Numerical prediction of crack growth and damage are long-standing problems in computational mechanics. The difficulties inherent in these problems arise from the basic incompatibility of cracks with the partial differential equations that are used in the classical theory of solid mechanics. The spatial derivatives needed for these partial differential equations to make sense do not exist on a crack tip or surface. Therefore, any numerical method derived from these equations inherits this difficulty in modeling cracks. The peridynamic theory of mechanics attempts to unite the mathematical modeling of continuous media, cracks, and particles within a single framework. It does this by replacing the partial differential equations of the classical theory of solid mechanics with integral or integro-differential equations. Theoretical and numerical analysis based on the deterministic and stochastic peridynamics model will be studied, relevant numerical experiments will be conducted to show the effectiveness of the proposed methods.
In this talk I describe our efforts to incorporate a real time fluid solver, using Smoothed Particle Hydrodynamics (SPH) into the game engine of Blender, a powerful open source framework. We implemented the algorithm on the GPU using OpenCL, an open standard and portable across a range of devices. I will cover the ideas behind our method and the issues we encountered working with OpenCL. I will also show a short demo or video along with benchmark results of our implementation against other existing CPU and GPU codes.
I also have a blog post that includes a video demonstrating the latest progress: http://enja.org/2010/12/16/particles-in-bge-fluids-in-real-time-with-opencl/
For many biological investigations, genetic material is collected from groups of individuals at multiple, somewhat isolated locations. It is unknown in advance whether individuals from different locations will commonly migrate and interbreed. To estimate the sizes of the separate breeding populations and the rate of migrations between them, we use a Bayesian approach. This requires a large scale integration using the Markov Chain Monte Carlo method. Model selection is critical for good results. Our choice has been to implement the Thermodynamic Integration Method to estimate the marginal likelihoods These are used to select for models. We have also created methods to split multiple data sets across computers, conduct the analysis in parallel, and combine the result afterwards.
In the context of the data assimilation of observations, observation operators model the mapping between the model state to observation space. In an operational numerical weather prediction (NWP) setting, these operators may be represented by large code-bases such as a radiative transfer model. Such operators may contain inherent discontinuities in the derivative of the operator - such is the case for data assimilation of satellite data that may include presence of clouds - or numerical discontinuities - such as those introduced by "on/off" parametrization switches. The present work addresses the challenge of the data assimilation of non-smooth observation operators of either type with a shallow water equations model. Smooth optimization techniques - such as conjugate gradient and L-BFGS - are compared to the non-smooth optimization Limited Memory Bundle Method (LMBM) in both the 4DVar (variational optimal control) and Maximum Likelihood Ensemble Filter (sequential, ensemble/variational hybrid) data assimilation approaches. Results show that in the presence of strong non-smoothness, the LMBM method has superior performance over optimization methods currently used in operational NWP centers.
We propose an accurate multi-step scheme on time-space grids for solving backward stochastic differential equations (BSDEs). For high accuracy, we preprocess the target equation before discretization, changing it from a stochastic problem to a deterministic problem. Then, we apply a multi-step method in time and the Gauss-Hermite quadrature approximation in space to construct the multi-step scheme. Error estimates are rigorously proved for the semi-discrete version of the proposed scheme for BSDEs with certain type of simplified generator functions. For efficiency, we improve our scheme by applying a new kind of random process, called the Gauss-Hermite process, by which the time-space domain needed for solving BSDEs at a particular point of interest can be significantly reduced compared to the domain used by the multi-step scheme. Error estimates are rigorously derived to verify that the improved scheme can keep essentially the same accuracy as that of the multi-step scheme. Furthermore, in order to break the curse of dimensionality, we apply hierarchical bases and associated sparse grids to effect the spatial interpolation. Combining these techniques, high dimensional BSDEs can be solved efficiently with acceptable accuracy.
You can return to the SIAM Student MiniConference 2011 web page.