A fast algorithm for parabolic PDE-based inverse problems based on Laplace transforms and flexible Krylov solvers

Journal Article · Thu Oct 15 04:00:00 EDT 2015 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2015.07.007· OSTI ID:22465674

Bakhos, Tania ^[1]; Saibaba, Arvind K. ^[2]; Kitanidis, Peter K. ^[1]

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1. Institute for Computational and Mathematical Engineering, Stanford University (United States)
2. Department of Electrical and Computer Engineering, Tufts University (United States)

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method.

• OSTI ID: 22465674
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 299; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
CALCULATION METHODS
HYDRAULICS
JACOBIAN FUNCTION
LAPLACE TRANSFORMATION
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
PRESSURE MEASUREMENT
PUMPING
SENSITIVITY
TIME DEPENDENCE
TOMOGRAPHY