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Full-waveform inversion via the scaled boundary finite element method

Alireza Daneshyar, Stefan Kollmannsberger

TL;DR

This work introduces a time-domain full-waveform inversion framework that leverages a scaled boundary finite element method with radial discretization to handle spatially varying density. By deriving a semi-weak SB-SBFEM formulation and condensing dynamic stiffness matrices on the fly, the authors enable a two-level optimization where coarse forward solves feed fine-gradient computations, achieving parallelizable and scalable performance. The approach is validated on two benchmark specimens, showing that the complete equation with density variation yields superior reconstructions and faster convergence than simplified formulations. The results demonstrate robust high-resolution imaging capabilities and practical potential for HPC-enabled FWI deployments in heterogeneous media, with an open-source data repository accompanying the work.

Abstract

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 Fréchet 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.

Full-waveform inversion via the scaled boundary finite element method

TL;DR

This work introduces a time-domain full-waveform inversion framework that leverages a scaled boundary finite element method with radial discretization to handle spatially varying density. By deriving a semi-weak SB-SBFEM formulation and condensing dynamic stiffness matrices on the fly, the authors enable a two-level optimization where coarse forward solves feed fine-gradient computations, achieving parallelizable and scalable performance. The approach is validated on two benchmark specimens, showing that the complete equation with density variation yields superior reconstructions and faster convergence than simplified formulations. The results demonstrate robust high-resolution imaging capabilities and practical potential for HPC-enabled FWI deployments in heterogeneous media, with an open-source data repository accompanying the work.

Abstract

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 Fréchet 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.
Paper Structure (15 sections, 108 equations, 11 figures, 2 tables)

This paper contains 15 sections, 108 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Inverse problem
  3. Adjoint method
  4. Scaled boundary discretization
  5. weak formulation
  6. Low-frequency expansion
  7. Radial discretization
  8. Numerical implementation
  9. Specimen with hole
  10. specimen with weak inclusion
  11. Conclusion
  12. Authorship contribution statement
  13. Declaration of competing interest
  14. Data Availability
  15. Acknowledgement

Figures (11)

  • Figure 1: A typical two-dimensional scaled boundary finite element.
  • Figure 2: Schematics of radial discretization and condensation procedure.
  • Figure 3: Configuration of the problems: rectangular specimen with a circular hole (left) and with a randomly shaped weak inclusion (right).
  • Figure 4: Sine burst impulse: curve of normalized amplitude $a(t)/a_0$ versus normalized time $tf_0$.
  • Figure 5: Centroidal Voronoi tessellations over the domain of problem with 1000, 2000, 4000, and 8000 elements.
  • ...and 6 more figures