Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically ...

SUMMARY Full-waveform inversion has become an essential technique for mapping geophysical subsurface structures. However, proper uncertainty quantification is often lacking in current applications. In theory, related to the inverse Hessian (or posterior covariance matrix). Even common problems its calculation beyond computational and storage capacities of largest high-performance computing syst...

The limited memory BFGS (L-BFGS) method is one of the popular methods for solving large-scale unconstrained optimization. Since standard L-BFGS uses a line search to guarantee its global convergence, it sometimes requires large number function evaluations. To overcome difficulty, we propose new with certain regularization technique. We show convergence under usual assumptions. In order make mor...

In this paper, a trust-region algorithm combining with the limited memory BFGS (L-BFGS) update is proposed for solving nonlinear equations, where the super relaxation technique(SRT) is used. We choose the next iteration point by SRT. The global convergence without the nondegeneracy assumption is obtained under suitable conditions. Numerical results show that this method is very effective for la...

The Broyden class of quasi-Newton updates for inverse Hessian approximation are transformed to the formal BFGS update, which makes possible to generalize the well-known Nocedal method based on the Strang recurrences to the scaled limited-memory Broyden family, using the same number of stored vectors as for the limited-memory BFGS method. Two variants are given, the simpler of them does not requ...