# Advanced techniques for multi-source, multi-parameter, and multi-physics inverse problems

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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: multi-source, multi-parameter, and multi-physics inverse problems. The computational cost of solving a multi-source 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 right-hand 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 A-optimal 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 optimize-then-discretize 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 A-optimal encoding weights. The proposed method is potentially attractive for real-time 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 multi-parameter inverse problem, also called joint inverse problem, as the simultaneous inference of multiple parameter fields. In this dissertation, we concentrate on two types of multi-parameter 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 multi-physics inverse problem. The regularization of a multi-parameter 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 cross-gradient, the normalized cross-gradient, 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 primal-dual Newton solver for joint inverse problems regularized with vecto- rial total variation. Introducing an auxiliary dual variable in the first-order 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. Multi-physics 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 full-waveform inversion, and the presence of local minima in its objective function when using high-frequency 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 multi-physics inverse problems is to allow an auxiliary physical phenomenon, e.g., electromagnetic waves, to provide the missing low-frequency information for full-waveform inversion. In this dissertation, we provide supporting evi- dence for this approach when using the vectorial total variation functional as a regularization.