Project title: Multiscale multilevel iterative substructuring
Dr. Bedřich Sousedík has been awarded a $199,920, three-year grant for the period 2015–18 from the National Science Foundation (DMS–1521563) to develop novel multiscale multilevel iterative substructuring methods with applications to flow in porous media. The award includes funds to support a Graduate Research Assistant for two years (with a possible summer internship in Sandia or LLNL) and also summer support for an undergraduate student.
The objective of this project is to develop novel algorithms for solving saddle-point linear systems combining numerical upscaling techniques with parallel, domain decomposition iterative solvers. The algorithms will be applied to the simulation of flow in porous media in real-world reservoir models. The simulation of flow in porous media finds applications in a number of areas, such as water management, oil and gas recovery, carbon dioxide (CO2) sequestration, or nuclear waste disposal, to name a few.
The setup of the underlying mathematical models and their efficient numerical solution is challenging for several reasons. The reservoirs are typically very large, so the discretized mathematical model leads to systems of equations with hundreds of millions of unknowns, they have irregular structure, which complicates the model geometry, and they consist of materials that significantly differ in geological properties, which translates in the model into jumps in coefficients over several orders of magnitude. Moreover, the geological formations quite often also contain fractures that alter the effective permeabilities, and therefore need to be be accurately incorporated in the numerical model. For example, the flow of water in granite rock, which represents one of the suitable sites for nuclear waste deposit, is conducted by the complex system of vugs, cavities and fractures with various topology and sizes. Alternatively, the fractures might result from the engineering technology, for example hydraulic fracturing (also known as fracking) used for the extraction of natural gas.
It is well known that these issues have an impact on the environment, for example the quality of drinking water. Therefore there is an urgent need to develop novel mathematical techniques that would be able to tackle all these difficulties and improve methods for solution of this class of problems arising in science and engineering.
There are many aspects of multiscale and domain decomposition methods that are quite well understood, but the major drawback of current methodologies is that they do not take full advantage of their potential by using the multiscale phenomena in the design of the solvers, which results in their inefficiency.
Multiscale methods frequently consist in fact only of two scales, whereas in a porous medium there are typically many scales. At the same time, advances in multicore architectures, networking, high end computers, and large data stores, are ushering in a new era of high performance parallel and distributed simulations. Naturally, with these new capabilities come new challenges in computing and system modeling. The goal of this project is to open new avenues to tackle these issues. In particular, we suggests a development of a multiscale method that allows for a multiple of scales, and uses the upscaling algorithm to build also a multilevel preconditioner for the iterative solver. The components of the method are thus recycled, which significantly decreases the computational cost. Moreover, this approach can be applied recursively and thus offers naturally a multilevel multiscale potential, unlike many traditional multiscale approaches that consist in fact only of two scales. It is expected that understanding of the issues related to design of multiscale and multilevel methods for extremely large problems will ultimately contribute to development of the next generation of parallel iterative solvers suitable for implementation on future exascale supercomputers.
