2,027 research outputs found
Anderson acceleration for geometry optimization and physics simulation
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
Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach
We consider the problem of minimizing a convex function over the intersection
of finitely many simple sets which are easy to project onto. This is an
important problem arising in various domains such as machine learning. The main
difficulty lies in finding the projection of a point in the intersection of
many sets. Existing approaches yield an infeasible point with an
iteration-complexity of for nonsmooth problems with no
guarantees on the in-feasibility. By reformulating the problem through exact
penalty functions, we derive first-order algorithms which not only guarantees
that the distance to the intersection is small but also improve the complexity
to and for smooth functions. For
composite and smooth problems, this is achieved through a saddle-point
reformulation where the proximal operators required by the primal-dual
algorithms can be computed in closed form. We illustrate the benefits of our
approach on a graph transduction problem and on graph matching
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