Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE’s

• Published in: Nonlinear analysis 2011-12, Vol.74 (18), p.7455-7473
• Main Authors: Attouch, H., Cabot, A., Frankel, P., Peypouquet, J.
• Format: Article
• Language: English
• Subjects: Alternating minimization; Calculus of variations and optimal control; Convex minimization; Domain decomposition for PDE’s; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Monotone inclusions; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Proximal algorithm; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Variational inequalities
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.07.066

Let X , Y , Z be real Hilbert spaces, let f : X → R ∪ { + ∞ } , g : Y → R ∪ { + ∞ } be closed convex functions and let A : X → Z , B : Y → Z be linear continuous operators. Let us consider the constrained minimization problem ( P ) min { f ( x ) + g ( y ) : A x = B y } . Given a sequence ( γ n ) which tends toward 0 as n → + ∞ , we study the following alternating proximal algorithm ( A ) { x n + 1 = argmin { γ n + 1 f ( ζ ) + 1 2 ‖ A ζ − B y n ‖ Z 2 + α 2 ‖ ζ − x n ‖ X 2 ; ζ ∈ X } y n + 1 = argmin { γ n + 1 g ( η ) + 1 2 ‖ A x n + 1 − B η ‖ Z 2 + ν 2 ‖ η − y n ‖ Y 2 ; η ∈ Y } , where α and ν are positive parameters. It is shown that if the sequence ( γ n ) tends moderately slowly toward 0 , then the iterates of ( A ) weakly converge toward a solution of ( P ) . The study is extended to the setting of maximal monotone operators, for which a general ergodic convergence result is obtained. Applications are given in the area of domain decomposition for PDE’s.