A Bayesian numerical homogenization method for elliptic multiscale inverse problems

A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings

Bayesian inference with optimal maps

We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Grant DE-SC0002517)United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Grant DE-SC0003908

Description Length Based Signal Detection in singular Spectrum Analysis

This paper provides an information theoretic analysis of the signal-noise separation problem in Singular Spectrum Analysis. We present a signal-plus-noise model based on the Karhunen-Loève expansion and use this model to motivate the construction of a minimum description length criterion that can be employed to select both the window length and the signal. We show that under very general regularity conditions the criterion will identify the true signal dimension with probability one as the sample size increases, and will choose the smallest window length consistent with the Whitney embedding theorem. Empirical results obtained using simulated and real world data sets indicate that the asymptotic theory is reflected in observed behaviour, even in relatively small samples.Karhunen-Loève expansion, minimum description length, signal-plus-noise model, Singular Spectrum Analysis, embedding

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysi

STOCHASTIC MATCHED FILTERS FOR SIGNAL DETECTION APPLICATIONS

The stochastic matched filter (SMF) is a variation of the matched filter that can detect stochastic signals in noisy environments. Some earlier studies suggest that the SMF can be extended to the detection of frequency time-variant (nonstationary) signals, namely wideband modulated sonar in shallow water. This thesis considers the SMF algorithm first proposed by J.-F. Cavasillas in signal detection and estimation scenarios, and investigates its application to narrowband and chirp signals imbedded in white noise. In medium to high signal to noise ratio (SNR) values, results indicate that the SMF is a viable technique for signal detection and estimation, and could be employed in passive, real-time signal detection and estimation scenarios.Lieutenant Commander, United States NavyApproved for public release. Distribution is unlimited

A sequential sampling strategy for extreme event statistics in nonlinear dynamical systems

We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems, using a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the 'next-best' data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach utilizes Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds utilizing the posterior distribution of the inferred map. The 'next-best' design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process utilizes only information from the inferred map it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size, i.e. order O(10). We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to three-dimensional irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using limited number of samples

A spectral surrogate model for stochastic simulators computed from trajectory samples

Stochastic simulators are non-deterministic computer models which provide a different response each time they are run, even when the input parameters are held at fixed values. They arise when additional sources of uncertainty are affecting the computer model, which are not explicitly modeled as input parameters. The uncertainty analysis of stochastic simulators requires their repeated evaluation for different values of the input variables, as well as for different realizations of the underlying latent stochasticity. The computational cost of such analyses can be considerable, which motivates the construction of surrogate models that can approximate the original model and its stochastic response, but can be evaluated at much lower cost. We propose a surrogate model for stochastic simulators based on spectral expansions. Considering a certain class of stochastic simulators that can be repeatedly evaluated for the same underlying random event, we view the simulator as a random field indexed by the input parameter space. For a fixed realization of the latent stochasticity, the response of the simulator is a deterministic function, called trajectory. Based on samples from several such trajectories, we approximate the latter by sparse polynomial chaos expansion and compute analytically an extended Karhunen-Lo\`eve expansion (KLE) to reduce its dimensionality. The uncorrelated but dependent random variables of the KLE are modeled by advanced statistical techniques such as parametric inference, vine copula modeling, and kernel density estimation. The resulting surrogate model approximates the marginals and the covariance function, and allows to obtain new realizations at low computational cost. We observe that in our numerical examples, the first mode of the KLE is by far the most important, and investigate this phenomenon and its implications

Numerical treatment of imprecise random fields in non-linear solid mechanics

The quantification and propagation of mixed uncertain material parameters in the context of solid mechanical finite element simulations is studied. While aleatory uncertainties appear in terms of spatial varying parameters, i.e. random fields, the epistemic character is induced by a lack of knowledge regarding the correlation length, which is therefore described by interval values. The concept and description of the resulting imprecise random fields is introduced in detail. The challenges occurring from interval valued correlation lengths are clarified. These include mainly the stochastic dimension, which can become very high under some circumstances, as well as the comparability of different correlation length scenarios with regard to the underlying truncation error of the applied Karhunen-Loève expansion. Additionally, the computation time can increase drastically, if the straightforward and robust double loop approach is applied. Sparse stochastic collocation method and sparse polynomial chaos expansion are studied to reduce the number of required sample evaluations, i.e. the computational cost. To keep the stochastic dimension as low as possible, the random fields are described by Karhunen-Loève expansion, using a modified exponential correlation kernel, which is advantageous in terms of a fast convergence while providing an analytic solution. Still, for small correlation lengths, the investigated approaches are limited by the curse of dimensionality. Furthermore, they turn out to be not suited for non-linear material models. As a straightforward alternative, a decoupled interpolation approach is proposed, offering a practical engineering estimate. For this purpose, the uncertain quantities only need to be propagated as a random variable and deterministically in terms of the mean values. From these results, the so-called absolutely no idea probability box (ani-p-box) can be obtained, bounding the results of the interval valued correlation length being between zero and infinity. The idea is, to interpolate the result of any arbitrary correlation length within this ani-p-box, exploiting prior knowledge about the statistical behaviour of the input random field corresponding to the correlation length. The new approach is studied for one- and two-dimensional random fields. Furthermore, linear and non-linear finite element models are used in terms of linear-elastic or elasto-plastic material laws, the latter including linear hardening. It appears that the approach only works satisfyingly for sufficiently smooth responses but an improvement by considering also higher order statistics is motivated for future research.DFG/SPP 1886/NA330/12-1/E

Ensemble Kalman filter for multiscale inverse problems

We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is provided, and convergence of the approximate conditional posterior distribution is proved with respect to the Wasserstein distance. A numerical experiment validates our methodology, with a particular emphasis on modelling error and computational cost