Farrell, P. E. (2015) Multiple local minima of PDEconstrained optimisation problems via deflation. Technical Report. Unspecified. (Submitted)

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Abstract
Nonconvex optimisation problems constrained by partial differential equations (PDEs) may permit distinct local minima. In this paper we present a numerical technique, called deflation, for computing multiple local solutions of such optimisation problems. The basic approach is to apply a nonlinear transformation to the KarushKuhnTucker optimality conditions that eliminates previously found solutions from consideration. Starting from some initial guess, Newton's method is used to find a stationary point of the Lagrangian; this solution is then deflated away, and Newton's method is initialised from the same initial guess to find other solutions. In this paper, we investigate how the Schur complement preconditioners widely used in PDEconstrained optimisation perform after deflation. We prove an upper bound on the number of new distinct eigenvalues of a matrix after an arbitrary additive perturbation; from this it follows that for diagonalisable operators the number of Krylov iterations required for exact convergence of the Newton step at most doubles compared to the undeflated problem. While deflation is not guaranteed to converge to all minima, these results indicate the approach scales to arbitrarydimensional problems if a scalable Schur complement preconditioner is available. The technique is demonstrated on a discretised nonconvex PDEconstrained optimisation problem with approximately ten million degrees of freedom.
Item Type:  Technical Report (Technical Report) 

Subjects:  H  N > Numerical analysis 
Research Groups:  Numerical Analysis Group 
ID Code:  1903 
Deposited By:  Helen Taylor 
Deposited On:  17 Sep 2015 06:52 
Last Modified:  17 Sep 2015 06:52 
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