Three-dimensional transient elastodynamic inversion using an error in constitutive relation functional

This work is concerned with large-scale three-dimensional (3D) inversion under transient elastodynamic conditions by means of the modified error in constitutive relation (MECR), an energy-based, cost functional. In contrast to quasi-static or frequency-domain contexts, time-domain formulations have so far seen very limited investigation. A peculiarity of time-domain MECR formulations is that each evaluation involves the solution of two elastodynamic problems (one forward, one backward), which moreover are coupled (unlike the case of L² misfit functionals, where the forward state does not depend on the adjoint state). This coupling creates a major computational bottleneck, making MECR-based inversion difficult for spatially 2D or 3D configurations. To overcome this obstacle, we propose an approach whose main ingredients are (a) setting the entire computational procedure in a consistent time-discrete framework that incorporates the chosen time-stepping algorithm, and (b) using an iterative successive over-relaxation-like method for the resulting stationarity equations. The resulting MECR-based inversion algorithm is formulated under quite general conditions, allowing for 3D transient elastodynamics, straightforward use of available parallel solvers, a wide array of time-stepping algorithms commonly used for transient structural dynamics, and flexible boundary condition and measurement settings. The proposed MECR algorithm is then demonstrated on computational experiments involving 2D and 3D transient elastodynamics and up to over 500 000 unknown elastic moduli.