Advanced techniques for multisource, multiparameter, and multiphysics inverse problems
Abstract
With the increase in compute power and the advent of the big data
era, inverse problems have grown more complex, attempting to extract more
information and to use more data. While this evolution manifests itself in
multiple forms, we focus in this dissertation on three specific aspects: multisource, multiparameter, and multiphysics inverse problems.
The computational cost of solving a multisource inverse problem in
creases linearly with the number of experiments. A recently proposed method
to decrease this cost uses only a small number of random linear combinations
of all experiments for solving the inverse problem. This approach applies to
inverse problems where the PDE solution depends linearly on the righthand
side function that models the experiment. As this method is stochastic in
essence, the quality of the obtained reconstructions can vary, in particular
when only a small number of combinations are used. We propose to replace
the random weights traditionally used in the linear combinations of the experiments, with deterministic weights (or, encoding weights). We approach the
computation of these weights as an optimal experimental design problem, and develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty.
We call these weights Aoptimal encoding weights. Our framework applies to
inverse problems where the governing PDE is nonlinear with respect to the
inversion parameter field. We formulate the problem in infinite dimensions
and follow the optimizethendiscretize approach, devoting special attention
to the discretization and the choice of numerical methods in order to achieve
a computational cost that is independent of the parameter discretization. We
elaborate our method for a Helmholtz inverse problem, and derive the adjoint
based expressions for the gradient of the objective function of the optimization
problem for finding the Aoptimal encoding weights. The proposed method
is potentially attractive for realtime monitoring applications, where one can
invest the effort to compute optimal weights offline, to later solve an inverse
problem repeatedly, over time, at a fraction of the initial cost.
We define a multiparameter inverse problem, also called joint inverse
problem, as the simultaneous inference of multiple parameter fields. In this
dissertation, we concentrate on two types of multiparameter inverse problems.
In the first case, we have at our disposal a single type of observations, generated
by a single physical phenomenon which depends on multiple parameters. In
the second case, we utilize multiple datasets generated from physical phenomena that depend on different parameters; when the data are generated from
different physics, this is a multiphysics inverse problem. The regularization
of a multiparameter inverse problem plays a critical role. It not only acts as
a regularizer to the inverse problem, but can also be used to impose coupling
between the inversion parameters when they are known to share similar structures. We compare four joint regularizations terms: the crossgradient, the normalized crossgradient, the vectorial total variation, and a novel regular
izer based on the nuclear norm of a gradient matrix. Following comprehensive
numerical investigations, we concluded that vectorial total variation leads to
the best reconstructions. We next devoted our attention to develop an efficient
primaldual Newton solver for joint inverse problems regularized with vecto
rial total variation. Introducing an auxiliary dual variable in the firstorder
optimality condition, which we then solve using Newton method, we were able
to reduce the nonlinearity in the inverse problem. Through an extensive nu
merical investigation, we showed that this solver is scalable with respect to the
mesh size, the hyperparameter, and the number of inversion parameters. We
also observed that it significantly outperforms the classical Newton method
and the popular lagged diffusivity method when fine convergence tolerances
are needed.
Multiphysics inverse problems are becoming more popular as a way
to enhance the quality of the reconstructions by combining the strengths of
multiple imaging modalities. In this dissertation, we specialize to the case
of fullwaveform inversion, and the presence of local minima in its objective
function when using highfrequency data. The most practical workaround to
day remains a continuation scheme over the frequency of the source term.
However, in a seismic exploration setting, modern equipment does not allow
to generate data of sufficiently low frequencies. One potential application of
multiphysics inverse problems is to allow an auxiliary physical phenomenon,
e.g., electromagnetic waves, to provide the missing lowfrequency information
for fullwaveform inversion. In this dissertation, we provide supporting evi
dence for this approach when using the vectorial total variation functional as
a regularization.
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