I am a computational mathematician whose research is focused at the interface between numerical analysis, applied mathematics, and computer science, with an emphasis on the design and analysis of algorithms for the numerical solution of partial differential equations that feature heterogeneous media.
Beginning with my thesis, my interest in developing such spectral methods for time-dependent, variable-coefficient PDE has been driven by the desire to extend, as much as possible, the high accuracy, stability and efficiency of the Fourier spectral method for constant-coefficient problems on simple domains to more general problems, that feature heterogeneity and complicated geometries. Over time, this effort has evolved into a broader, two-pronged mission that continues courses first charted by my two advisors, Gene Golub and Joseph Oliger: to realize the full potential and applicability of (1) the powerful techniques developed by several pioneers, including Golub, for approximating Riemann-Stieltjes integrals by Gaussian quadrature, which is the foundation of my work on spectral methods, and (2) the insightful analysis of Charles Fefferman and others of eigensystems of self-adjoint variable-coefficient differential operators, for the purpose of designing new computational methods for problems involving such operators that are based on local analysis of symbols.
I am also drawn to the myriad daunting challenges of simulating multi-phase flow in porous media, as well as the importance of such simulations to the energy industry, research pertaining to climate change, remediation of soil contaminants, and other application areas. These problems, with their high degrees of nonlinearity and coupling, their extreme heterogeneity, and large domains in both space and time, require the very best efforts of computational mathematicians, as well as integration of effective techniques for discretization, gridding, upscaling, geostatistical modeling, solution of large, sparse linear systems and systems of nonlinear equations, programming for parallel and vector architectures, and more. Building an effective simulator, therefore, calls for interdisciplinary work on many levels, among different areas of expertise within scientific computing as well as other disciplines, as well as a creative approach to integrating these techniques that maximizes accuracy, efficiency and robustness.
In spite of significant advances in computing power in recent years, mathematical modeling in the context of industrial applications remains impractical without computational algorithms and programming practices, including those that I adopted during my years as a software engineer, that use this power as efficiently as possible. As such, my research is guided by the goal of developing algorithms that are useful in computing environments that feature high-resolution discretization, large computational domains (in terms of both space and time), and parallel architectures. By maintaining relationships with researchers in industry, as I am with companies such as Chevron, KLA-Tencor, and ENI, I can ensure that my efforts remain consistent with this goal, and therefore relevant to problems that arise in industry.
Krylov subspace spectral (KSS) methods, introduced in my thesis, are a generalization of the Fourier method for time-dependent, variable-coefficient PDE. They approximate Fourier components of the solution by Gaussian quadrature in the spectral domain, using techniques due to Golub and Meurant for computing elements of functions of matrices by treating them as Riemann-Stieltjes integrals. Because the measures of these integrals are frequency-dependent, high-order temporal accuracy is achieved for high- and low-frequency components alike, resulting in a best-of-both-worlds approach to time-stepping, in that KSS methods are explicit, but possess the stability of implicit methods, and are even unconditionally stable in some cases. In spite of the frequency dependence of the quadrature rules, they can still be computed very efficiently by treating recursion coefficients generated by Lanczos iteration as functions of the frequency.
KSS methods have been demonstrated to be highly accurate, robust and efficient methods for solving both parabolic and hyperbolic PDE. Current and future work will explore the application of KSS methods to the following problems of particular interest:
To accomplish planned generalizations to problems with absorbing boundary conditions, complicated geometry, or problematic heterogeneity, techniques recently developed by Emmanuel Candè Lexing Ying, and Laurent Demanet for working with symbols of differential operators, including fast application of Fourier integral operators and operations from discrete symbol calculus, will play a prominent role, by allowing rapid computation of sets of moments of differential operators, and facilitating application of preconditioning similarity transformations that homogenize leading-order coefficients, thus allowing KSS methods to perform most effectively.
In collaboration with Prof. Margot Gerritsen at Stanford, Dr. Bradley Mallison of Chevron ETC, and Daniele Fragola of ENI, I have developed methods for coarse-scale modeling of multi-phase flow in porous media that integrate upscaling, finite-volume discretization based on multi-point flux approximations (MPFA), and adaptive mesh refinement. We are primarily interested in using this integrated approach for the simulation of enhanced oil recovery (EOR) by gas injection. Such processes take place in highly heterogeneous media in which rock permeability varies sharply across several orders of magnitude, and highly mobile fluids tend to seek narrow high-permeability flow paths that can be difficult to adequately resolve using simple coarse-scale models that use Cartesian grids or two-point flux approximations (TPFA) that cannot accurately resolve flow paths with varying orientation.
The centerpiece of this integration is Variable Compact Multi-Point Upscaling (VCMP). For each face in the coarse grid, it constructs a multi-point stencil by solving a constrained optimization problem, the objective function for which provides a balance between accuracy and robustness by matching local fine-scale flows while also minimizing the deviation of the stencils from TPFA. A modification of this method, called the M-fix, adds additional constraints that guarantee that the resulting discretization of the pressure equation is an M-matrix, thus guaranteeing that pressure fields satisfy the discrete maximum principle, in exchange for a small sacrifice of accuracy. Recent work has included the generalization of VCMP to 3-D, and combination with various global and local-global upscaling methods.
Because uniform grids are not well suited to resolving fine-scale effects due to narrow high-permeability channels, we combine our upscaling and discretization techniques with an algorithm for automatically generating adapted grids in such a way as to accurately capture fine-scale behavior. For simplicity of implementation, we use Cartesian cell-based anisotropically refined (CCAR) grids, although extension to other types of grids will be considered in future work. The algorithm proceeds by first constructing a base grid by refining near large contrasts in permeability. Then, the grid is adapted further based on information acquired from flow simulations about the magnitude and direction of flow across interfaces, for which the weights of multi-point stencils generated by VCMP provide a simple indication of flow direction.
Recent work has involved the application of our integrated scheme, originally developed for single-phase flow, to multi-phase flow and transport. For such simulations, saturation gradients serve as an additional guide for adaptivity, so that variation in velocity and saturation is limited near each face in the coarse grid. This allows the stencils that VCMP constructs for single-phase flow to be used for multi-phase flow by evaluation of multi-phase parameters at representative saturation values, instead of incurring the computational expense of pseudo functions normally used in multi-phase upscaling. The result is a dynamically adaptive coarse-scale model that can be used to efficiently and accurately simulate multi-phase flow and transport, even in channelized media.
One long-term goal of this work is to continue this trend of tightly integrating the phases of the simulation process, considering not only upscaling, gridding, and discretization, but also other preceding phases, such as geostatistical modeling that provides our fine-scale data, and succeeding phases, such as linear and nonlinear solvers in which our discretizations would be used. More concretely, in the petroleum industry, gridding and upscaling is an extraordinarily time-consuming process due to lack of automation, and therefore working with multiple realizations of geological features, such as faults and fractures, is impractical. Our goal is to develop effective automated gridding and upscaling algorithms for use in applications such as reservoir simulation that allow all available geostatistical data to be used. On the other hand, since multigrid solvers are a natural choice for the linear systems that our algorithms construct, it is natural to draw parallels between our grid generation algorithms and the coarsening employed by multigrid methods in order to improve the effectiveness of both the discretization and solution phases of the simulation process.
A second long-term goal is to add mathematical formalism to the body of work on upscaling for flow in highly heterogeneous media. While such formalism already exists for homogeneous, smoothly varying, or periodic media, generalization to realistic permeability fields such as those that arise in reservoir simulation is much more difficult. However, it is our hope that through the addition of not just adaptive mesh refinement but well-conceived adaptivity criteria, we can establish results concerning error control in order to provide assurances about the accuracy of coarse-scale models for such challenging problems.
The third, and most important, goal is to explore the application of our coarse-scale modeling techniques to other applications involving simulation of fluid flow and transport in highly heterogeneous media. Immediate applications within our research group, SUPRI-C, include oil recovery by in situ combustion (ISC) and carbon capture and sequestration (CCS), but we will also consider applications such as remediation of soil contaminants, as well as problems from computational biology such as modeling of extracorporeal shock therapy.
The Perona-Malik equation was developed for de-noising and de-blurring images. Because it exhibits both forward and backward diffusion, it is not well-posed, but numerical solutions are surprisingly well-behaved. With Prof. Patrick Guidotti at UC Irvine, I am investigating numerical schemes for two modified equations, one in divergence form with periodic boundary conditions, and one in non-divergence form with Neumann boundary conditions, in which the nonlinearity is slightly weakened in order to ensure well-posedness of these modified equations, while preserving the original intent of the Perona-Malik equation is preserved.
Like the original Perona-Malik equation, the modified equations admit certain piecewise smooth functions as stationary solutions, thus preventing blurring of the image that can result from regularization of Perona-Malik. At the same time, the weakened nonlinearity prevents the staircasing effect that is discretizations of Perona-Malik tend to exhibit. Thus, our modified methods preserve the desirable properties of Perona-Malik that make it an effective denoising tool, while overcoming both its theoretical and practical limitations.
Our use of a pseudospectral discretization of the spatial differential operators in our modified equations yield an efficient implementation of implicit time-stepping schemes that employ standard iterative linear solvers such as MINRES or GMRES. On the other hand, this problem is well-suited for the time-stepping approach behind KSS methods, which employ different approximation methods for different Fourier components, thus allowing backward diffusion to be limited to low-frequency components. Current work includes (1) the investigation of similar modifications of fourth-order PDE in order to overcome the limitations of second-order models, which tend to produce patchy results, (2) modifying our new models to achieve de-blurring of images, and (3) processing of color images.
By reconsidering the approach to dealing with the backward diffusion exhibited by the Perona-Malik equation, we are able to overcome the limitations of standard regularizations and accomplish their intended purpose without sacrificing an important property of denoising methods, the ability to preserve edges. This is a prime example of how there is always room for the kind of new approach that a fresh perspective can bring. In all of my research to date, I have made it a point to exploit my broad span of interests, and my desire to seek new connections between existing ideas, in order to discover such new approaches, and as a faculty member, I will continue to do so in order to fulfill my long-term goals of adding not only substantial contributions to the large body of work on numerical methods for solving partial differential equations in heterogeneous media, but also building substantial new bridges between this area of research and many others.