Duality based domain decomposition with proportioning for the solution of free boundary problems
The author proposes an algorithm for solving the quadratic programming problem \(\min\{f(x): x\in {\mathcal B}\}\), where \(f(x)={1\over 2} x^T Ax-x^Tb\) and \({\mathcal B}=\{x:Bx\leq c\}\), \(b\in \mathbb{R}^n\), \(c\in \mathbb{R}^m\). Furthermore, \(B\) denotes an \(m\times n\) full rank matrix and the symmetric positive semidefinite matrix \(A\) is given as \(A=\text{diag}(A_1,\dots,A_p)\). The author assumes that the problem arises from a discretization of free boundary problems (contact problems of a system of \(p\) elastic bodies) with a spatial domain comprising \(p\) subdomains (\(A_i\) are band matrices, \(m\ll n\)). After the introduction of the dual problem and a brief description of the domain decomposition in Section 2, Section 3 is concerned with the discussion of the algorithm and its main properties applied to the original problem. The algorithm is characterized by a sequence of auxiliary problems, where the precision of their solution is controlled by the norm of the so-called unbalanced contact residuals. For details the reader is referred to another paper by the author. In Section 4 the application of the algorithm to the dual problem is discussed and finally numerical results for a model problem are presented to illustrate the good properties of the proposed algorithm.
scientific article; zbMATH DE number 903498