. HUANG (U Penn) and S. REICH (Potsdam).
Wednesday, June 14, 15:00 ~ 15:30
Nonlinear SPDE models of particle systems: analysis and numerics
Ana Djurdjevac
FU Berlin, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Interacting particle systems provide flexible and powerful models that are useful in many application areas. The motivation that we will address comes from the opinion dynamics. In particular, we are interested in particle systems with large number of particles, hence they are very complex from the computational point of view. A common strategy is to derive effective equations that describe the time evolution of the empirical particle density. The equation that is obtained has the form of the Dean-Kawasaki equation which is well-known for its singular structure that results in difficulty of its rigorous mathematical interpretation. Our aim is to consider non-linear SPDE models that provide approximation of the Dean-Kawasaki equation. We will discuss the well-posedness of these equations and the preservation of physical constraints of the particle system. Finally, we will also approach the question of its approximation and in particular of preservation of positivity at the discrete level.
Joint work with H. Kremp (TU Wien) and N. Perkowski (FU Berlin).
Wednesday, June 14, 15:30 ~ 16:00
The aromatic bicomplex for the study of integrators that exactly preserve the invariant measure
Adrien Laurent
Universitetet i Bergen, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
For the approximation of ergodic stochastic differential equations (SDEs), there exists a methodology for the creation of high-order schemes for sampling the invariant measure. One then wonders if there exists a scheme that preserves the invariant measure exactly, in the spirit of exact volume preservation for ODEs. While B-series (resp. exotic B-series) are used to represent the Taylor expansion of the solution of ODEs (resp. SDEs), aromatic B-series (resp. exotic aromatic B-series) appear in the creation of integrators that preserve the invariant measure of ODEs (resp. ergodic SDEs). In this talk, we define aromatic forms and the aromatic bicomplex, in the spirit of the variational bicomplex in differential geometry. We prove the exactness of this bicomplex and use it to give an explicit description of volume-preserving methods. We use this description to show that no aromatic modification of Runge-Kutta methods preserves volume in general and to discuss the possible ansatz for creating methods that preserve the invariant measure exactly.
Joint work with Robert I. McLachlan (Massey University, New Zealand), Hans Z. Munthe-Kaas (Universitetet i Bergen, Norway) and Olivier Verdier (Western Norway University of Applied Sciences, Norway).
Wednesday, June 14, 16:30 ~ 17:00
On the numerical approximation of one-dimensional continuous Markov processes
Thomas Kruse
University of Wuppertal, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We propose a new approach for approximating one-dimensional continuous Markov processes in law. Special cases include stochastic differential equations with irregular coefficients and processes with sticky features. In particular, we prove a functional limit theorem (FLT) for weak approximation of the paths of arbitrary continuous Markov processes. Based on this result we propose a new scheme, called EMCEL, which satisfies the assumption of the FLT and thus allows to approximate every one-dimensional continuous Markov process. We determine its convergence speed in terms of Wasserstein distances. Further we present various properties of the EMCEL scheme, analyze its differences from the Euler scheme and discuss several examples.
Joint work with Stefan Ankirchner (University of Jena, Germany), Mikhail Urusov (University of Duisburg-Essen, Germany) and Wolfgang Löhr (University of Duisburg-Essen, Germany).
Wednesday, June 14, 17:00 ~ 17:30
Asymptotic preserving schemes for some multiscale stochastic systems
Charles-Edouard Bréhier
Université de Pau et des Pays de l'Adour, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss how to construct temporal discretization schemes for some classes of multiscale stochastic ordinary and partial differential equations. While well-known averaging or diffusion approximation principles are obtained for the considered systems, it is challenging to obtain numerical schemes which are efficient with time-step size independent of the scale separation parameter, and can capture the limiting evolution. I will present examples of such asymptotic preserving schemes and of associated uniform accuracy results.
Wednesday, June 14, 17:30 ~ 18:00
Mean estimation for Randomized Quasi Monte Carlo method
Emmanuel Gobet
Ecole polytechnique, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We are given a simulation budget of $B$ points to calculate an expectation $\mu=\mathbb{E}(F(U))$. A Monte Carlo method achieves a root mean squared risk of order $1/\sqrt B$, while a Randomized Quasi Monte Carlo (RQMC) method achieves an accuracy $\sigma_B \ll 1/\sqrt B$. The question we address in this work is, given a budget $B$ and a confidence level $\delta$, what is the optimal size of error tolerance such that $\mathbb{P}(|{\tt Est}-\mu| \gt {\tt TOL})\leq \delta$ for an estimator ${\tt Est}$ to be determined? We show that a judicious choice of ``robust'' aggregation methods coupled with RQMC methods allows to reach the best ${\tt TOL}$. This study is supported by numerical experiments, ranging from bounded $F(U)$ to heavy-tailed $F(U)$.
Joint work with Matthieu Lerasle ( CREST, ENSAE, Institut Polytechnique de Paris) and David Métivier (INRAE Montpellier).
Wednesday, June 14, 18:00 ~ 18:30
Adaptive stochastic optimizers, neural nets and diffusion generative models
Sotirios Sabanis
University of Edinburgh, National Technical University of Athens and The Alan Turing Institute, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
A new class of stochastic (adaptive) optimization algorithms with superior performance in the training of artificial neural networks, has emerged recently due to the fundamental progress in the theory of numerical methods for SDEs with superlinear coefficients in recent years. Such stochastic optimizers can successfully address known shortcomings in the training of neural networks, which are known as 'exploding' and 'vanishing' gradients, both theoretically and practically.
Key findings of this new methodology will be reviewed and their links to diffusion generative models will be highlighted.
Posters
Similarity-Based Data Selection to Improve Automatic Acoustic Target Classification
Pala Ahmet
Department of Mathematics, University of Bergen, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
Acoustic surveys provide important data for fisheries management. During the surveys, ship-mounted echo sounders send acoustic signals into the water and measure the arrival time of the reflection, so-called backscatter. Acoustic target classification aims to identify backscatters by categorizing them into specific groups, e.g., sandeel, mackerel, and background (as bottom and plankton). Convolutional neural networks typically perform well for acoustic target classification but fail in cases where the background class is similar to the foreground class. In this poster, we discuss how to address the challenge of class imbalance in the sampling of training and validation data for deep convolutional neural networks. The proposed strategy seeks to equally sample areas containing all different classes while prioritizing background data that have similar characteristics to the foreground class. The Near-Miss algorithm is used to select these tricky areas from the background class in order to detect regions where misclassification is more likely. The poster contains an introduction to acoustic target classification, the description of the deep learning model, the challenges, and the results of the Near Miss algorithm, along with a comparison to the baseline results.
Joint work with Ahmet Pala (University of Bergen, Norway), Anna Oleynik (University of Bergen, Norway), Nils Olav Handegard (Institute of Marine Research, Norway) and Guttorm Alendal (University of Bergen).
Kiefer-Wolfowitz algorithm under quasi-associated random errors in the functional case
Lydia Amzal
University of Bejaia, Algeria - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we investigate the Kiefer-Wolfowitz algorithm under quasi-associative random errors in the functional case. We establish its complete convergence and obtain an exponential bound. Furthermore, we study the rate of convergence of the algorithm. Numerical examples are provided to confirm the theoretical results and demonstrate the accuracy of the algorithm.
Joint work with Tabti Hadjila ( university of Bejaia ) and Zerouati Halima ( university of Bejaia ).
Mann’s iteration with functional errors for Volterra integral equation
Samir Arezki
University of Bejaia, Algeria - This email address is being protected from spambots. You need JavaScript enabled to view it.
This work deals with stochastic Mann iteration under the influence of functional random errors. First, we establish an exponential Brenstien-Frechet inequality that allows to prove the almost complete convergence to the fixed point of a contracting application in a Banach space, which is the solution of the Volterra integral equation.
Then, we will study the rate of convergence of this algorithm, which represents the speed at which the algorithm converges to the fixed point.
In sum, this research work contributes to improving our understanding of stochastic Mann iteration and its behavior under specific conditions. The results of this research can be useful for optimizing the algorithm in various practical applications.
Joint work with Barache Bahia ( university of Bejaia) and Zerouati Halima ( university of Bejaia ).
Weak numerical convergence rates for stochastic differential equations with nonglobally monotone coefficients
Martin Chak
Sorbonne University, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Order one weak convergence rates are presented for SDEs with nonglobally monotone coefficients approximated by the stopped increment-tamed Euler-Maruyama scheme. Regularity results for associated Kolmogorov equations are also presented but a more direct argument requiring weaker assumptions is given based on the It\^o-Alekseev-Gr\"obner formula.
Modified equations and backward error analysis for stochastic Hamiltonian systems
Stefano Di Giovacchino
University of L'Aquila, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we address our attention on long-term analysis of numerical discretizations to stochastic Hamiltonian systems of Ito and Stratonovich types. Specifically, it is well-known that, in Ito scenario, the averaged Hamiltonian computed along the flow of such systems grows linearly in time; instead, for Stratonovich systems, the Hamiltonian remains pathwise constant. Hence, our aim is to detect how numerical Hamiltonians behave, in terms of being able to show the characteristic behavior of the exact flows along the numerical dynamics, over long time windows. Then, we perform a weak backward error analysis for such systems based on the construction of weak modified differential equations. These equations allow us to understand the trend of Hamiltonians along numerical trajectories. It turns out that, in general, extra error terms arise and, in particular, in the Stratonovich case, an exponential growth of the error is exhibited, after time intervals of length $O(\Delta t^{-p})$, being $p$ the weak order of the analyzed method. Finally, selected numerical experiments are given to confirm the theoretical analysis. The results are based on a joint work with Raffaele D'Ambrosio (University of L'Aquila).
Joint work with Raffaele D'Ambrosio (University of L'Aquila).
Refined descriptive sampling with dependent variables
Siham Kebaili
University Bejaia, Algeria - This email address is being protected from spambots. You need JavaScript enabled to view it.
This paper deals with Monte Carlo simulation in case of dependent input random vari- ables. We propose an algorithm to generate refined descriptive samples from dependent ran- dom variables for estimation of expectations of functions of output variables using the Iman and Conover algorithm to transform the dependent variables to independent ones. There- fore, such estimates obtained through a chosen mathematical model are compared with those obtained using the simple random sampling method, which proved that the former are the most e¢ cient. Besides, using already published work on independent input variables, we can deduce in case of dependent input random variables, that asymptotically the variance of the RDS estimator is less than that of SRS estimator for any simulation function having finite second moment.
Key Words: Simulation; Monte Carlo Methods; Variance reduction; Iman and Conover
Stochastic and Deterministic Computation in the regulation of Decision – Making Genetic circuits by Energy Availability.
Rajneesh Kumar
University of Bergen, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
All living organisms require energy because it is a driven, non-equilibrium system at all scales, from the cellular to the evolutionary. Like human society, a consistent energy supply is essential for cells to process information and take actions, and when cellular energy supplies are challenged, disorders can result. Here we use a mathematical model of gene expression in a bistable decision-making regulatory network to explore cellular bifurcation behaviour when we vary the energy availability. We impose that the rates of the associated gene expression processes in our model are reliant on an ATP concentration parameters since we know that each step in transcription and translation requires energy in the form of ATP. We discuss both a deterministic model, to explore the emergence of different attractors under different ATP concentration parameters, and a stochastic simulation case to explore how ATP influences the noisy dynamics and magnitude of protein expression and importantly whether switching between two protein concentration become more or less common.
Joint work with Iain G. Johnston (University of Bergen, Norway).