Stochastic Subspace Correction in Hilbert Space

Abstract

We consider an incremental approximation method for solving variational problems in infinite-dimensional separable Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is solved. We show that convergence rates for the expectation of the squared error can be guaranteed under weaker conditions than previously established in [9].

Stochastic subspace correction methods and fault tolerance

“…In Figure 3 (right), we show similar results for the accelerated method (12)(13), see also Table 1 for the recorded iteration counts to termination. The parameters ξ = 0.3 and η = 0.577 were determined by experiment, and are near-optimal in the sense that for them the iteration count of the additive Schwarz method for the given problem and error reduction level ε 0 is close to minimal.…”

“….. Here,κ =λ /λ is an upper bound for the condition κ of the space splitting. If in addition B holds then the algorithm (12)(13) converges in expectation for any u ∈ V , and…”

“…and e That, in contrast to the recursion (4), the coefficient α m in the recursion formula (12) for the accelerated scheme depends on the size p m of the random index set I m is not a problem as long as I m is known before the subproblems needed for the update steps (12) and (13) are solved. However, in the applications discussed in the next section this is not the case: The appropriate set I m is known only after the required subproblem solves are executed.…”

“…The attractive feature of stochastic subspace correction schemes in this respect is the fact that hard faults such as compute node failure or communication losses (as long as they are detectable) can be modeled as a random process of selecting the index set I m of acceptable subproblem solves in each iteration step (4). This random process often fits the independence assumption B that is crucial in order to obtain the convergence rates in Theorem 1, similarly for the accelerated version (12)(13) and Theorem 2.…”

Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

“…This strategy is generalized in [11] to the collective setting (with α k = 1 − (k + 1) −1 and β k minimizing the norm of the residual), and is proved to achieve similar convergence properties as the collective OMP algorithm.…”