This paper provides the first proof that Anderson acceleration (AA) improves the convergence rate of general fixed point iterations. AA has been used for decades to speed up nonlinear solvers in many applications, however a rigorous mathematical justification of the improved convergence rate has remained lacking. The key ideas of the analysis presented here are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is that AA improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step. In addition to improving the convergence rate, our results indicate that AA increases the radius of convergence. Lastly, our estimate shows that while the linear convergence rate is improved, additional quadratic terms arise in the estimate, which shows why AA does not typically improve convergence in quadratically converging fixed point iterations. Results of several numerical tests are given which illustrate the theory.

Natural product and synthetic macrocycles are chemically and topologically diverse. An efficient, accurate, and general method for generating macrocycle conformations would enable structure-based design of macrocycle drugs or host–guest complexes. Computational sampling also provides insight into transiently populated states, complementing crystallographic and NMR data. Here, we report a new algorithm, BRIKARD, that addresses this challenge through computational algebraic geometry and inverse kinematics together with local energy minimization. BRIKARD is demonstrated on 67 diverse macrocycles with structural data, encompassing various ring topologies. We find this approach enumerates diverse structures with macrocyclic RMSD < 1.0 Å to the experimental conformation for 85% of our data set in contrast to success rates of 67–75% with other approaches, while for the subset of 21 more challenging compounds in the data set, these rates are 57% and 10–29%, respectively. Because the algorithm can be efficiently run in parallel on many processors, exhaustive conformational sampling of complex cycles can be obtained in minutes rather than hours: with a 40 processor implementation protocol, BRIKARD samples the conformational diversity of a potential energy landscape in a median of 1.3 minutes of wallclock time, much faster than 3.1–10.3 hours necessary with current programs. By rigorously testing BRIKARD on a broad range of scaffolds with highly complex ring systems, we push the frontiers of macrocycle sampling to encompass multiring compounds, including those with more than 50 ring atoms and up to seven interlaced flexible rings.

Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated algorithms for nonlinear dimensionality reduction for representation of trans, trans-1,2,4-trifluorocyclo-octane conformation-a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space. We describe an efficient approach for a deterministic enumeration of ring conformations. We demonstrate a drastic improvement in dimensionality reduction with the use of nonlinear methods. We discuss the use of dimensionality reduction algorithms for estimating intrinsic dimensionality and the relationship to the Whitney embedding theorem. Additionally, we investigate the influence of the choice of high-dimensional encoding on the reduction. We show for the case studied that, in terms of reconstruction error root mean square deviation, Cartesian coordinate representations and encodings based on interatom distances provide better performance than encodings based on a dihedral angle representation.

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.