Finite difference method

Abstract: We have applied the finite-difference contrast-source inversion (FDCSI) method to seismic full-waveform inversion problems. The FDCSI method is an iterative nonlinear inversion algorithm. However, unlike the nonlinear conjugate gradient method and the Gauss-Newton method, FDCSI does not solve any full forward problem explicitly in each iterative step of the inversion process. This feature makes the method very efficient in solving large-scale computational problems. It is shown that FDCSI, with a significant lower computation cost, can produce inversion results comparable in quality to those produced by the Gauss-Newton method and better than those produced by the nonlinear conjugate gradient method. Another attractive feature of the FDCSI method is that it is capable of employing an inhomogeneous background medium without any extra or special effort. This feature is useful when dealing with time-lapse inversion problems where the objective is to reconstruct changes between the baseline and the monitor model. By using the baseline model as the background medium in crosswell seismic monitoring problems, high quality time-lapse inversion results are obtained.

Abstract: We study locally mass conservative approximations of coupled Darcy and Stokes flows on polygonal and polyhedral meshes. The discontinuous Galerkin (DG) finite element method is used in the Stokes region and the mimetic finite difference method is used in the Darcy region. DG finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces. Optimal convergence estimates for the numerical scheme are derived. Results from computational experiments supporting the theory are presented.

Abstract: A finite-difference method for integro-differential equations arising from Le´vy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be second-order accurate for a relevant parameter range determining the degree of the singularity in the Le´vy measure. The singularity is dealt with by means of an integration by parts technique. An application of the fast Fourier transform gives the overall amount of work $O(N_t N\log N)$, rendering the method fast.