Anderson acceleration for geometry optimization and physics simulation

Abstract

Many computer graphics problems require computing geometric shapes subject to certain constraints. This often results in non-linear and non-convex optimization problems with globally coupled variables, which pose great challenge for interactive applications. Local-global solvers developed in recent years can quickly compute an approximate solution to such problems, making them an attractive choice for applications that prioritize efficiency over accuracy. However, these solvers suffer from lower convergence rate, and may take a long time to compute an accurate result. In this paper, we propose a simple and effective technique to accelerate the convergence of such solvers. By treating each local-global step as a fixed-point iteration, we apply Anderson acceleration, a well-established technique for fixed-point solvers, to speed up the convergence of a local-global solver. To address the stability issue of classical Anderson acceleration, we propose a simple strategy to guarantee the decrease of target energy and ensure its global convergence. In addition, we analyze the connection between Anderson acceleration and quasi-Newton methods, and show that the canonical choice of its mixing parameter is suitable for accelerating local-global solvers. Moreover, our technique is effective beyond classical local-global solvers, and can be applied to iterative methods with a common structure. We evaluate the performance of our technique on a variety of geometry optimization and physics simulation problems. Our approach significantly reduces the number of iterations required to compute an accurate result, with only a slight increase of computational cost per iteration. Its simplicity and effectiveness makes it a promising tool for accelerating existing algorithms as well as designing efficient new algorithms.

Accelerating ADMM for efficient simulation and optimization

“…We apply our methods to a variety of ADMM solvers in graphics. We implement Anderson acceleration following the source code released by the authors of [Peng et al 2018] 1 . The source code of our implementation is available at https://github.com/bldeng/ AA-ADMM.…”

“…We follow this convention throughout this paper. Previously, Anderson acceleration has been applied to speed up local-global solvers in computer graphics [Peng et al 2018].…”

“…8 compares the effectiveness of these two approaches in enforcing hard constraints while decreasing the original target function. For the quadratic penalty method, we use ShapeUp [Bouaziz et al 2012] with Anderson acceleration as described in [Peng et al 2018], and solve three problem instances with different penalty weights for hard constraints and fixed weights for the other terms. Each solver is run to full convergence for comparison.…”

A Proof That Anderson Acceleration Improves the Convergence Rate in Linearly Converging Fixed-Point Methods (But Not in Those Converging Quadratically)

“…Anderson in 1965 in the context of integral equations [2]. It has recently been used in many applications, including multisecant methods for fixed-point iterations in electronic structure computations [5], geometry optimization problems [12], various types of flow problems [11,13], radiation diffusion and nuclear physics [1,16], molecular interaction [14], machine learning [6], improving the alternating projections method for computing nearest correlation matrices [7], and on a wide range of nonlinear problems in the context of generalized minimal residual (GMRES) methods in [17]. We further refer readers to [8,10,11,17] and references therein for detailed discussions on both practical implementation and a history of the method and its applications.…”

A fast numerical solver for local barycentric coordinates

“…Although our reformulation comes with improved scalability, its asymptotic convergence rate is still the same as other ADMM solvers. Recently, there are research efforts to accelerate the convergence of such first-order solvers Wang (2015); Peng et al (2018). These techniques are complementary to our gradient-based approach, and as a future work they can be integrated to further improve the efficiency of LBC computation.…”