An Efficient Iteration Based on Reduced Basis Method for Time-Dependent Problems With Random Inputs

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline--Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

With the aim of efficiently estimating the upper bound of time-dependent failure probability for the time-dependent reliability problem involving random and interval inputs (RI-TDFP), an extended crossing rate method is proposed. The proposed method first converts the time-dependent hybrid reliability analysis into a temporal and spatial multi-parameter problem. Then, it uses the extended crossing rate strategy to estimate the RI-TDFP. The proposed strategy is a single-loop strategy, which is more efficient than the nested double-loop strategies. The proposed method avoids using discretization or optimization strategies to estimate the global minimum of the time-dependent performance function with respect to the time parameter and interval inputs, and thus the computational cost is further reduced. Finally, the proposed method avoids using the first order reliability method (FORM) for reliability analysis with respect to the random inputs, and thus the proposed method is more accurate than the FORM.

In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuska, F. Nobile, and R. Tempone, SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrat...

The nonintrusive generalized polynomial chaos (gPC) method is a popular computational approach for solving partial differential equations with random inputs. The main hurdle preventing its efficient direct application for high-dimensional input parameters is that the size of many parametric sampling meshes grows exponentially in the number of inputs (the “curse of dimensionality''). In this paper, we design a weighted version of the reduced basis method (RBM) for use in the nonintrusive gPC framework. We construct an RBM surrogate that can rigorously achieve a user-prescribed error tolerance and ultimately is used to more efficiently compute a gPC approximation nonintrusively. The algorithm is capable of speeding up traditional nonintrusive gPC methods by orders of magnitude without degrading accuracy, assuming that the solution manifold has low Kolmogorov width. Numerical experiments on our test problems show that the relative efficiency improves as the parametric dimension increases, demonstrating the potential of the method in delaying the curse of dimensionality. Theoretical results as well as numerical evidence justify these findings.

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk

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

In the paper, a reduced basis (RB) method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed. Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation (PDE) systems, model reduction for nonlocal systems becomes more critical. The method of snapshots and greedy (MOS-greedy) algorithm of RB method is developed for nonlocal problems with random inputs, which provides an efficient and reliable approximation of the solution. A major challenge lies in the excessive influence of the time domain on the model reduction process. To address this, the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation, where variable frequencies are independent. This enables parallel computation for approximating the solution in the frequency domain. Finally, the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems. Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.

We study methods that aim to reduce the dimension of a finite dimensional solution space, in which the solution corresponding to a certain parametrized Optimal Control Problems governed by environmental models, e.g. Quasi-Geostrophic flow, is sought. The parameter is modeled as a random variable to incorporate possible uncertainty, for example in parametric measurements. For such a reduction to be useful, it should be guaranteed, for every possible parameter value, that it results in an acceleration of the solution process while maintaining an accurate approximate solution. In order to do this, conditions are formulated, and under those conditions, several versions of a specific reduction method known as Proper Orthogonal Decomposition are implemented. We consider examples and show that a simplification of the general state of the art reduction method performs equally well.

A new reliability analysis method is proposed for time-dependent problems with explicit in time limit-state functions of input random variables and input random processes using the total probability theorem and the concept of composite limit state. The input random processes are assumed Gaussian. They are expressed in terms of standard normal variables using a spectral decomposition method. The total probability theorem is employed to calculate the time-dependent probability of failure using time-dependent conditional probabilities which are computed accurately and efficiently in the standard normal space using the first-order reliability method (FORM) and a composite limit state of linear instantaneous limit states. If the dimensionality of the total probability theorem integral is small, we can easily calculate it using Gauss quadrature numerical integration. Otherwise, simple Monte Carlo simulation (MCS) or adaptive importance sampling are used based on a Kriging metamodel of the conditional probabilities. An example from the literature on the design of a hydrokinetic turbine blade under time-dependent river flow load demonstrates all developments.

The task of repeatedly solving parametrized partial differential equations (pPDEs) in optimization, control, or interactive applications makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold on which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However this decomposition, usually implemented with aid from the empirical interpolation method (EIM) for nonlinear and/or parametric-nonaffine PDEs, can be challenging to implement, or results in severely degraded online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding efficiency degradation exhibited in traditional applications of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced approximation space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values whose snapshots span the reduced approximation space. Together, these two ingredients ensure that the proposed L1-ROC scheme is both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.

An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation

This thesis presents the development and the implementation of an uncertainty propagation algorithm based on the concept of spectral expansion. The first part of the thesis is dedicated to the study of uncertainty propagation methodologies and to the analysis of spectral techniques. The concepts introduced within this preliminary analysis are successively used for the derivation of the spectral algorithm. In Chapter 2 we discuss the application of higher order adjoint perturbation theory for coupled problems. This method is relatively easy to implement once the first order adjoint problem is defined, however it is computationally expensive. It is shown, for example, that the number of additional adjoint calculations needed to build the Hessian matrix of a response corresponds, for nonlinear problems, to two times the number of input parameters. It is also shown that for linear problems this number can be halved. It is also discussed that for linear problems it is possible to perform a ranking of the higher order perturbation components, while for nonlinear ones this is not the case. In general, higher order adjoint perturbation theory can be a useful tool to understand uncertainty propagation phenomena. In Chapter 3 an overview of spectral techniques for uncertainty quantification is presented. The mathematical backgrounds of two approaches, defined as intrusive and non-intrusive, are discussed. These approaches are applied to perform uncertainty quantification of a simplified coupled time-dependent problem. The illustrative example shows how non-intrusive approaches are relatively easy to apply while intrusive approaches are quite challenging from the implementation point of view. The curse of dimensionality affecting spectral techniques is also discussed. The example also demonstrates that for time-dependent problems, the convergence of spectral expansions required to represent stochastic outputs varies considerably during the transient. From this point of view, non-intrusive approaches allow the usage of different expansion orders at different times, thereby reducing the computational requirements. Using these initial conclusions as a starting point, an algorithm based on the definition of Polynomial Chaos Expansion is developed. Chapter 4 introduces this new algorithm for the application of quadrature based spectral techniques. This algorithm is based on the notion of sparse grid and its application is divided into two main steps. Firstly, the algorithm adds quadrature points exclusively along the main axes of the stochastic domain. During this phase the convergence of the PCE is assessed and a reduced multi-dimensional PCE is defined. Secondly, this reduced PCE is then used within the second part of the algorithm which focuses on the addition of higher dimensional sub-grids to the final quadrature rule. The adaptive sparse grid algorithm is tested for a reference stochastic case defined by using a simple source detector problem. The algorithm is first validated by comparing it to another sparse grid integration approach found in literature. It is successively shown how the particular construction of the spectral basis, based on a convergence check performed considering each random direction to be independent, can further reduce the number of realizations needed to build the spectral outputs. In Chapter 5 two cost reduction techniques which take advantage of the peculiar definition of the algorithm are presented. These techniques are proven to be effective in the reduction of quadrature points needed to reach convergence. Two uncertainty propagation examples are also considered. The method has been proven to be particularly effective for reactor physics applications, mainly because of the fact that higher order propagation phenomena are usually dominated by a limited set of input parameters. It is also shown, with the first example, that the convergence rate of the adaptive quadrature algorithm directly depends on the differentiability of the response surface. Chapter 6 shows another application of the adaptive sparse grid algorithm, this time to a time-dependent multi-physics problem. This problem is formulated in order to reproduce the type of system that arises when performing safety analysis. Two reference transients simulating an accident scenario of fast reactors are considered. Even in this case the adaptive algorithm proves to be very effective, being capable of reproducing all the stochastic outputs of interest with a relatively low number of realizations. In conclusion, adaptive spectral methods represent a computationally efficient uncertainty quantification technique when in presence of a moderately large set of random input parameters. However, this number strongly depends on the regularity of the response surface. Several strategies could be adopted in order to increase this number and make the method more appealing for a larger set of problems. An overview of these possibilities is presented in the final recommendation section of the thesis.

In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.