Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)
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MultiIndex Monte Carlo and stochastic collocation methods for random PDEs(20160109)In this talk we consider the problem of computing statistics of the solution of a partial differential equation with random data, where the random coefficient is parametrized by means of a finite or countable sequence of terms in a suitable expansion. We describe and analyze a MultiIndex Monte Carlo (MIMC) and a MultiIndex Stochastic Collocation method (MISC). the former is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using firstorder differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles s MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence, O(TOL2). On the same vein, MISC is a deterministic combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. Provided enough mixed regularity, MISC can achieve better complexity than MIMC. Moreover, we show that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a onedimensional spatial problem. We propose optimization procedures to select the most effective mixed differences to include in MIMC and MISC. Such optimization is a crucial step that allows us to make MIMC and MISC computationally effective. We finally show the effectiveness of MIMC and MISC with some computational tests, including tests with a infinite countable number of random parameters.

Two numerical methods for meanfield games(20160109)Here, we consider numerical methods for stationary meanfield games (MFG) and investigate two classes of algorithms. The first one is a gradient flow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including onedimensional periodic MFG, congestion problems, and higherdimensional models.

Computable error estimates for Monte Carlo finite element approximation of elliptic PDE with lognormal diffusion coefficients(20160109)The Monte Carlo (and Multilevel Monte Carlo) finite element method can be used to approximate observables of solutions to diffusion equations with lognormal distributed diffusion coefficients, e.g. modeling ground water flow. Typical models use lognormal diffusion coefficients with H´ older regularity of order up to 1/2 a.s. This low regularity implies that the high frequency finite element approximation error (i.e. the error from frequencies larger than the mesh frequency) is not negligible and can be larger than the computable low frequency error. We address how the total error can be estimated by the computable error.

On the predictive capabilities of multiphase Darcy flow models(20160109)Darcy s law is a widely used model and the limit of its validity is fairly well known. When the flow is sufficiently slow and the porosity relatively homogeneous and low, Darcy s law is the homogenized equation arising from the Stokes and Navier Stokes equations and depends on a single effective parameter (the absolute permeability). However when the model is extended to multiphase flows, the assumptions are much more restrictive and less realistic. Therefore it is often used in conjunction with empirical models (such as relative permeability and capillary pressure curves), derived usually from phenomenological speculations and experimental data fitting. In this work, we present the results of a Bayesian calibration of a twophase flow model, using highfidelity DNS numerical simulation (at the porescale) in a realistic porous medium. These reference results have been obtained from a NavierStokes solver coupled with an explicit interphasetracking scheme. The Bayesian inversion is performed on a simplified 1D model in Matlab by using adaptive spectral method. Several data sets are generated and considered to assess the validity of this 1D model.

Uncertainty quantification for mean field games in social interactions(20160109)We present an overview of mean field games formulation. A comparative analysis of the optimality for a stochastic McKeanVlasov process with timedependent probability is presented. Then we examine meanfield games for social interactions and we show that optimizing the longterm wellbeing through effort and social feeling state distribution (meanfield) will help to stabilize couple (marriage). However , if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We then examine the influence of society on a couple using mean field sentimental games. We show that, in meanfield equilibrium, the optimal effort is always higher than the oneshot optimal effort. Finally we introduce the Wiener chaos expansion for the construction of solution of stochastic differential equations of MckeanVlasov type. The method is based on the CameronMartin version of the Wiener Chaos expansion and allow to quantify the uncertainty in the optimality system.

A potpourri of results from the KAUST SRIUQ(20160108)As the KAUST Strategic Research Initiative for Uncertainty Quantification completes its fourth year of existence we recall several results produced during its exciting journey of discovery. These include, among others, contributions on Multilevel and Multiindex sampling techniques that address both direct and inverse problems. We may discuss also several techniques for Bayesian Inverse Problems and Optimal Experimental Design.

Optimal mesh hierarchies in Multilevel Monte Carlo methods(20160108)I will discuss how to choose optimal mesh hierarchies in Multilevel Monte Carlo (MLMC) simulations when computing the expected value of a quantity of interest depending on the solution of, for example, an Ito stochastic differential equation or a partial differential equation with stochastic data. I will consider numerical schemes based on uniform discretization methods with general approximation orders and computational costs. I will compare optimized geometric and nongeometric hierarchies and discuss how enforcing some domain constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. I will also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. This talk presents joint work with N.Collier, A.L.HajiAli, F. Nobile, and R. Tempone.

Adaptive stochastic Galerkin FEM with hierarchical tensor representations(20160108)PDE with stochastic data usually lead to very highdimensional algebraic problems which easily become unfeasible for numerical computations because of the dense coupling structure of the discretised stochastic operator. Recently, an adaptive stochastic Galerkin FEM based on a residual a posteriori error estimator was presented and the convergence of the adaptive algorithm was shown. While this approach leads to a drastic reduction of the complexity of the problem due to the iterative discovery of the sparsity of the solution, the problem size and structure is still rather limited. To allow for larger and more general problems, we exploit the tensor structure of the parametric problem by representing operator and solution iterates in the tensor train (TT) format. The (successive) compression carried out with these representations can be seen as a generalisation of some other model reduction techniques, e.g. the reduced basis method. We show that this approach facilitates the efficient computation of different error indicators related to the computational mesh, the active polynomial chaos index set, and the TT rank. In particular, the curse of dimension is avoided.

Bayesian techniques for fatigue life prediction and for inference in linear time dependent PDEs(20160108)In this talk we introduce first the main characteristics of a systematic statistical approach to model calibration, model selection and model ranking when stresslife data are drawn from a collection of records of fatigue experiments. Focusing on Bayesian prediction assessment, we consider fatiguelimit models and random fatiguelimit models under different a priori assumptions. In the second part of the talk, we present a hierarchical Bayesian technique for the inference of the coefficients of time dependent linear PDEs, under the assumption that noisy measurements are available in both the interior of a domain of interest and from boundary conditions. We present a computational technique based on the marginalization of the contribution of the boundary parameters and apply it to inverse heat conduction problems.

A study of Monte Carlo methods for weak approximations of stochastic particle systems in the meanfield?(20160108)I discuss using single level and multilevel Monte Carlo methods to compute quantities of interests of a stochastic particle system in the meanfield. In this context, the stochastic particles follow a coupled system of Ito stochastic differential equations (SDEs). Moreover, this stochastic particle system converges to a stochastic meanfield limit as the number of particles tends to infinity. I start by recalling the results of applying different versions of Multilevel Monte Carlo (MLMC) for particle systems, both with respect to time steps and the number of particles and using a partitioning estimator. Next, I expand on these results by proposing the use of our recent Multiindex Monte Carlo method to obtain improved convergence rates.

Bayesian optimal experimental design for priors of compact support(20160108)In this study, we optimize the experimental setup computationally by optimal experimental design (OED) in a Bayesian framework. We approximate the posterior probability density functions (pdf) using truncated Gaussian distributions in order to account for the bounded domain of the uniform prior pdf of the parameters. The underlying Gaussian distribution is obtained in the spirit of the Laplace method, more precisely, the mode is chosen as the maximum a posteriori (MAP) estimate, and the covariance is chosen as the negative inverse of the Hessian of the misfit function at the MAP estimate. The model related entities are obtained from a polynomial surrogate. The optimality, quantified by the information gain measures, can be estimated efficiently by a rejection sampling algorithm against the underlying Gaussian probability distribution, rather than against the true posterior. This approach offers a significant error reduction when the magnitude of the invariants of the posterior covariance are comparable to the size of the bounded domain of the prior. We demonstrate the accuracy and superior computational efficiency of our method for shocktube experiments aiming to measure the model parameters of a key reaction which is part of the complex kinetic network describing the hydrocarbon oxidation. In the experiments, the initial temperature and fuel concentration are optimized with respect to the expected information gain in the estimation of the parameters of the target reaction rate. We show that the expected information gain surface can change its shape dramatically according to the level of noise introduced into the synthetic data. The information that can be extracted from the data saturates as a logarithmic function of the number of experiments, and few experiments are needed when they are conducted at the optimal experimental design conditions.

Multilevel ensemble Kalman filtering(20160108)The ensemble Kalman filter (EnKF) is a sequential filtering method that uses an ensemble of particle paths to estimate the means and covariances required by the Kalman filter by the use of sample moments, i.e., the Monte Carlo method. EnKF is often both robust and efficient, but its performance may suffer in settings where the computational cost of accurate simulations of particles is high. The multilevel Monte Carlo method (MLMC) is an extension of classical Monte Carlo methods which by sampling stochastic realizations on a hierarchy of resolutions may reduce the computational cost of moment approximations by orders of magnitude. In this work we have combined the ideas of MLMC and EnKF to construct the multilevel ensemble Kalman filter (MLEnKF) for the setting of finite dimensional state and observation spaces. The main ideas of this method is to compute particle paths on a hierarchy of resolutions and to apply multilevel estimators on the ensemble hierarchy of particles to compute Kalman filter means and covariances. Theoretical results and a numerical study of the performance gains of MLEnKF over EnKF will be presented. Some ideas on the extension of MLEnKF to settings with infinite dimensional state spaces will also be presented.

Quasipotential and TwoScale Large Deviation Theory for Gillespie Dynamics(20160107)The construction of energy landscape for biodynamics is attracting more and more attention recent years. In this talk, I will introduce the strategy to construct the landscape from the connection to rare events, which relies on the large deviation theory for Gillespietype jump dynamics. In the application to a typical genetic switching model, the twoscale large deviation theory is developed to take into account the fast switching of DNA states. The comparison with other proposals are also discussed. We demonstrate different diffusive limits arise when considering different regimes for genetic translation and switching processes.

Scalable algorithms for optimal control of stochastic PDEs(20160107)We present methods for the optimal control of systems governed by partial differential equations with infinitedimensional uncertain parameters. We consider an objective function that involves the mean and variance of the control objective, leading to a riskaverse optimal control formulation. To make the optimal control problem computationally tractable, we employ a local quadratic approximation of the objective with respect to the uncertain parameter. This enables computation of the mean and variance of the control objective analytically. The resulting riskaverse optimization problem is formulated as a PDEconstrained optimization problem with constraints given by the forward and adjoint PDEs for the first and secondorder derivatives of the quantity of interest with respect to the uncertain parameter, and with an objective that involves the trace of a covariancepreconditioned Hessian (of the objective with respect to the uncertain parameters) operator. A randomized trace estimator is used to make tractable the trace computation. Adjointbased techniques are used to derive an expression for the infinitedimensional gradient of the riskaverse objective function via the Lagrangian, leading to a quasiNewton method for solution of the optimal control problem. A specific problem of optimal control of a linear elliptic PDE that describes flow of a fluid in a porous medium with uncertain permeability field is considered. We present numerical results to study the consequences of the local quadratic approximation and the efficiency of the method.

Hierarchical lowrank approximation for high dimensional approximation(20160107)Tensor methods are among the most prominent tools for the numerical solution of highdimensional problems where functions of multiple variables have to be approximated. Such highdimensional approximation problems naturally arise in stochastic analysis and uncertainty quantification. In many practical situations, the approximation of highdimensional functions is made computationally tractable by using rankstructured approximations. In this talk, we present algorithms for the approximation in hierarchical tensor format using statistical methods. Sparse representations in a given tensor format are obtained with adaptive or convex relaxation methods, with a selection of parameters using crossvalidation methods.

Static models, recursive estimators and the zerovariance approach(20160107)When evaluating dependability aspects of complex systems, most models belong to the static world, where time is not an explicit variable. These models suffer from the same problems than dynamic ones (stochastic processes), such as the frequent combinatorial explosion of the state spaces. In the Monte Carlo domain, on of the most significant difficulties is the rare event situation. In this talk, we describe this context and a recent technique that appears to be at the top performance level in the area, where we combined ideas that lead to very fast estimation procedures with another approach called zerovariance approximation. Both ideas produced a very efficient method that has the right theoretical property concerning robustness, the Bounded Relative Error one. Some examples illustrate the results.

Estimation of parameter sensitivities for stochastic reaction networks(20160107)Quantification of the effects of parameter uncertainty is an important and challenging problem in Systems Biology. We consider this problem in the context of stochastic models of biochemical reaction networks where the dynamics is described as a continuoustime Markov chain whose states represent the molecular counts of various species. For such models, effects of parameter uncertainty are often quantified by estimating the infinitesimal sensitivities of some observables with respect to model parameters. The aim of this talk is to present a holistic approach towards this problem of estimating parameter sensitivities for stochastic reaction networks. Our approach is based on a generic formula which allows us to construct efficient estimators for parameter sensitivity using simulations of the underlying model. We will discuss how novel simulation techniques, such as tauleaping approximations, multilevel methods etc. can be easily integrated with our approach and how one can deal with stiff reaction networks where reactions span multiple timescales. We will demonstrate the efficiency and applicability of our approach using many examples from the biological literature.

An efficient forwardreverse expectationmaximization algorithm for statistical inference in stochastic reaction networks(20160107)In this work, we present an extension of the forwardreverse representation introduced in Simulation of forwardreverse stochastic representations for conditional diffusions , a 2014 paper by Bayer and Schoenmakers to the context of stochastic reaction networks (SRNs). We apply this stochastic representation to the computation of efficient approximations of expected values of functionals of SRN bridges, i.e., SRNs conditional on their values in the extremes of given timeintervals. We then employ this SRN bridgegeneration technique to the statistical inference problem of approximating reaction propensities based on discretely observed data. To this end, we introduce a twophase iterative inference method in which, during phase I, we solve a set of deterministic optimization problems where the SRNs are replaced by their reactionrate ordinary differential equations approximation; then, during phase II, we apply the Monte Carlo version of the ExpectationMaximization algorithm to the phase I output. By selecting a set of overdispersed seeds as initial points in phase I, the output of parallel runs from our twophase method is a cluster of approximate maximum likelihood estimates. Our results are supported by numerical examples.

Tight Error Bounds for Fourier Methods for Option Pricing for Exponential Levy Processes(20160106)Prices of European options whose underlying asset is driven by the L´evy process are solutions to partial integrodifferential Equations (PIDEs) that generalise the BlackScholes equation by incorporating a nonlocal integral term to account for the discontinuities in the asset price. The Levy Khintchine formula provides an explicit representation of the characteristic function of a L´evy process (cf, [6]): One can derive an exact expression for the Fourier transform of the solution of the relevant PIDE. The rapid rate of convergence of the trapezoid quadrature and the speedup provide efficient methods for evaluationg option prices, possibly for a range of parameter configurations simultaneously. A couple of works have been devoted to the error analysis and parameter selection for these transformbased methods. In [5] several payoff functions are considered for a rather general set of models, whose characteristic function is assumed to be known. [4] presents the framework and theoretical approach for the error analysis, and establishes polynomial convergence rates for approximations of the option prices. [1] presents FTrelated methods with curved integration contour. The classical flat FTmethods have been, on the other hand, extended for option pricing problems beyond the European framework [3]. We present a methodology for studying and bounding the error committed when using FT methods to compute option prices. We also provide a systematic way of choosing the parameters of the numerical method, minimising the error bound and guaranteeing adherence to a predescribed error tolerance. We focus on exponential L´evy processes that may be of either diffusive or pure jump in type. Our contribution is to derive a tight error bound for a Fourier transform method when pricing options under riskneutral Levy dynamics. We present a simplified bound that separates the contributions of the payoff and of the process in an easily processed and extensible product form that is independent of the asymptotic behaviour of the option price at extreme prices and at strike parameters. We also provide a proof for the existence of optimal parameters of the numerical computation that minimise the presented error bound.