Applied Mathematics & Information Sciences

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In this work we are concerned with the full waveform inversion problem. The problem is formulated as one of minimizing a nonlinear least squares functional. Assuming Fr\{e}chet differentiability we use the adjoint state approach to compute the gradient. To approximate local minima, we develop a discrete framework for descent methods in a finite difference lattice. We describe the methods of Gradient descent with line search and the positive definite secant update (BFGS) for computation in the lattice. To illustrate the methods numerical solutions of several examples in 1D are presented. In this case we carry out some analysis and provide a simple proof for identifiability of wave speeds using the spread and shrink argument. It is argued that we may build on this work and apply techniques such as regularization or bayesian inference in future investigations.

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