Termination criteria for inexact fixed‐point schemes

Abstract

We analyze inexact fixed point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed point iteration. Important applications are the Picard iteration and partitioned fluid structure interaction. We prove that the iteration converges irrespective of how accurate the inner systems are solved, provided that a specific relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. For the analysis, the iteration is modelled as a perturbed fixed point iteration and existing analysis is extended to the nested case x = F(S(x)).

Approaches for mitigating over‐solving in multiphysics simulations

“…Thus, each method obtains the same answer (within tolerance) despite approaching the solution from a different path. For all of the problems that were investigated, the results from different methods converged to the same values exactly as predicted elsewhere [36]. So the optimal coupling method is the one that obtains the answer in the fewest steps.…”

“…There are a few examples where this concept has been implemented. For example, Birken has developed a nonstandard convergence criterion to reduce over‐solving given that Picard iteration over a multiphysics problem can be analyzed as a nested fixed‐point iteration. That is, coupled physics problems y = F ( x ) and x = G ( y ) can be expressed as x = G ( F ( x )), for which convergence can be proved under the appropriate conditions.…”

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A low-rank solver for the stochastic unsteady Navier–Stokes problem

“…In the numerical experiments, we use TT-Toolbox [22] for tensor train computations. Z ← reshape(Z, [κ j−1 n j , n j+1 · · · n d ]) 4:…”

“…Let (3.14) be denoted as L z (i) = r (i−1) , and define the residual norm s k 2 = r (i−1) − L z (i) k 2 for an approximate solution z (i) k . It was shown in [4] that if the stopping criterion for the linear solve (line 2 of Algorithm 4.2) is given as…”