Inverse Hessian by stochastic projection and application to system identification in nonlinear mechanics of solids

Abstract

Parameter identification in nonlinear mechanics of solids, a class of non-convex optimization problems of significant interest, often has to balance fast convergence with enhanced exploration. Local optimization schemes based on Newton or quasi-Newton updates cannot tread this fine balance. A similar limitation also applies to many derivative-free global evolutionary searches that typically end up with slower convergence. Notwithstanding a relatively rapid convergence around a local optimum, schemes using classical gradients require that the objective function be smooth. Thus, applied to real-world scenarios, these algorithms face challenges such as non-smoothness including discontinuities in the objective function and costly evaluations of gradients or Hessians, whenever possible. Based on a novel stochastic projection to estimate inverse Hessians, this work aims at generalizing a fast Newton-type local search whilst retaining capabilities of noise-assisted exploration in non-convex optimization. In principle, such a meeting of the best of both worlds in optimization should provide for a powerful paradigm shift, offering a versatile and robust tool for solving complex identification and variational problems arising in nonlinear mechanics of solids. The approach is well-suited even for cases where the objective function is implicitly defined and not amenable to explicit evaluation. In a first implementation of our proposal, we assess its performance against a few benchmark non-convex problems. Finally, we solve a parameter estimation problem for an isotropic and incompressible hyperelastic material.