High-dimensional Random Inputs Research Articles

A clustering-based partially stratified sampling for high-dimensional structural reliability assessment

Assessing structural reliability problem with high-dimensional random inputs is still challenging due to the “curse of dimensionality”. In this paper, this challenge is addressed by extending the Generalized Distribution Reconstruction Method via Characteristic Function Inversion (GDRM-CFI). Specifically, a clustering-based partially stratified sampling method is proposed for selecting high-dimensional points to numerically evaluate the characteristic function (CF) curve of complex high-dimensional problems. An improved number-theoretical method (i-NTM) is used to establish a uniform, efficient point set, ensuring determinism and reducing variability. Subsequently, a partial stratification approach partitions the high-dimensional space into orthogonal two-dimensional subspaces. The fundamental point set is projected into each subspace, and the k-means clustering algorithm identifies centroids within each, acting as representative points. The complete set of representative points from all subspaces formulates the high-dimensional point set. Numerical examples are investigated, which demonstrate the proposed method is effective for high-dimensional structural reliability assessment.

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

Stochastic virtual element methods for uncertainty propagation of stochastic linear elasticity

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

Stochastic collocation for optimal control problems with stochastic PDE constraints by meshless techniques

In this paper, we investigate the stochastic collocation method with meshless techniques for an optimal control problem of elliptic equations with random inputs. The proposed method does not depend on the mesh partition and only requires the solution of uncoupled deterministic problem at each interpolation point, which can be competitive for high-dimensional random inputs and handle easily a wider range of situations such as random variables that are unbounded and input data that depend nonlinearly on the random variables. A priori error estimates are provided for the control, state and co-state variables. The stochastic collocation method based on gradient projection is presented, and some numerical examples are carried out to illustrate the performance of the new scheme.

Probabilistic Power Flow of AC/DC Hybrid Grids With Addressing Boundary Issue of Correlated Uncertainty Sources

In practical AC/DC hybrid grids, the uncertainty sources such as wind speeds and loads are massive and bounded intrinsically, the latter feature of which is neglected in most probabilistic power flow (PPF) analyses unfortunately. In this paper, a scaled unscented transformation (SUT)-based PPF on the large AC/DC hybrid grids, in particular for addressing the boundary issue of these high-dimensional random input variables under Pearson correlation, is proposed and investigated. The empirical formula used for determining the scaling parameter of the SUT method is designed to obtain the sample points, with a purpose of capturing sufficient probability information from the preliminary standard normal distributions. Boundary inverse transformation (BIT) is developed to transform these samples into the original probability space via handling each dimensional boundary value, and to guarantee that all the sample points of SUT fall into a reasonable, physical interval. The effectiveness and advantage of the proposed method are validated by using a set of test results on the modified IEEE 1354-bus (PEGASE) system.

An adaptive polynomial skewed-normal transformation model for distribution reconstruction and reliability evaluation with rare events

Evaluating the failure probabilities with rare events remains a challenging task, especially when dealing with complex limit state functions with high-dimensional random inputs. To handle this problem, an adaptive polynomial skewed-normal transformation (A-PSNT) model is proposed in this paper for evaluating the probability distribution of the limit state function and failure probability. First, the polynomial transformation models on the basis of normal distribution are introduced, where the limitations are also pointed out. Then, the skewed-normal distribution is presented, which serves as the basis of the proposed polynomial transformation model. The PSNT with a given order can be initiated, whereby the unknown coefficients can be determined by using the probability weighted moments (PWMs)-matching technique. Further, a tail error criterion is proposed to select the most appropriate order for the PSNT model, which makes the proposed method an adaptive one. To handle the low- to high-dimensional random inputs, two low-discrepancy sampling methods are employed to generate samples for unbiasedly estimating the PWMs of the limit state function. Once the probability distribution of the limit state function is reconstructed by the proposed A-PSNT model, the failure probability, especially for the rare one, can be determined accordingly. The proposed method is validated through numerical examples involving both static and dynamic, low- to high-dimensional problems, where some classical reliability analysis methods are also adopted for comparisons. The results show that the proposed A-PSNT model not only can effectively reconstruct the probability distribution, but also is able to accurately predict the failure probability with rare events.

A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.

Unified reliability assessment for problems with low- to high-dimensional random inputs using the Laplace transform and a mixture distribution

Efficient evaluation of the failure probability of systems with low- to high-dimensional random inputs in a unified way is still a challenging task since the methods that work in low dimensions usually become inefficient in high dimensions and vice versa. In this paper, a unified method is proposed to address this challenge. First, the Laplace transform (LT) is introduced to characterize the output variable of the limit state function (LSF). Two fifth-degree cubature formulae are employed to numerically approximate the LT when the input parameter space is low/moderate-dimensional, whereas a low-discrepancy sampling technique is adopted for high-dimensional problems. A mixture of skew normal distributions, is then developed to recover the probability distribution of the LSF from the knowledge of its LT. By matching with discrete values of the LT, the parameters of the mixture distribution are identified and the probability distribution of the LSF can be reconstructed. Five numerical examples are investigated to verify and exemplify the proposed method, where some standard reliability analysis methods are also conducted for comparison. The results indicate that the proposed method can efficiently recover the probability distribution of the LSF and estimate the failure probability for problems with low- to high-dimensional random inputs within a unified framework. The source code is readily available at: https://github.com/Chao-Dang/Reliability-Analysis-Using-Laplace-Transform-and-Mixture-Distribtution.

ANOVA Gaussian process modeling for high-dimensional stochastic computational models

In this paper we present a novel analysis of variance Gaussian process (ANOVA-GP) emulator for models governed by partial differential equations (PDEs) with high-dimensional random inputs. The Gaussian process (GP) is a widely used surrogate modeling strategy, but it can become invalid when the inputs are high-dimensional. In this new ANOVA-GP strategy, high-dimensional inputs are decomposed into unions of local low-dimensional inputs, and principal component analysis (PCA) is applied to provide dimension reduction for each ANOVA term. We then systematically build local GP models for PCA coefficients based on ANOVA decomposition to provide an emulator for the overall high-dimensional problem. We present a general mathematical framework of ANOVA-GP, validate its accuracy and demonstrate its efficiency with numerical experiments.

Stress-based topology optimization under uncertainty via simulation-based Gaussian process

Gaussian processes (GP) form a well-established predictive tool which provides a natural platform for tackling high-dimensional random input data in challenging simulations. This paper introduces a generic framework for integrating Gaussian Processes with risk-based structural optimization. We solve robust and reliability-based design problems in the context of stress-based topology optimization under imperfections in geometry and material properties, and loading variability. We construct independent GPs for primal and adjoint quantities, namely the global maximum von Mises stress and its sensitivity where we enhance the computational efficiency by leveraging the information from multiresolution finite element simulations. The GP framework naturally lends itself to modeling noise in data. We investigate the effect of numerical modeling error in high-fidelity simulations via a noisy GP emulator and provide a pareto curve that shows the robustness of optimal design with respect to the noise level. We provide a posteriori error estimates that quantify the discrepancy between the noisy emulator and true simulations, and verify them with a numerical study. We demonstrate our approach on a benchmark L-shape structure which exhibits stress concentration, a compliant mechanism design and a heat sink design. We also provide practical guidelines for estimation of failure probability and its sensitivity to facilitate reliability-based topology optimization.

Rank adaptive tensor recovery based model reduction for partial differential equations with high-dimensional random inputs

This work proposes a systematic model reduction approach based on rank adaptive tensor recovery for partial differential equation (PDE) models with high-dimensional random parameters. Since the standard outputs of interest of these models are discrete solutions on given physical grids which are high-dimensional, we use kernel principal component analysis to construct stochastic collocation approximations in reduced dimensional spaces of the outputs. To address the issue of high-dimensional random inputs, we develop a new efficient rank adaptive tensor recovery approach to compute the collocation coefficients. Novel efficient initialization strategies for non-convex optimization problems involved in tensor recovery are also developed in this work. We present a general mathematical framework of our overall model reduction approach, analyze its stability, and demonstrate its efficiency with numerical experiments.

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

ABSTRACTIn this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.

A generalized multi-resolution expansion for uncertainty propagation with application to cardiovascular modeling

Computational models are used in a variety of fields to improve our understanding of complex physical phenomena. Recently, the realism of model predictions has been greatly enhanced by transitioning from deterministic to stochastic frameworks, where the effects of the intrinsic variability in parameters, loads, constitutive properties, model geometry and other quantities can be more naturally included. A general stochastic system may be characterized by a large number of arbitrarily distributed and correlated random inputs, and a limited support response with sharp gradients or event discontinuities. This motivates continued research into novel adaptive algorithms for uncertainty propagation, particularly those handling high dimensional, arbitrarily distributed random inputs and non-smooth stochastic responses.In this work, we generalize a previously proposed multi-resolution approach to uncertainty propagation to develop a method that improves computational efficiency, can handle arbitrarily distributed random inputs and non-smooth stochastic responses, and naturally facilitates adaptivity, i.e., the expansion coefficients encode information on solution refinement. Our approach relies on partitioning the stochastic space into elements that are subdivided along a single dimension, or, in other words, progressive refinements exhibiting a binary tree representation. We also show how these binary refinements are particularly effective in avoiding the exponential increase in the multi-resolution basis cardinality and significantly reduce the regression complexity for moderate to high dimensional random inputs. The performance of the approach is demonstrated through previously proposed uncertainty propagation benchmarks and stochastic multi-scale finite element simulations in cardiovascular flow.

Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs

In this paper we present a reduced basis ANOVA approach for partial deferential equations (PDEs) with random inputs. The ANOVA method combined with stochastic collocation methods provides model reduction in high-dimensional parameter space through decomposing high-dimensional inputs into unions of low-dimensional inputs. In this work, to further reduce the computational cost, we investigate spatial low-rank structures in the ANOVA-collocation method, and develop efficient spatial model reduction techniques using hierarchically generated reduced bases. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression

Independent sampling of orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models, using Polynomial Chaos (PC) expansions. It is known that bounding the spectral radius of a random matrix consisting of PC samples, yields a bound on the number of samples necessary to identify coefficients in the PC expansion via solution to a least-squares regression problem. We present a related analysis which guarantees a mean square convergence using a coherence parameter of the sampled PC basis that may be both analytically bounded and computationally estimated. Utilizing asymptotic results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under each respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the PC basis. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all such sampling schemes with identical support, and which guarantees recovery with a number of samples that is, up to logarithmic factors, linear in the number of basis functions considered. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In all observed cases the coherence-optimal sampling leads to similar or considerably improved accuracy over the other considered sampling distributions.

Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In this paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.

Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ1-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.

A weighted [formula omitted]-minimization approach for sparse polynomial chaos expansions

This work proposes a method for sparse polynomial chaos (PC) approximation of high-dimensional stochastic functions based on non-adapted random sampling. We modify the standard ℓ1-minimization algorithm, originally proposed in the context of compressive sampling, using a priori information about the decay of the PC coefficients, when available, and refer to the resulting algorithm as weightedℓ1-minimization. We provide conditions under which we may guarantee recovery using this weighted scheme. Numerical tests are used to compare the weighted and non-weighted methods for the recovery of solutions to two differential equations with high-dimensional random inputs: a boundary value problem with a random elliptic operator and a 2-D thermally driven cavity flow with random boundary condition.

Partitioned treatment of uncertainty in coupled domain problems: A separated representation approach

This work is concerned with the propagation of uncertainty across coupled domain problems with high-dimensional random inputs. A stochastic model reduction approach based on low-rank separated representations is proposed for the partitioned treatment of the uncertainty space. The construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub-domain and not the combined dimensionality, hence drastically reducing the overall computational cost. The coupling between the sub-domain solutions is done via the classical finite element tearing and interconnecting (FETI) method, thus providing a well suited framework for parallel computing. Two high-dimensional stochastic problems, a 2D elliptic PDE with random diffusion coefficient and a stochastic linear elasticity problem, have been considered to study the performance and accuracy of the proposed stochastic coupling approach.

Accurate and efficient evaluation of failure probability for partial different equations with random input data

Several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis. When the dimension of the uncertainties becomes high, well established direct numerical methods can not be employed because of the “curse-of-dimensionality”. Many surrogate models have been proposed with the aim of reducing computational effort. However, most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness. In addition, for a stochastic system modeled by partial differential equations (PDEs) with random input, only a limited number of the underlying PDEs (order of a few tens) are affordable to solve in practice due to the considerable computational effort, therefore preventing the application of many numerical methods especially for high dimensional random inputs. In this work we develop hybrid and goal-oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic PDE. The curse-of-dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal-oriented adaptation. Moreover, an accurate evaluation of the failure probability for PDE system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error. At the end of this paper we suitably extend our proposed method to deal with more general PDE models. Finally we perform several numerical experiments to illustrate its computational accuracy and efficiency.