Sifr Optimizer Waveform Inversion Hessian Bypass

The patent extracts and utilizes only the essential second-order derivative information (curvature) needed for optimization while discarding the full Hessian matrix computation. This is achieved through the Sifr optimizer which computes a simplified update direction that captures the beneficial effects of second-order methods without the computational burden of forming or factorizing the complete Hessian matrix, thus resolving the contradiction between convergence rate improvement and computational complexity reduction.

The patent changes the optimization parameters and update formulation to inherently address cycle skipping. The Sifr optimizer modifies the update direction computation to incorporate curvature information in a way that prevents cycle skipping artifacts, even when low-frequency components are absent from the data. This parameter change in the optimization approach simultaneously maintains computational efficiency while improving robustness.

The patent employs a computationally inexpensive approximation of second-order optimization updates that does not require the expensive iterative procedures of BFGS or Hessian-free methods. The Sifr optimizer computes a simplified update direction using direct formulas that avoid multiple inner iterations or memory-intensive matrix operations, thereby achieving good optimization performance with significantly reduced computation time compared to existing second-order methods.