In this paper, we describe tensor methods for large systems of nonlinear equations based on Krylov subspace techniques for approximately solving the linear systems that are required in each tensor iteration. We refer to a method in this class as a tensor-Krylov algorithm. We describe comparative testing for a tensor-Krylov implementation versus an analogous implementation based on a Newton-Kryl...

We explain an interface for the implementation of rate-independent elastoplasticity which separates the pointwise evaluation of the elastoplastic material law and the global solution of the momentum balance equation. The elastoplastic problem is discretized in time by the implicit Euler method and every time step is solved with a Newton iteration. For the discretization in space the material pa...

In this paper we study new preconditioners to be used within the Nonlinear Conjugate Gradient (NCG) method, for large scale unconstrained optimization. The rationale behind our proposal draws inspiration from quasi– Newton updates, and its aim is to possibly approximate in some sense the inverse of the Hessian matrix. In particular, at the current iteration of the NCG we consider some precondit...

In this paper we present a primal interior-point hybrid proximal extragradient (HPE) method for solving a monotone variational inequality over a closed convex set endowed with a selfconcordant barrier and whose underlying map has Lipschitz continuous derivative. In contrast to the method of [7] in which each iteration required an approximate solution of a linearized variational inequality over ...

Newton’s method for the inverse matrix pth root, A−1/p, has the attraction that it involves only matrix multiplication. We show that if the starting matrix is cI for c ∈ R then the iteration converges quadratically to A−1/p if the eigenvalues of A lie in a wedge-shaped convex set containing the disc { z : |z−cp| < cp }. We derive an optimal choice of c for the case where A has real, positive ei...

Current successful methods for solving semidefinite programs, SDP, are based on primal-dual interior-point approaches. These usually involve a symmetrization step to allow for application of Newton’s method followed by block elimination to reduce the size of the Newton equation. Both these steps create ill-conditioning in the Newton equation and singularity of the Jacobian of the optimality con...

By making use of duality mappings and the Bregman distance, we propose a regularizing Levenberg-Marquardt scheme to solve nonlinear inverse problems in Banach spaces, which is an extension of the one proposed in [6] in Hilbert space setting. The method consists of two components: an outer Newton iteration and an inner scheme. The inner scheme involves a family of convex minimization problems in...

This work presents a global radial basis collocation combining with the quasiNewton iteration method for solving semilinear elliptic partial differential equations. A convergence analysis for such a meshfree discretization has been established. The main result is that there exists an exponential convergence rate with respect to the number and the shape of the radial basis functions. In addition...

Full Waveform Inversion (FWI) is a powerful seismic imaging method, based on the iterative minimization of the distance between simulated and recorded wavefields. The inverse Hessian operator related to this misfit function plays an important role in the reconstruction scheme. As conventional methods use direct approximations of this operator, we investigate an alternative optimization scheme: ...

This paper reports theoretical and empirical investigations on the use of quasi-Newton methods to minimize the Optimal Bellman Residual (OBR) of zero-sum two-player Markov Games. First, it reveals that state-of-the-art algorithms can be derived by the direct application of Newton’s method to different norms of the OBR. More precisely, when applied to the norm of the OBR, Newton’s method results...