Title: Introducing the Uncertain Grids Method An advanced, meshless Lagrangian technique for the solution of multidimensional PDEs  

Speaker: Dr. Oleg Diyankov and Dr. Sergey Terekhov, Sparsix Corporation

Date/Time: Monday, February 23, 2009, 9:00 am

Location: CSRI Building, Room 90 (Sandia NM)

Brief Abstract: Sparsix Corporation, along with its partner Neurok Software, will deliver a presentation on a proposed novel approach to the solution of multidimensional partial differential equations called the Uncertain Grids Method (UGM). UGM will significantly improve upon the accuracy and performance of existing methods for modeling time-dependent flows with large deformations or stationary processes in complicated regions, including flows with moving boundaries.

The presentation will consist of two parts followed by an open, group discussion. The first part will provide an overview of the presenters’ expertise in the solution of applied mathematical problems. The presenters will share how their experience developing solutions for the Russian and American national laboratories and multinational corporate clients convinced them of the need for a new approach to the solution of complex, multidimensional PDEs and motivated them to pursue the development of UGM. The second, and principal, part of the presentation will provide a detailed overview of the concepts and mathematics on which UGM is based. The method will be illustrated through the numerical solution of sample 2D gas dynamics and Poisson equations. The presentation will expand on the following ideas:

The most commonly used numerical methods for the solution of PDEs are grid-based methods, notably finite difference (FD) and finite element (FE), and meshless methods, notably particle-in-cell (PIC) and free-Lagrange (FL). Each of these methods has advantages and disadvantages, but none of them is able to accurately model some of the most difficult problems encountered in advanced scientific research. The failure of these methods to provide a usable, robust and accurate numerical method for modeling large deformations flows was the catalyst for a new idea – the Uncertain Grids Method. UGM builds on the advantages of meshless methods, allowing the modeling of large deformations, but corrects the fundamental flaws of these methods, such as the nonmonotonicity and loss of accuracy of derived numerical solutions. The key to UGM is its approximation of the surface and volume integrals of the unknown functions in a particle as sums of these functions’ values in the points (particles) which are closest to that particle. To obtain the constitutive equations, one should substitute the Taylor expansion in the corresponding approximation and null the corresponding members. The obtained relationships should be complemented by the condition of non-negativeness of the coefficients, thereby leading to a linear programming formulation for the coefficients of the gradient approximation. So each particle has something like its own cloud of particles around it which is used to construct the approximation. The power of this cloud (equal to the number of particles it contains) can vary without significantly decreasing the quality of the approximation. The term “particle” is a useful description, but in reality what we have is a volume which is not strictly bounded but about which we know certain physical values (density, energy, velocity, etc.) that are necessary for modeling a specific applied problem. The unbounded nature of this volume, or cloud of particles, gives “Uncertain Grids Method” its name.

The presentation will highlight several key topics for the open panel discussion to follow. In order to broaden the discussion to include some important complementary algorithms and technologies, a brief review of machine learning methods and their relevance to mathematical modeling will be given during the discussion. The review will address the following topics:

The basic goal of contemporary approaches to mathematical modeling is to improve the reliability and efficiency of numerical computations. Linear solvers themselves consume significant computational resources, but the coefficients embedded into a model’s equations and into the sources and boundary conditions also can require significant computing power. For example, in hydrodynamics simulations valuable amounts of computation time are spent on the evaluation of equations of state, especially for phase transitions, composite mixtures and multiphase flows. The same is true for problems with complicated chemical reactions.

Contemporary approaches generally use tabular representations of complicated coefficients, but doing so has the major drawback of dramatically growing the sizes of stored tables in the case of many independent variables. One possible alternative is to represent complicated static and even dynamic coefficients, sources, and boundary condition functions by means of machine learning algorithms.

In modern practice, a single run of a multi-dimensional simulator for a specific formulation of initial and boundary conditions rarely gets a researcher to their final objective. There are many sources of uncertainty and ambiguity in the formulation of problems and the description of material properties. To understand the consequences of these uncertainties, arising both from imprecise measurements of experimental coefficients and an incomplete understanding of the physics or chemistry involved, many simulations should be evaluated simultaneously and combined to create the final picture. Machine learning and stochastic process methods can be applied to facilitate the process of estimating and quantifying these uncertainties.

To further investigate these concepts, Sparsix proposes to equip a simulation environment with algorithmic methods of multidimensional stochastic field generation. Independent realizations of stochastic fields will describe deviations of material properties and sources as well as the initial geometry in different simulator runs. In addition, Sparsix proposes to develop and utilize the perfect approximation quality of neural networks to simulated data fusion when the results of several perturbed simulations are combined in one probabilistic model covering all the results in one aggregated predictive model.

CSRI POC: Randall Summers (505) 844-6296