Anderson acceleration (AA) has a long history of use and a strong recent interest due to its potential ability to dramatically improve the linear convergence of the fixed-point iteration. Most authors are simply using and analyzing the stationary version of Anderson acceleration (sAA) with a constant damping factor or without damping. Little attention has been paid to nonstationary algorithms. However, damping can be useful and is sometimes crucial for simulations in which the underlying fixed-point operator is not globally contractive. The role of this damping factor has not been fully understood. In the present work, we consider the non-stationary Anderson acceleration algorithm with optimized damping (AAoptD) in each iteration to further speed up linear and nonlinear iterations by applying one extra inexpensive optimization. We analyze the convergence rate this procedure and develop an efficient and inexpensive implementation scheme. We show by extensive numerical experiments that the proposed non-stationary Anderson acceleration with optimized damping procedure often converges much faster than stationary AA with constant damping, adaptive damping or without damping, especially in the cases larger window sizes are needed. We also observe that simple strategies like using constant damping factors and adaptive damping factors, sometimes, work very well for some problems while sometimes they are even worse than AA without damping. Our proposed method is usually more robust than AA with constant damping and adaptive damping. Moreover, we also observed from our numerical results that damping can be good, but choosing the wrong damping factors may slow down the convergence rate. Theoretical analysis of the effects of damping factors are needed and important.

Constructing an analysis-suitable parameterization for the computational domain from its boundary representation plays a crucial role in the isogeometric design-through-analysis pipeline. PDE-based elliptic grid generation is an effective method for generating high-quality parameterizations with rapid convergence properties for the planar case. However, it may generate non-uniform grid lines, especially near the concave/convex parts of the boundary. In the present work, we introduce a novel scaled discretization of harmonic mappings in the Sobolev space H¹ to remit it. Analytical Jacobian matrices for the involved nonlinear equations are derived to accelerate the computation. To enhance the numerical stability and the speed of convergence, we propose a simple and yet effective preconditioned Anderson acceleration framework instead of using computationally expensive Newton-type iteration. Three preconditioning strategies are suggested, namely diagonal Jacobian, block-diagonal Jacobian, and full Jacobian. Furthermore, we discuss a delayed update strategy of the preconditioner, i.e., the preconditioner is updated every few steps to reduce the computational cost per iteration. Numerical experiments demonstrate the effectiveness and efficiency of our improved parameterization approach and the computational efficiency of our preconditioned Anderson acceleration scheme.