Accelerating implicit integration in multi-body dynamics using GPU computing

A new direct linear equation solver is proposed for GPUs. The proposed solver is applied to mechanical system analysis. In contrast to the DFS post-order traversal which is widely used for conventional implementation of supernodal and multifrontal methods, the BFS reverse-level order traversal has been adopted to obtain more parallelism and a more adaptive control of data size. The proposed implementation allows solving large problems efficiently on many kinds of GPUs. Separators are divided into smaller blocks to further improve the parallel efficiency. Numerical experiments show that the proposed method takes smaller factorization time than CHOLMOD in general and has better operational availability than SPQR. Mechanical dynamic analysis has been carried out to show the efficiency of the proposed method. The computing time, memory usage, and solution accuracy are compared with those obtained from DSS included in MKL. The GPU has been accelerated about 2.5–5.9 times during the numerical factorization step and approximately 1.9–4.7 times over the whole analysis process, compared to an experimental CPU device.     

Performance of implicit A-stable time integration methods for multibody system dynamics

Multibody System Dynamics - This paper illustrates the performance of several representative implicit A-stable time integration methods with algorithmic dissipation for multibody system dynamics,...     

Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme

We focus on a simple implicit time integration scheme for the transient response solution of structures when large deformations and long time durations are considered. Our aim is to have a practical method of implicit time integration for analyses in which the widely used Newmark time integration procedure is not conserving energy and momentum, and is unstable. The method of time integration discussed in this paper is performing well and is a good candidate for practical analyses.     

Fast and accurate surface normal integration on non-rectangular domains

Computational Visual Media - The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However, even nowadays it is...     

Fast and accurate pricing of discretely monitored barrier options by numerical path integration

Barrier options are financial derivative contracts that are activated or deactivated according to the crossing of specified barriers by an underlying asset price. Exact models for pricing barrier options assume continuous monitoring of the underlying dynamics, usually a stock price. Barrier options in traded markets, however, nearly always assume less frequent observation, e.g. daily or weekly. These situations require approximate solutions to the pricing problem. We present a new approach to pricing such discretely monitored barrier options that may be applied in many realistic situations. In particular, we study daily monitored up-and-out call options of the European type with a single underlying stock. The approach is based on numerical approximation of the transition probability density associated with the stochastic differential equation describing the stock price dynamics, and provides accurate results in less than one second whenever a contract expires in a year or less. The flexibility of the method permits more complex underlying dynamics than the Black and Scholes paradigm, and its relative simplicity renders it quite easy to implement.     

Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs

The methods for the dynamical simulation of multi-body systems in real-time applications have to guarantee that the time integration of the equations of motion is always successfully completed within an a priori fixed sampling time interval, typically in the range of 1.0–10.0 ms. Model structure, model complexity and numerical solution methods have to be adapted to the needs of real-time simulation. Standard solvers for stiff and for constrained mechanical systems are implicit and cannot be used straightforwardly in real-time applications because of their iterative strategies to solve the nonlinear corrector equations and because of adaptive strategies for stepsize and order selection. As an alternative, we consider in the present paper noniterative fixed stepsize time integration methods for stiff ordinary differential equations (ODEs) resulting from tree-structured multi-body system models and for differential algebraic equations (DAEs) that result from multi-body system models with loop-closing constraints.     

Fast and accurate global motion estimation algorithm using pixel subsampling

Global motion generally describes the motion of a camera, although it may comprise motions of large objects. Global motions are often modeled by parametric transformations of two-dimensional images. The process of estimating the motions parameters is called global motion estimation (GME). GME is widely employed in many applications such as video coding, image stabilization and super-resolution. To estimate global motion parameters, the Levenburg–Marquardt algorithm (LMA) is typically used to minimize an objective function iteratively. Since the region of support for the global motion representation consists of the entire image frame, the minimization process tends to be very expensive computationally by involving all the pixels within an image frame. In order to significantly reduce the computational complexity of the LMA, we proposed to select only a small subset of the pixels for estimating the motion parameters, based on several subsampling patterns and their combinations. Simulation results demonstrated that the proposed method could speed up the conventional GME approach by over ten times, with only a very slight loss (less than 0.1 dB) in estimation accuracy. The proposed method was also found to outperform several state-of-the-art fast GME methods in terms of the speed/accuracy tradeoffs.     

Jacobian-free newton-krylov methods for the accurate time integration of stiff wave systems

Stiff wave systems are systems which exhibit a slow dynamical time scale while possessing fast wave phenomena. The physical effects of this fast wave may be important to the system, but resolving the fast time scale may not be required. When simulating such phenomena one would like to use time steps on the order of the dynamical scale for time integration. Historically, Semi-Implicit (SI) methods have been developed to step over the stiff wave time scale in a stable fashion. However, SI methods require some linearization and time splitting, and both of these can produce additional time integration errors. In this paper, the concept of using SI methods as preconditioners to Jacobian-Free Newton-Krylov (JFNK) methods is developed. This algorithmic approach results in an implicitly balanced method (no linearization or time splitting). In this paper, we provide an overview of SI methods in a variety of applications, and a brief background on JFNK methods. We will present details of our new algorithmic approach. Finally, we provide an overview of results coming from problems in geophysical fluid dynamics (GFD) and magnetohydrodynamics (MHD).     

Fast and accurate texture placement

Mapping from parameter space to texture space separates textures from objects, enabling control of textured images without modifying the object or texture. Although conceptually straightforward, γ-mapping turns out to be very useful in practical texture mapping implementation. It logically separates the texture from the object and makes the texture mapping process flexible and easier to use. By modifying the γ-mapping, texture placement on the object is efficiently controlled without modifying either the texture image or the underlying geometry. We concentrate on efficiency issues related to γ-mapping, as well as its practical applications to parametric surfaces