Search

An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator T on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation x ˙ (t) + T λ (t) (x (t) − α (t) T λ (t) (x (t))) = 0 , t ≥ t 0 ≥ 0 , to the solution set T − 1 (0) of a generalized equation 0 ∈ T (x). With smart choices of parameters λ (t) and α (t) , we prove the weak convergence of the trajectory to some point of T − 1 (0) with ‖ x ˙ (t) ‖ ≤ O (1 / t) as t → + ∞. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory x h (t) → x (t) on interval [ 0 , + ∞) is demonstrated when the step size h → 0. [ABSTRACT FROM AUTHOR]

Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated. [ABSTRACT FROM AUTHOR]