Composing Scalable Nonlinear Algebraic Solvers

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table

A parallel multigrid solver for multi-patch Isogeometric Analysis

Isogeometric Analysis (IgA) is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then. If large spline degrees are considered, one obtains the approximation power of a high-order method, but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers (like Gauss Seidel) does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree. In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF) for the financial support through the DK W1214-04, while the second author was supported by the FWF grant NFN S117-0

Opt: A Domain Specific Language for Non-linear Least Squares Optimization in Graphics and Imaging

Many graphics and vision problems can be expressed as non-linear least squares optimizations of objective functions over visual data, such as images and meshes. The mathematical descriptions of these functions are extremely concise, but their implementation in real code is tedious, especially when optimized for real-time performance on modern GPUs in interactive applications. In this work, we propose a new language, Opt (available under http://optlang.org), for writing these objective functions over image- or graph-structured unknowns concisely and at a high level. Our compiler automatically transforms these specifications into state-of-the-art GPU solvers based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate different variations of the solver, so users can easily explore tradeoffs in numerical precision, matrix-free methods, and solver approaches. In our results, we implement a variety of real-world graphics and vision applications. Their energy functions are expressible in tens of lines of code, and produce highly-optimized GPU solver implementations. These solver have performance competitive with the best published hand-tuned, application-specific GPU solvers, and orders of magnitude beyond a general-purpose auto-generated solver

Evaluating the Impact of SDC on the GMRES Iterative Solver

Increasing parallelism and transistor density, along with increasingly tighter energy and peak power constraints, may force exposure of occasionally incorrect computation or storage to application codes. Silent data corruption (SDC) will likely be infrequent, yet one SDC suffices to make numerical algorithms like iterative linear solvers cease progress towards the correct answer. Thus, we focus on resilience of the iterative linear solver GMRES to a single transient SDC. We derive inexpensive checks to detect the effects of an SDC in GMRES that work for a more general SDC model than presuming a bit flip. Our experiments show that when GMRES is used as the inner solver of an inner-outer iteration, it can "run through" SDC of almost any magnitude in the computationally intensive orthogonalization phase. That is, it gets the right answer using faulty data without any required roll back. Those SDCs which it cannot run through, get caught by our detection scheme