Full-waveform inversion via the scaled boundary finite element method

We begin by addressing the time-domain full-waveform inversion using the adjoint method. Next, we derive the scaled boundary semi-weak form of the scalar wave equation in heterogeneous media through the Galerkin method. Unlike conventional formulations, the resulting system incorporates variable density and two additional terms involving its spatial derivative. As a result, the coefficient matrices are no longer constant and depend on the radial coordinate, rendering the common solution methods inapplicable. Thus, we introduce a radial discretization scheme within the framework of the scaled boundary finite element method. We employ finite difference approximation, yet the choice underlying our ansatz is made for demonstration purposes and remains flexible. Next, we introduce an algorithmic condensation procedure to compute the dynamic stiffness matrices on the fly. Therefore, we maneuver around the need to introduce auxiliary unknowns. As a result, the optimization problem is structured in a two-level hierarchy. We obtain the Fr\'echet kernel by computing the zero-lag cross-correlations of the forward and adjoint wavefields, and solve the minimization problems iteratively by moving downhill on the cost function hypersurface through the limited-memory BFGS algorithm. The numerical results demonstrate the effectiveness and robustness of the new formulation and show that using the simplified differential equation along with the conventional formulation is highly inferior to applying the complete form of the differential equation. This approach effectively decomposes the computational load into independent local problems and a single coupled global system, making the solution method highly parallelizable. We demonstrate that, with a simple OpenMP implementation using 12 threads on a personal laptop, the new formulation outperforms the existing approach in terms of computation time.