Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2015)
Recent Submissions

Multilevel quadrature of elliptic PDEs with lognormal diffusion(20150107) [Poster]We apply multilevel quadrature methods for the moment computation of the solution of elliptic PDEs with lognormally distributed diffusion coefficients. The computation of the moments is a difficult task since they appear as high dimensional Bochner integrals over an unbounded domain. Each function evaluation corresponds to a deterministic elliptic boundary value problem which can be solved by finite elements on an appropriate level of refinement. The complexity is thus given by the number of quadrature points times the complexity for a single elliptic PDE solve. The multilevel idea is to reduce this complexity by combining quadrature methods with different accuracies with several spatial discretization levels in a sparse grid like fashion.

Efficient approximation of random fields for numerical applications(20150107) [Poster]We consider the rapid computation of separable expansions for the approximation of random fields. We compare approaches based on techniques from the approximation of nonlocal operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. We provide an aposteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples validate and quantify the considered methods.

Bayesian Inversion for Large Scale Antarctic Ice Sheet Flow(20150107) [Presentation]The flow of ice from the interior of polar ice sheets is the primary contributor to projected sea level rise. One of the main difficulties faced in modeling ice sheet flow is the uncertain spatiallyvarying Robin boundary condition that describes the resistance to sliding at the base of the ice. Satellite observations of the surface ice flow velocity, along with a model of ice as a creeping incompressible shearthinning fluid, can be used to infer this uncertain basal boundary condition. We cast this illposed inverse problem in the framework of Bayesian inference, which allows us to infer not only the basal sliding parameters, but also the associated uncertainty. To overcome the prohibitive nature of Bayesian methods for largescale inverse problems, we exploit the fact that, despite the large size of observational data, they typically provide only sparse information on model parameters. We show results for Bayesian inversion of the basal sliding parameter field for the full Antarctic continent, and demonstrate that the work required to solve the inverse problem, measured in number of forward (and adjoint) ice sheet model solves, is independent of the parameter and data dimensions

Computational error estimates for Monte Carlo finite element approximation with log normal diffusion coefficients(20150107) [Presentation]The Monte Carlo (and Multilevel Monte Carlo) finite element method can be used to approximate observables of solutions to diffusion equations with log normal distributed diffusion coefficients, e.g. modelling ground water flow. Typical models use log normal 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. This talk will address how the total error can be estimated by the computable error.

Information Theoretic Tools for Parameter Fitting in Coarse Grained Models(20150107) [Poster]We study the application of information theoretic tools for model reduction in the case of systems driven by stochastic dynamics out of equilibrium. The model/dimension reduction is considered by proposing parametrized coarse grained dynamics and finding the optimal parameter set for which the relative entropy rate with respect to the atomistic dynamics is minimized. The minimization problem leads to a generalization of the force matching methods to non equilibrium systems. A multiplicative noise example reveals the importance of the diffusion coefficient in the optimization problem.

MetropolisHastings Algorithms in Function Space for Bayesian Inverse Problems(20150107) [Presentation]We consider Markov Chain Monte Carlo methods adapted to a Hilbert space setting. Such algorithms occur in Bayesian inverse problems where the solution is a probability measure on a function space according to which one would like to integrate or sample. We focus on MetropolisHastings algorithms and, in particular, we introduce and analyze a generalization of the existing pCNproposal. This new proposal allows to exploit the geometry or anisotropy of the target measure which in turn might improve the statistical efficiency of the corresponding MCMC method. Numerical experiments for a realworld problem confirm the improvement.

Simulation of conditional diffusions via forwardreverse stochastic representations(20150107) [Presentation]We derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval,conditioned on the terminal state. The conditioning can be with respect to a fixed measurement point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced by Milstein, Schoenmakers and Spokoiny in the context of density estimation. The corresponding Monte Carlo estimators have essentially rootN accuracy, and hence they do not suffer from the curse of dimensionality. We also present an application in statistics, in the context of the EM algorithm.

Discrete least squares polynomial approximation with random evaluations  application to PDEs with Random parameters(20150107) [Presentation]We consider a general problem F(u, y) = 0 where u is the unknown solution, possibly Hilbert space valued, and y a set of uncertain parameters. We specifically address the situation in which the parametertosolution map u(y) is smooth, however y could be very high (or even infinite) dimensional. In particular, we are interested in cases in which F is a differential operator, u a Hilbert space valued function and y a distributed, space and/or time varying, random field. We aim at reconstructing the parametertosolution map u(y) from random noisefree or noisy observations in random points by discrete least squares on polynomial spaces. The noisefree case is relevant whenever the technique is used to construct metamodels, based on polynomial expansions, for the output of computer experiments. In the case of PDEs with random parameters, the metamodel is then used to approximate statistics of the output quantity. We discuss the stability of discrete least squares on random points show convergence estimates both in expectation and probability. We also present possible strategies to select, either apriori or by adaptive algorithms, sequences of approximating polynomial spaces that allow to reduce, and in some cases break, the curse of dimensionality

Transport maps and dimension reduction for Bayesian computation(20150107) [Presentation]We introduce a new framework for efficient sampling from complex probability distributions, using a combination of optimal transport maps and the MetropolisHastings rule. The core idea is to use continuous transportation to transform typical Metropolis proposal mechanisms (e.g., random walks, Langevin methods) into nonGaussian proposal distributions that can more effectively explore the target density. Our approach adaptively constructs a lower triangular transport map—an approximation of the KnotheRosenblatt rearrangement—using information from previous MCMC states, via the solution of an optimization problem. This optimization problem is convex regardless of the form of the target distribution. It is solved efficiently using a Newton method that requires no gradient information from the target probability distribution; the target distribution is instead represented via samples. Sequential updates enable efficient and parallelizable adaptation of the map even for large numbers of samples. We show that this approach uses inexact or truncated maps to produce an adaptive MCMC algorithm that is ergodic for the exact target distribution. Numerical demonstrations on a range of parameter inference problems show orderofmagnitude speedups over standard MCMC techniques, measured by the number of effectively independent samples produced per target density evaluation and per unit of wallclock time. We will also discuss adaptive methods for the construction of transport maps in high dimensions, where use of a nonadapted basis (e.g., a total order polynomial expansion) can become computationally prohibitive. If only samples of the target distribution, rather than density evaluations, are available, then we can construct highdimensional transformations by composing sparsely parameterized transport maps with rotations of the parameter space. If evaluations of the target density and its gradients are available, then one can exploit the structure of the variational problem used for map construction. In both settings, we will show links to recent ideas for dimension reduction in inverse problems.

NonIntrusive Solution of Stochastic and Parametric Equations(20150107) [Presentation]Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “nonintrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto the subspace spanned by the approximating functions. Usually this will involve minimising some norm of the difference between the true parametric solution and the approximation. Such methods are sometimes called pseudospectral projections, or regression solutions. On the other hand, methods which try to ensure that the approximation satisfies the parametric equation as well as possible are often based on a RayleighRitz or Galerkin type of “ansatz”, which leads to a coupled system for the unknown coefficients. This is often taken as an indication that the original solver can not be used, i.e. that these methods are “intrusive”. But in many circumstances these methods may as well be used in a nonintrusive fashion. Some very effective new methods based on lowrank approximations fall in the class of “not obviously nonintrusive” methods; hence it is important to show here how this may be computed nonintrusively.

Adaptive Surrogate Modeling for Response Surface Approximations with Application to Bayesian Inference(20150107) [Presentation]The need for surrogate models and adaptive methods can be best appreciated if one is interested in parameter estimation using a Bayesian calibration procedure for validation purposes. We extend here our latest work on error decomposition and adaptive refinement for response surfaces to the development of surrogate models that can be substituted for the full models to estimate the parameters of Reynoldsaveraged NavierStokes models. The error estimates and adaptive schemes are driven here by a quantity of interest and are thus based on the approximation of an adjoint problem. We will focus in particular to the accurate estimation of evidences to facilitate model selection. The methodology will be illustrated on the SpalartAllmaras RANS model for turbulence simulation.

Numerical Solution of Stochastic Nonlinear Fractional Differential Equations(20150107) [Poster]Using WienerHermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the GrunwaldLetnikov approximation in case of fractional order and with Coimbra approximation in case of variableorder damping. The numerical solver was tested with the analytic solution and with MonteCarlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.

Low Complexity Beampattern Design in MIMO Radars Using Planar Array(20150107) [Poster]In multipleinput multipleoutput radar systems, it is usually desirable to steer transmitted power in the regionofinterest. To do this, conventional methods optimize the waveform covariance matrix, R, for the desired beampattern, which is then used to generate actual transmitted waveforms. Both steps require constrained optimization, therefore, use iterative and expensive algorithms. In this paper, we provide a closedform solution to design covariance matrix for the given beampattern using the planar array, which is then used to derive a novel closedform algorithm to directly design the finitealphabet constantenvelope (FACE) waveforms. The proposed algorithm exploits the twodimensional fastFouriertransform. The performance of our proposed algorithm is compared with existing methods that are based on semidefinite quadratic programming with the advantage of a considerably reduced complexity.

Scalable Hierarchical Algorithms for stochastic PDEs and UQ(20150107) [Poster]Hmatrices and Fast Multipole (FMM) are powerful methods to approximate linear operators coming from partial differential and integral equations as well as speed up computational cost from quadratic or cubic to loglinear (O(n log n)), where n number of degrees of freedom in the discretization. The storage is reduced to the loglinear as well. This hierarchical structure is a good starting point for parallel algorithms. Parallelization on shared and distributed memory systems was pioneered by Kriemann [1,2]. Since 2005, the area of parallel architectures and software is developing very fast. Progress in GPUs and ManyCore Systems (e.g. XeonPhi with 64 cores) motivated us to extend work started in [1,2,7,8].

Multilevel Hybrid Chernoff TauLeap(20150107) [Poster]Markovian pure jump processes can model many phenomena, e.g. chemical reactions at molecular level, protein transcription and translation, spread of epidemics diseases in small populations and in wireless communication networks, among many others. In this work [6] we present a novel multilevel algorithm for the Chernoffbased hybrid tauleap algorithm. This variance reduction technique allows us to: (a) control the global exit probability of any simulated trajectory, (b) obtain accurate and computable estimates for the expected value of any smooth observable of the process with minimal computational work.

Hierarchical matrix approximation of large covariance matrices(20150107) [Poster]We approximate large nonstructured covariance matrices in the Hmatrix format with a loglinear computational cost and storage O(n log n). We compute inverse, Cholesky decomposition and determinant in Hformat. As an example we consider the class of Matern covariance functions, which are very popular in spatial statistics, geostatistics, machine learning and image analysis. Applications are: kriging and optimal design

Comparison of quasioptimal and adaptive sparsegrids for groundwaterflowproblems(20150107) [Poster]

Quasioptimal sparsegrid approximations for random elliptic PDEs(20150107) [Poster]

Response Surface in Tensor Train Format for Uncertainty Quantification(20150107) [Poster]

A Polynomial Chaos Expansion technique for correlated variables(20150107) [Poster]