Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the regularity of the solution's dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel collocation and multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution's mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory

Probabilistic analysis of a gas storage cavity mined in a spatially random rock salt medium

In most engineering problems the material parameters spread over spatial extents but this variability is commonly neglected. Analyses mostly assign the mean value of a variable to the entire medium, while in the case of heterogeneous materials as geomaterials, this may lead to an unreliable design. The existing scatter in such materials can be represented in the design procedure using the random field concept. In this paper, the random field method is used in a probabilistic analysis of a gas storage cavern in rock salt. The rock salt formation, as a porous media with low permeability and particular creep features, has been used for decades as the host rock for the hydrocarbon storage. To achieve a reliable design, a probabilistic model is presented to compute the failure probability of a cavern mined in a spatially varying salt dome. Here, the nodilatant region around the cavity is regarded as the failure criterion. In this regard, a thermo-mechanical model of a natural gas storage in rock salt, employing BGRa creep law, is developed. Afterwards, the most effective input variable on the model response is identified, using global sensitivity analysis. The Karhunen-Loève expansion is introduced to generate random field. In the following, the subset simulation methodology is utilised to facilitate the execution of Monte-Carlo method. The findings of this study emphasize that considering spatial variability in rock properties significantly affects the reliability of a solution-mined cavity

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

Metamodel-based importance sampling for structural reliability analysis

Structural reliability methods aim at computing the probability of failure of systems with respect to some prescribed performance functions. In modern engineering such functions usually resort to running an expensive-to-evaluate computational model (e.g. a finite element model). In this respect simulation methods, which may require $10^{3-6}$ runs cannot be used directly. Surrogate models such as quadratic response surfaces, polynomial chaos expansions or kriging (which are built from a limited number of runs of the original model) are then introduced as a substitute of the original model to cope with the computational cost. In practice it is almost impossible to quantify the error made by this substitution though. In this paper we propose to use a kriging surrogate of the performance function as a means to build a quasi-optimal importance sampling density. The probability of failure is eventually obtained as the product of an augmented probability computed by substituting the meta-model for the original performance function and a correction term which ensures that there is no bias in the estimation even if the meta-model is not fully accurate. The approach is applied to analytical and finite element reliability problems and proves efficient up to 100 random variables.Comment: 20 pages, 7 figures, 2 tables. Preprint submitted to Probabilistic Engineering Mechanic

The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem

We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire (Arch Rational Mech Anal 113(3): 209-259,1990. doi: 10.1007/BF00375065 , Arch Rational Mech Anal 113(3): 261-298, 1990. doi: 10.1007/BF00375066 ) with a lognormal random permeability. Specifically, the permeability is assumed to be a lognormal random field taking values in the symmetric matrices of size $$d\times d$$ , where $$d$$ denotes the spatial dimension of the physical domain $$D$$ . We prove that the solutions admit bounded moments of any finite order with respect to the random input's Gaussian measure. We present a Mixed Finite Element discretization in the physical domain $$D$$ , which is uniformly stable with respect to the realization of the lognormal permeability field. Based on the error analysis of this mixed finite element method (MFEM), we develop a multi-level Monte Carlo (MLMC) discretization of the stochastic Brinkman problem and prove that the MLMC-MFEM allows the estimation of the statistical mean field with the same asymptotical accuracy versus work as the MFEM for a single instance of the stochastic Brinkman problem. The robustness of the MFEM implies in particular that the present analysis also covers the Darcy diffusion limit. Numerical experiments confirm the theoretical result

Quadrature methods for elliptic PDEs with random diffusion

In this thesis, we consider elliptic boundary value problems with random diffusion coefficients. Such equations arise in many engineering applications, for example, in the modelling of subsurface flows in porous media, such as rocks. To describe the subsurface flow, it is convenient to use Darcy's law. The key ingredient in this approach is the hydraulic conductivity. In most cases, this hydraulic conductivity is approximated from a discrete number of measurements and, hence, it is common to endow it with uncertainty, i.e. model it as a random field. This random field is usually characterized by its mean field and its covariance function. Naturally, this randomness propagates through the model which yields that the solution is a random field as well. The thesis on hand is concerned with the effective computation of statistical quantities of this random solution, like the expectation, the variance, and higher order moments. In order to compute these quantities, a suitable representation of the random field which describes the hydraulic conductivity needs to be computed from the mean field and the covariance function. This is realized by the Karhunen-Loeve expansion which separates the spatial variable and the stochastic variable. In general, the number of random variables and spatial functions used in this expansion is infinite and needs to be truncated appropriately. The number of random variables which are required depends on the smoothness of the covariance function and grows with the desired accuracy. Since the solution also depends on these random variables, each moment of the solution appears as a high-dimensional Bochner integral over the image space of the collection of random variables. This integral has to be approximated by quadrature methods where each function evaluation corresponds to a PDE solve. In this thesis, the Monte Carlo, quasi-Monte Carlo, Gaussian tensor product, and Gaussian sparse grid quadrature is analyzed to deal with this high-dimensional integration problem. In the first part, the necessary regularity requirements of the integrand and its powers are provided in order to guarantee convergence of the different methods. It turns out that all the powers of the solution depend, like the solution itself, anisotropic on the different random variables which means in this case that there is a decaying dependence on the different random variables. This dependence can be used to overcome, at least up to a certain extent, the curse of dimensionality of the quadrature problem. This is reflected in the proofs of the convergence rates of the different quadrature methods which can be found in the second part of this thesis. The last part is concerned with multilevel quadrature approaches to keep the computational cost low. As mentioned earlier, we need to solve a partial differential equation for each quadrature point. The common approach is to apply a finite element approximation scheme on a refinement level which corresponds to the desired accuracy. Hence, the total computational cost is given by the product of the number of quadrature points times the cost to compute one finite element solution on a relatively high refinement level. The multilevel idea is to use a telescoping sum decomposition of the quantity of interest with respect to different spatial refinement levels and use quadrature methods with different accuracies for each summand. Roughly speaking, the multilevel approach spends a lot of quadrature points on a low spatial refinement and only a few on the higher refinement levels. This reduces the computational complexity but requires further regularity on the integrand which is proven for the considered problems in this thesis

An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains

A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding Karhunen-Loeve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problem-dependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems

Numerical approximation and simulation of the stochastic wave equation on the sphere

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schr\uf6dinger equation on the unit sphere

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm

Probabilistic micromechanical spatial variability quantification in laminated composites

SN and SS are grateful for the support provided through the Lloyd’s Register Foundation Centre. The Foundation helps to protect life and property by supporting engineering-related education, public engagement and the application of research.Peer reviewedPostprin