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

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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.