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AuthorTempone, Raul (32)Alouini, Mohamed-Slim (11)Nobile, Fabio (9)Bagci, Hakan (7)Vilanova, Pedro (7)View MoreDepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division (69)Applied Mathematics and Computational Science Program (44)Electrical Engineering Program (19)Physical Sciences and Engineering (PSE) Division (7)Mechanical Engineering Program (3)View MoreSubjectWireless (11)Bayesian (9)Sampling (9)CEM (7)SDE (7)View MoreTypePoster (57)Presentation (35)Meetings and Proceedings (1)Year (Issue Date)2016 (93)Item AvailabilityOpen Access (93)

Now showing items 1-10 of 93

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On the predictive capabilities of multiphase Darcy flow models

Icardi, Matteo; Prudhomme, Serge (2016-01-09) [Presentation]

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 two-phase flow model, using high-fidelity DNS numerical simulation (at the pore-scale) in a realistic porous medium. These reference results have been obtained from a Navier-Stokes solver coupled with an explicit interphase-tracking 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.

Adaptive stochastic Galerkin FEM with hierarchical tensor representations

Eigel, Martin (2016-01-08) [Presentation]

PDE with stochastic data usually lead to very high-dimensional 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.

Optimal mesh hierarchies in Multilevel Monte Carlo methods

Von Schwerin, Erik (2016-01-08) [Presentation]

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 non-geometric 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.Haji-Ali, F. Nobile, and R. Tempone.

Bayesian optimal experimental design for priors of compact support

Long, Quan (2016-01-08) [Presentation]

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 shock-tube 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.

Estimation of parameter sensitivities for stochastic reaction networks

Gupta, Ankit (2016-01-07) [Presentation]

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 continuous-time 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 tau-leaping approximations, multi-level methods etc. can be easily integrated with our approach and how one can deal with stiff reaction networks where reactions span multiple time-scales. We will demonstrate the efficiency and applicability of our approach using many examples from the biological literature.

Scalable algorithms for optimal control of stochastic PDEs

Ghattas, Omar; Alexanderian, Alen; Petra, Noemi; Stadler, Georg (2016-01-07) [Presentation]

We present methods for the optimal control of systems governed by partial differential equations with infinite-dimensional uncertain parameters. We consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse 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 risk-averse optimization problem is formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs for the first and second-order derivatives of the quantity of interest with respect to the uncertain parameter, and with an objective that involves the trace of a covariance-preconditioned Hessian (of the objective with respect to the uncertain parameters) operator. A randomized trace estimator is used to make tractable the trace computation. Adjoint-based techniques are used to derive an expression for the infinite-dimensional gradient of the risk-averse objective function via the Lagrangian, leading to a quasi-Newton 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.

Static models, recursive estimators and the zero-variance approach

Rubino, Gerardo (2016-01-07) [Presentation]

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 zero-variance 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.

Multi-Index Monte Carlo and stochastic collocation methods for random PDEs

Nobile, Fabio; Haji Ali, Abdul Lateef; Tamellini, Lorenzo; Tempone, Raul (2016-01-09) [Presentation]

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 Multi-Index Monte Carlo (MIMC) and a Multi-Index 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(TOL-2). 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 one-dimensional 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.

Bayesian techniques for fatigue life prediction and for inference in linear time dependent PDEs

Scavino, Marco (2016-01-08) [Presentation]

In this talk we introduce first the main characteristics of a systematic statistical approach to model calibration, model selection and model ranking when stress-life data are drawn from a collection of records of fatigue experiments. Focusing on Bayesian prediction assessment, we consider fatigue-limit models and random fatigue-limit 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.

Estimation of uncertain parameters of large Matern covariance functions with using hierarchical matrix technique

Litvinenko, Alexander; Genton, Marc G.; Sun, Ying; Keyes, David E. (2016-01-09) [Presentation]

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