A Novel Variable-Separation Method Based on Sparse and Low Rank Representation for Stochastic Partial Differential Equations

Li, Qiuqi; Jiang, Lijian

doi:10.1137/16m1100010

Cited by 14 publications

(11 citation statements)

References 40 publications

(45 reference statements)

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“…where p ( ) and f q ( ) are stochastic functions with respect to , a p ∶ × −→ R is a symmetric bilinear form, and b q ∶ −→ R is continuous functional; they are independent of . If (x; ) and f(x, t; ) are not affine with respect to , the empirical interpolation method 41,42 can be used, and the NVS method 25 can also be used to obtain their affine forms.…”

Section: Nvs Methods For Time-dependent Equationmentioning

confidence: 99%

“…The inner product (·,·; ξ ) is defined as

(w, v; ξ) = {(w (ξ), v (ξ))}_{L^{2} (𝒪)} .

We assume that the coefficient κ ( x ; ξ ) and the source term f ( x , t ; ξ ) have the variable‐separated form and that the bilinear form a (·,·; ξ ) and the associated linear form b (·; t , ξ ) are affine with respect to ξ , ie,

({, \begin{matrix} array a (w, v; ξ) = ∑_{p = 1}^{m_{a}} κ_{p} (ξ) a_{p} (w, v), \forall v, w \in V, \forall ξ \in Ω, \\ array b (v; t, ξ) = ∑_{q = 1}^{m_{b}} f_{q} (ξ) b_{q} (v; t), \forall v \in V, \forall ξ \in Ω, \end{matrix})

where κ p ( ξ ) and f q ( ξ ) are stochastic functions with respect to ξ ,

a_{p} : 𝒱 \times 𝒱 ⟶ double-struckR

is a symmetric bilinear form, and

b_{q} : 𝒱 ⟶ double-struckR

is continuous functional; they are independent of ξ . If κ ( x ; ξ ) and f ( x , t ; ξ ) are not affine with respect to ξ , the empirical interpolation method can be used, and the NVS method can also be used to obtain their affine forms.…”

Section: Proposed Approachmentioning

confidence: 99%

“…The proof of the inequation is provided in the Appendix. || • U L − U L N || is small given that • U L is the realization of the full low-fidelity models; when ||(C S L ) −1 ||||C L || is bounded, (25) suggests an accurate bi-fidelity error estimate. We will demonstrate the boundedness of ||(C S L ) −1 ||||C L || by numerical experiments in our paper.…”

Section: High-fidelity Approximation Via Basis Updatementioning

confidence: 99%

“…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. The novel variable‐separation (NVS) method proposed in the work of Li and Jiang provides a variable‐separated form approximation of the functions dependent on the stochastic inputs, where the stochastic basis functions can be expressed explicitly, escaping from the limitation of polynomial basis functions. Although recursive formulations can be derived for stochastic functions, it would be extremely time consuming when the degree of freedom for the model or the number of the separated term is large.…”

Section: Introductionmentioning

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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Section: Nvs Methods For Time-dependent Equationmentioning

confidence: 99%

“…The inner product (·,·; ξ ) is defined as

(w, v; ξ) = {(w (ξ), v (ξ))}_{L^{2} (𝒪)} .

({, \begin{matrix} array a (w, v; ξ) = ∑_{p = 1}^{m_{a}} κ_{p} (ξ) a_{p} (w, v), \forall v, w \in V, \forall ξ \in Ω, \\ array b (v; t, ξ) = ∑_{q = 1}^{m_{b}} f_{q} (ξ) b_{q} (v; t), \forall v \in V, \forall ξ \in Ω, \end{matrix})

where κ p ( ξ ) and f q ( ξ ) are stochastic functions with respect to ξ ,

a_{p} : 𝒱 \times 𝒱 ⟶ double-struckR

is a symmetric bilinear form, and

b_{q} : 𝒱 ⟶ double-struckR

Section: Proposed Approachmentioning

confidence: 99%

Section: High-fidelity Approximation Via Basis Updatementioning

confidence: 99%

Section: Introductionmentioning

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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“…As noted before, exploration of the posterior distribution requires repeated evaluations of the forward operator, and a large number of samples are needed to ensure reliable estimates of the inference. One attempts at accelerating Bayesian inference in computationally intensive inverse problems have relied on reduction of the stochastic forward model [2], the approaches of building stochastic surrogates include generalized polynomial chaos (gPC)-based stochastic method [38,26,43,39,20], Gaussian process [33,3] or projection-type reduced order models [18,28,29,23], for multiple solutions, proper orthogonal decomposition (POD) [18] reduced-order model is incorporated into the ensemble-based method [21] to significantly reduce the costs [17], etc. Large numbers of forward model simulations are required for these methods, especially when the dimension of stochastic parameter is high.…”

mentioning

confidence: 99%

Bayesian identification of discontinuous fields with an ensemble-based variable separation multiscale method

Ou,

Lin,

Jiang

2018

Preprint

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This work presents a multiscale model reduction approach to discontinuous fields identification problems in the framework of Bayesian inference. An ensemble-based variable separation (VS) method is proposed to approximate multiscale basis functions used to build a coarse model. The variable-separation expression is constructed for stochastic multiscale basis functions based on the random field, which is treated Gauss process as prior information. To this end, multiple local inhomogeneous Dirichlet boundary condition problems are required to be solved, and the ensemblebased method is used to obtain variable separation forms for the corresponding local functions. The local functions share the same interpolate rule for different physical basis functions in each coarse block. This approach significantly improves the efficiency of computation. We obtain the variable separation expression of multiscale basis functions, which can be used to the models with different boundary conditions and source terms, once the expression constructed. The proposed method is applied to discontinuous field identification problems where the hybrid of total variation and Gaussian (TG) densities are imposed as the penalty. We give a convergence analysis of the approximate posterior to the reference one with respect to the Kullback-Leibler (KL) divergence under the hybrid prior. The proposed method is applied to identify discontinuous structures in permeability fields. Two patterns of discontinuous structures are considered in numerical examples: separated blocks and nested blocks.

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