1,530 research outputs found
Deflation for semismooth equations
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.Comment: 24 pages, 3 figure
Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis
We show how the discovery of robust scalable numerical solvers for arbitrary
bounded linear operators can be automated as a Game Theory problem by
reformulating the process of computing with partial information and limited
resources as that of playing underlying hierarchies of adversarial information
games. When the solution space is a Banach space endowed with a quadratic
norm , the optimal measure (mixed strategy) for such games (e.g. the
adversarial recovery of , given partial measurements with
, using relative error in -norm as a loss) is a
centered Gaussian field solely determined by the norm , whose
conditioning (on measurements) produces optimal bets. When measurements are
hierarchical, the process of conditioning this Gaussian field produces a
hierarchy of elementary bets (gamblets). These gamblets generalize the notion
of Wavelets and Wannier functions in the sense that they are adapted to the
norm and induce a multi-resolution decomposition of that is
adapted to the eigensubspaces of the operator defining the norm .
When the operator is localized, we show that the resulting gamblets are
localized both in space and frequency and introduce the Fast Gamblet Transform
(FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT
can be used to solve and diagonalize arbitrary PDEs with constant coefficients,
the FGT can be used to decompose a wide range of continuous linear operators
(including arbitrary continuous linear bijections from to or
to ) into a sequence of independent linear systems with uniformly bounded
condition numbers and leads to
solvers and eigenspace adapted Multiresolution Analysis (resulting in near
linear complexity approximation of all eigensubspaces).Comment: 142 pages. 14 Figures. Presented at AFOSR (Aug 2016), DARPA (Sep
2016), IPAM (Apr 3, 2017), Hausdorff (April 13, 2017) and ICERM (June 5,
2017
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi--Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces
The main purpose of this paper is by using a hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a infinite family of total quasi--asymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results announced by some authors
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