Stochastic Collocation Algorithms Using $l_1$-Minimization for Bayesian Solution of Inverse Problems

Yan, Liang; Guo, Ling

doi:10.1137/140965144

Cited by 37 publications

(27 citation statements)

References 33 publications

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“…In the online stage, we evaluate ||ê n ( )|| and, hence, Δ k ( ) for any using Equations (16) and (13).…”

Section: Nvs Methods For Time-dependent Equationmentioning

confidence: 99%

“…One attempt to accelerate Bayesian inference in computationally intensive inverse problems is to construct surrogates 13 of the stochastic forward model. The surrogate may be obtained by the generalized polynomial chaos-based stochastic method, [14][15][16][17][18][19] the Gaussian process, [20][21][22] or projection-type reduced-order models. 23,24 We note that the number of polynomial basis functions grows with an exponential rate as the dimension of the unknowns increases; although the number of forward model simulations required is less than the number of polynomial basis functions for sparsity-constricted stochastic collocation methods, it increases as the dimension of the unknowns increases to pursue accuracy.…”

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confidence: 99%

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A new bi‐fidelity model reduction method for Bayesian inverse problems

Jiang

Lin

2019

Numerical Meth Engineering

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Summary This work presents a new bi‐fidelity model reduction approach to the inverse problem under the framework of Bayesian inference. A low‐rank approximation is introduced to the solution of the corresponding forward problem and admits a variable‐separation form in terms of stochastic basis functions and physical basis functions. The calculation of stochastic basis functions is computationally predominant for the low‐rank expression. To significantly improve the efficiency of constructing the low‐rank approximation, we propose a bi‐fidelity model reduction based on a novel variable‐separation method, where a low‐fidelity model is used to compute the stochastic basis functions and a high‐fidelity model is used to compute the physical basis functions. The low‐fidelity model has lower accuracy but efficient to evaluate compared with the high‐fidelity model; it accelerates the derivative of recursive formulation for the stochastic basis functions. The high‐fidelity model is computed in parallel for a few samples scattered in the stochastic space when we construct the high‐fidelity physical basis functions. The required number of forward model simulations in constructing the basis functions is very limited. The bi‐fidelity model can be constructed efficiently while retaining good accuracy simultaneously. In the proposed approach, both the stochastic basis functions and physical basis functions are calculated using the model information. This implies that a few basis functions may accurately represent the model solution in high‐dimensional stochastic spaces. The bi‐fidelity model reduction is applied to Bayesian inverse problems to accelerate posterior exploration. A few numerical examples in time‐fractional derivative diffusion models are carried out to identify the smooth field and channel‐structured field in porous media in the framework of Bayesian inverse problems.

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“…In the online stage, we evaluate ||ê n ( )|| and, hence, Δ k ( ) for any using Equations (16) and (13).…”

Section: Nvs Methods For Time-dependent Equationmentioning

confidence: 99%

mentioning

confidence: 99%

A new bi‐fidelity model reduction method for Bayesian inverse problems

Jiang

Lin

2019

Numerical Meth Engineering

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“…In order to illustrate the accuracy and efficiency of the RB-EKI approach for solving the time fractional diffusion inverse problems, in this section, we present numerical experiments with two type different inverse problems. The first example, adapted from [38,39], considers a heat source inversion problem. The second example is the problem of inferring the spatially-varying diffusion coefficient [27,40].…”

Section: Numerical Examplesmentioning

confidence: 99%

A Non-Intrusive Reduced Basis EKI for Time Fractional Diffusion Inverse Problems

Yang

Yan

2019

Acta Math. Appl. Sin. Engl. Ser.

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In this study we consider an ensemble Kalman inversion (EKI) for the numerical solution of time-fractional diffusion inverse problems (TFDIPs). Computational challenges in the EKI arise from the need for repeated evaluations of the forward model. We address this challenge by introducing a non-intrusive reduced basis (RB) method for constructing surrogate models to reduce computational cost. In this method, a reduced basis is extracted from a set of full-order snapshots by the proper orthogonal decomposition (POD), and a doubly stochastic radial basis function (DSRBF) is used to learn the projection coefficients. The DSRBF is carried out in the offline stage with a stochastic leave-one-out cross-validation algorithm to select the shape parameter, and the outputs for new parameter values can be obtained rapidly during the online stage. Due to the complete decoupling of the offline and online stages, the proposed non-intrusive RB method -referred to as POD-DSRBF -provides a powerful tool to accelerate the EKI approach for TFDIPs. We demonstrate the practical performance of the proposed strategies through two nonlinear time-fractional diffusion inverse problems. The numerical results indicate that the new algorithm can achieve significant computational gains without sacrificing accuracy.

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“…LS-SCM inherits the merits from stochastic Galerkin methods and collocation methods. The recent work [39] employs a stochastic collocation algorithm using l1-minimization to construct stochastic sparse models with limited number of nodes, and their strategy has been applied to the Bayesian approach to handle nonlinear problems. The authors of [17] present a basis selection method that used with l1-minimization to adaptively determine the large coefficients of polynomial chaos expansions.…”

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confidence: 99%

Bayesian Inference Using Intermediate Distribution Based on Coarse Multiscale Model for Time Fractional Diffusion Equations

Jiang¹,

Ou²

2018

Multiscale Model. Simul.

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In the paper, we present a strategy for accelerating posterior inference for unknown inputs in time fractional diffusion models. In many inference problems, the posterior may be concentrated in a small portion of the entire prior support. It will be much more efficient if we build and simulate a surrogate only over the significant region of the posterior. To this end, we construct a coarse model using Generalized Multiscale Finite Element Method (GMsFEM), and solve a least-squares problem for the coarse model with a regularizing Levenberg-Marquart algorithm. An intermediate distribution is built based on the approximate sampling distribution. For Bayesian inference, we use GMsFEM and least-squares stochastic collocation method to obtain a reduced coarse model based on the intermediate distribution. To increase the sampling speed of Markov chain Monte Carlo, the DREAM ZS algorithm is used to explore the surrogate posterior density, which is based on the surrogate likelihood and the intermediate distribution. The proposed method with lower gPC order gives the approximate posterior as accurate as the the surrogate model directly based on the original prior. A few numerical examples for time fractional diffusion equations are carried out to demonstrate the performance of the proposed method with applications of the Bayesian inversion. ).Here the α is the regularization parameter and I is the identity matrix of size n z × n z , J is the sensitivity matrix, whose transpose is defined byThe gradient of G ri (z k ), i = 1, · · · , n d can be approximated by a difference method, e.g., each column of J can be computed by J(:, j) = G r (z k + hη j ) − G r (z k ) h

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Stochastic Collocation Algorithms Using $l_1$-Minimization for Bayesian Solution of Inverse Problems

Cited by 37 publications

References 33 publications

A new bi‐fidelity model reduction method for Bayesian inverse problems

A new bi‐fidelity model reduction method for Bayesian inverse problems

A Non-Intrusive Reduced Basis EKI for Time Fractional Diffusion Inverse Problems

Bayesian Inference Using Intermediate Distribution Based on Coarse Multiscale Model for Time Fractional Diffusion Equations

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