Deflation for semismooth equations

Abstract

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite-and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics. KEYWORDSDeflation, semismooth Newton, variational inequalities, complementarity problems. AMS CLASSIFICATION65K15 Numerical methods for variational inequalities and related problems, 65P30 Bifurcation problems, 65H10 Systems of equations, 35M86 Nonlinear unilateral problems and nonlinear variational inequalities of mixed type, 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions).

Control of Bifurcation Structures using Shape Optimization

“…The bifurcation diagrams displayed in this section are computed with deflated continuation [23,24], complemented with arclength continuation. These techniques have been successfully applied to a wide range of physical problems such as the deformation of a hyperelastic beam [25], liquid crystals [20,78], Bose--Einstein condensates [10,13,14], and fluid dynamics [11]. However, our optimization strategy is not tied to a specific algorithm for bifurcation analysis, and other options may be used.…”

Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization

“…Deflation is a mechanism to systemically discover multiple solutions of a nonlinear system with a Newton-like algorithm [24,25]. Let Z and Y be Banach spaces.…”

“…In recent work, Papadopoulos et al [47] developed an algorithm, called the deflated barrier method, that can systematically discover multiple stationary points of topology optimization problems formulated using a density approach. The deflated barrier method combines the techniques of barrier methods [31,32,34,54,55,60,63], primal-dual active set solvers [10], and deflation [24,25]. In one example [47,Fig.…”

Numerical analysis of a discontinuous Galerkin method for the Borrvall-Petersson topology optimization problem

“…The global nonlinear convergence is aided by the continuation of barrier terms. A key feature of the deflated barrier method is that it can systematically discover multiple solutions of topology optimization problems by utilizing the deflation technique [25,26]. In the BDM discretization, the linear systems arising in the deflated barrier method are solved with FGMRES [50] preconditioned with block preconditioning techniques and the Schur complements are controlled with an augmented Lagrangian term [44].…”