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Exploiting Sparsity in Solving PDE-Constrained Inverse Problems: Application to Subsurface Flow Model Calibration

  • Azarang Golmohammadi
  • M-Reza M. Khaninezhad
  • Behnam Jafarpour
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)

Abstract

Inverse problems are frequently encountered in many areas of science and engineering where observations are used to estimate the parameters of a system. In several practical applications, the dynamic processes that take place in a physical system are described using a set of partial differential equations (PDEs), which are typically nonlinear and coupled. The inverse problems that arise in those systems ought to be constrained to honour the governing PDEs. In this chapter, we consider high-dimensional PDE-constrained inverse problems in which, because of spatial patterns and correlations in the distribution of physical properties of a system, the underlying parameters tend to reside in (usually unknown) low-dimensional manifolds, thus have sparse (low-rank) representations. The sparsity of the parameters is amenable to an effective and flexible regularization form that can be exploited to improve the solution of such inverse problems. In applications where prior training data are available, sparse manifold learning methods can be adopted to tailor parameter representations to the specific requirements of the prior data. However, a major risk in employing prior training data is the significant uncertainty about the underlying conceptual models and assumptions used to develop the prior. A group-sparsity formulation is discussed for addressing the uncertainty in the prior training data when multiple distinct, but plausible, prior scenarios are encountered. Examples from geosciences application are presented where images of rock material properties are reconstructed from limited nonlinear fluid flow measurements.

Notes

Acknowledgements

The content of this chapter is based on research partially funded by the US Department of Energy, Foundation CMG, and American Chemical Society.

Appendix 1: k-SVD Dictionary Learning

The k-SVD algorithm is used to construct learned sparse dictionaries from a training dataset. The algorithm is similar to the k-means clustering method and is designed to find a dictionary \(\boldsymbol {\Phi } \in \mathbb {R}^{n\times k}\) containing k elements that sparsely represent each of the training samples in Un×L = [u1uiuL]. To achieve this goal, the algorithm attempts to solve the following minimization problem:
$$\displaystyle \begin{aligned} \hat{\mathbf{V}},\hat{\boldsymbol{\Phi}}={\text{argmin}}_{\mathbf{V},\boldsymbol{\Phi}}\quad {\sum_{i=1}^{L}{\lVert {\mathbf{u}}_i- \boldsymbol{\Phi}{\mathbf{v}}_i \rVert}_2^2}\quad \quad \text{s.t.,}\quad \quad {\lVert {\mathbf{v}}_i \rVert}_0\leq S \quad \text{for} \quad i\in1:L\end{aligned} $$
(31)
where Vk×L = [v1vivL] are the expansion coefficients corresponding to the training data. Given the NP-hard nature of the problem, the k-SVD algorithm uses a heuristic greedy solution technique by dividing the above optimization problem into two subproblems: (i) sparse coding and (ii) dictionary update. In the sparse coding step, for the current dictionary, a basis pursuit algorithm is used to find the sparse representation for each member of the training dataset. In the dictionary update step, the sparse representation obtained in the first step is fixed and the dictionary elements are updated to reduce the sparse approximation error. These two steps are repeated until convergence. Table 2 summarizes the k-SVD algorithm. Further details about the k-SVD algorithm may be found in [2]. We note that for high-dimensional training data the k-SVD dictionary learning can be computationally expensive. The computational complexity of each iteration of k-SVD is O(L(2nk + S2k + 7Sk + S3 + 4Sn) + 5nk2), where S is the sparsity level. One strategy to improve the computational efficiency of the algorithm includes using segmentation or approximate low-rank representations of the training data (to reduce n).
Table 2

k-SVD algorithm

Initialization: Initialize dictionary with \(\boldsymbol {\Phi }^{(0)} \in \mathbb {R}^ {n \times k}\). Set j = 1.

REPEAT until stopping criteria is met

a. Sparse Coding Step:

-Using a pursuit algorithm (e.g. OMP) compute \({\mathbf {V}}_{k \times L}^{(j)}=[{\mathbf {v}}_1 {\mathbf {v}}_2\ldots {\mathbf {v}}_L]\) as the solution of

\({\mathbf {V}}^{(j)}={\text{argmin}}_{{\mathbf {v}}_i} \quad {\lVert {\mathbf {u}}_i- \boldsymbol {\Phi }^{(j-1)}{\mathbf {v}}_i \rVert }_2^2\quad \quad \text{s.t.,}\quad \quad {\lVert {\mathbf {v}}_i \rVert }_0\leq S \quad \text{for} \quad i\in 1:L\)

b. Dictionary Update Step:

For each column c = 1, 2, …, k in Φ(j−1)

-Define the group of prior model instances that use this element

ωc = {i|1 ≤ i ≤ L, V(j)(c, i) ≠ 0}

-Compute the residual matrix \(\mathbf {{E}}_c=\mathbf {{U}}-\sum _{i\neq c}^{}\boldsymbol {\phi }_i{{\mathbf {v}}_c}^{T}\), where vcT is the cth row of V(j)

-Restrict Ec by choosing columns corresponding to ωc , i.e. find \(\mathbf {{E}}_c^{\omega }\)

-Apply rank-1 SVD decomposition \(\mathbf {{E}}_c^{\omega }=\mathbf {{A}}\boldsymbol {\Delta }\mathbf {{B}}\)

-Update the dictionary element ϕc = a1 and the sparse representation vc by \({\mathbf {v}}_c^{\omega }=\boldsymbol {\Delta }{\mathbf {b}}_1\)

-END

Appendix 2: IRLS Algorithm

We use the IRLS algorithm [14] to solve the 1-norm regularized least-square minimization problem, that is:
$$\displaystyle \begin{aligned} \underset{\mathbf{v}}{\text{min}} \quad J(\mathbf{v})={\lVert \mathbf{v} \rVert}_1 + \lambda^2{\lVert \mathbf{d}- \mathbf{g}(\boldsymbol{\Phi}\mathbf{v}) \rVert}_2^2\end{aligned} $$
(32)
At iteration n of the IRLS algorithm, the 1-norm is approximated using a weighted 2-norm as follows:
$$\displaystyle \begin{aligned} \underset{{\mathbf{v}}^{(n)}}{\text{min}} \quad J({\mathbf{v}}^{(n)})=\sum_{i}^{}w_i^{(n)}{v_i^{(n)}}^2+ \lambda^2{\lVert \mathbf{d}- \mathbf{g}(\boldsymbol{\Phi}{\mathbf{v}}^{(n)}) \rVert}_2^2\end{aligned} $$
(33)
where \(w_i^{(n)}=\frac {1}{({v_i^{(n-1)}}^2+\epsilon ^{(n)})^{0.5}}\), (n) stands for the iteration n, and 𝜖(n) is a sequence of small numbers (that converge to zero with increasing n). Using this approximation of the objective function, and a first-order Taylor expansion for g( Φv(n)), the objective function in (33) takes the form:
$$\displaystyle \begin{aligned} \underset{{\mathbf{v}}^{(n)}}{\text{min}} \quad J({\mathbf{v}}^{(n)})=\sum_{i}^{}w_i^{(n)}{v_i^{(n)}}^2+ \lambda^2{\lVert \mathbf{d}- \mathbf{g}(\boldsymbol{\Phi}{\mathbf{v}}^{(n-1)})- {{\mathbf{G}}_{\mathbf{v}}}^{(n)}({\mathbf{v}}^{(n)}-{\mathbf{v}}^{(n-1)}) \rVert}_2^2 \end{aligned} $$
(34)
Here, Gv(n) is the Jacobian matrix of g(.) with respect to v at v = v(n−1). The updated solution at iteration n can be easily found by taking the derivative of the above convex function w.r.t. v(n) and setting it to zero.

Appendix 3: Group-Sparsity Inversion

The objective function for group-sparsity regularization can be expressed as:
$$\displaystyle \begin{aligned} \underset{\mathbf{v}}{\text{min}} \quad J(\mathbf{v})=\sum_{i=1}^{p}{\lVert {\mathbf{v}}_i \rVert}_2 + \lambda^2{\lVert \mathbf{d}- \mathbf{g}(\boldsymbol{\Phi}\mathbf{v}) \rVert}_2^2\end{aligned} $$
(35)
where the notations are discussed in the text. At iteration n, using the Gauss-Newton method and the first-order Taylor series for g( Φv), the linearized version of the above function takes the form:
$$\displaystyle \begin{aligned} \underset{{\mathbf{v}}^{(n)}}{\text{min}} \quad J({\mathbf{v}}^{(n)})=\sum_{i=1}^{p}(\sum_{j=1}^{s_i}({v_i^{j}}^{(n)})^{2})^{\frac{1}{2}} + \lambda^2{\lVert \mathbf{d}- \mathbf{g}(\boldsymbol{\Phi}{\mathbf{v}}^{(n-1)})- {{\mathbf{G}}_{\mathbf{v}}}^{(n)}({\mathbf{v}}^{(n)}-{\mathbf{v}}^{(n-1)}) \rVert}_2^2\end{aligned} $$
(36)
where Gv(n) is the Jacobian matrix of g(v), and \({v_i^{j}}\) is the jth basis in the ith group. Denoting Δd(n) = d −g( Φv(n−1)) + Gv(n)v(n−1), (36) can be simplified to:
$$\displaystyle \begin{aligned} \underset{{\mathbf{v}}^{(n)}}{\text{min}} \quad J({\mathbf{v}}^{(n)})=\sum_{i=1}^{p}(\sum_{j=1}^{s_i}({v_i^{j}}^{(n)})^{2})^{\frac{1}{2}} + \lambda^2{\lVert \boldsymbol{\Delta}{\mathbf{d}}^{(n)}-{{\mathbf{G}}_{\mathbf{v}}}^{(n)}{\mathbf{v}}^{(n)} \rVert}_2^2\end{aligned} $$
(37)
The derivative of the regularization term with respect to \({v_i^{j}}^{(n)}\) can be approximated as:
$$\displaystyle \begin{aligned} \frac{{v_i^{j}}^{(n)}}{(\sum_{k=1}^{s_i}({v_i^{k}}^{(n)})^{2})^{\frac{1}{2}}}\approx \frac{{v_i^{j}}^{(n)}}{(\sum_{k=1}^{s_i}({v_i^{k}}^{(n-1)})^{2}+{\epsilon_i}^{(n)})^{\frac{1}{2}}}\end{aligned} $$
(38)
where 𝜖i(n) is a small positive number that is used to avoid zero denominators. Note that \({v_i^{k}}^{(n)}\) in the denominator is approximated as \({v_i^{k}}^{(n-1)}\). Choosing 𝜖 such that 0 < 𝜖i(n) < 𝜖i(n−1) and \( \underset {n\rightarrow \infty }{\text{lim}}{\epsilon _i}^{(n)}=0\), it can be shown that this approximation does not change the solution of the original minimization problem. The iterative solution of (37) can now be derived as:
$$\displaystyle \begin{aligned} ( \boldsymbol{\Lambda}^{(n)}+\alpha { {{\mathbf{G}}_{\mathbf{v}}}^{(n)}}^{T} {{\mathbf{G}}_{\mathbf{v}}}^{(n)}) {\mathbf{v}}^{(n)} = \alpha { {{\mathbf{G}}_{\mathbf{v}}}^{(n)}}^{T}\boldsymbol{\Delta}{\mathbf{d}}^{(n)}\end{aligned} $$
(39)
where α = 2λ2, and Λ(n) is a diagonal matrix with diagonal entries \(\frac {1}{(\sum _{k=1}^{s_i}({v_i^{k}}^{(n-1)})^{2}+{\epsilon _i}^{(n)})^{\frac {1}{2}}}\).

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Authors and Affiliations

  • Azarang Golmohammadi
    • 1
  • M-Reza M. Khaninezhad
    • 1
  • Behnam Jafarpour
    • 1
    • 2
  1. 1.Ming Hsieh Department of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Mork Family Department of Chemical Engineering and Material ScienceUniversity of Southern CaliforniaLos AngelesUSA

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