Alternating proximal algorithms with asymptotically vanishing coupling. Application to domain decomposition for PDE's

Abstract

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. Given a sequence (γ _n ) which increases towards infinity as n → +∞, we study the following alternating proximal algorithm:

where α and ν are positive parameters. If the sequence (γ _n ) increases moderately slowly towards infinity, the algorithm (𝒜) tends to minimize the function over the set C = argmin f × argmin g (assumed to be nonempty). An illustration of this result is given in the area of domain decomposition for PDE's.

Keywords:

• convex minimization
• alternating minimization
• proximal algorithm
• hierarchical minimization
• domain decomposition for PDE's

AMS Subject Classifications::

• 65K05
• 65K10
• 49J40
• 90C25

Acknowledgements

The authors express their gratitude to the anonymous referees for their careful reading of this article. Their valuable suggestions and critical comments made numerous improvements throughout.