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Title: One-step convergence of inexact Anderson acceleration for contractive and non-contractive mappings
Award ID(s):
Publication Date:
Journal Name:
ETNA - Electronic Transactions on Numerical Analysis
Page Range or eLocation-ID:
285 to 309
Sponsoring Org:
National Science Foundation
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  1. Heterobiaryls composed of pyridine and diazine rings are key components of pharmaceuticals and are often central to pharmacological function. We present an alternative approach to metal-catalyzed cross-coupling to make heterobiaryls using contractive phosphorus C–C couplings, also termed phosphorus ligand coupling reactions. The process starts by regioselective phosphorus substitution of the C–H bonds para to nitrogen in two successive heterocycles; ligand coupling is then triggered via acidic alcohol solutions to form the heterobiaryl bond. Mechanistic studies imply that ligand coupling is an asynchronous process involving migration of one heterocycle to the ipso position of the other around a central pentacoordinate P(V) atom. The strategy can be applied to complex drug-like molecules containing multiple reactive sites and polar functional groups, and also enables convergent coupling of drug fragments and late-stage heteroarylation of pharmaceuticals.

  2. Abstract A one-step analysis of Anderson acceleration with general algorithmic depths is presented. The resulting residual bounds within both contractive and noncontractive settings reveal the balance between the contributions from the higher and lower order terms, which are both dependent on the success of the optimization problem solved at each step of the algorithm. The new residual bounds show the additional terms introduced by the extrapolation produce terms that are of a higher order than was previously understood. In the contractive setting these bounds sharpen previous convergence and acceleration results. The bounds rely on sufficient linear independence of the differences between consecutive residuals, rather than assumptions on the boundedness of the optimization coefficients, allowing the introduction of a theoretically sound safeguarding strategy. Several numerical tests illustrate the analysis primarily in the noncontractive setting, and demonstrate the use of the method, the safeguarding strategy and theory-based guidance on dynamic selection of the algorithmic depth, on a p-Laplace equation, a nonlinear Helmholtz equation and the steady Navier–Stokes equations with high Reynolds number in three spatial dimensions.