Kevin Long's Research

Overview

Broadly speaking, I work on computational simulations of physical problems. Doing an accurate and efficient simulation involves several challenges:

Student involvement in research

Computational mathematics provides many opportunities for students to become involved in research at all levels: even early undergraduate students can contribute by managing simulation runs and postprocessing results; more advanced undergraduate students and early graduate students can contribute in innumerable ways, such as development of new simulation methods, development of realistic but computationally tractable models, and detailed analysis of simulation results against experimental data and theoretical predictions. Doctoral students in computational mathematics will of course need to do a project putting together a combination of theoretical development of computable models, algorithm development, extensive simulation, and comparison to experiment.

Metanumerical computing

Traditionally, scientific computing has focused on numerical analysis: the art of solving mathematical problems accurately and efficiently on computers. A newer aspect of scientific computing is metanumerical computing, which is the use of techniques from mathematics and computer science to organize automatically an efficient combination of numerical algorithms. Some pioneering efforts in metanumerical computing are Computers were invented to automate tedious and error-prone numerical computations; ironically, programming computers is itself a tedious and error-prone task. Given the importance and expense of developing scientific simulation programs, it is worth studying whether computers can improve the development, as well as the execution, of these programs. This project focuses on the open-source project Sundance, which automates transition from high-level mathematical abstractions to high-performance, parallel partial differential equation (PDE) simulation code, freeing users from the burden of low-level programming. This approach can reduce simulator development time from months or years to days or even hours. Less obvious, but equally important, benefits of basing software firmly on mathematical abstractions are that internal performance improvements can be automated, and that intrusive algorithms -- algorithms that require transformation of the equation set to produce nonstandard operators -- can be implemented much more easily because such transformations can be carried out automatically. The combination of automated performance optimizations and the ready availability of efficient intrusive algorithms for preconditioning, sensitivity analysis, and PDE-constrained optimization makes it possible for a high-level, general-purpose tool such as Sundance to actually outperform hand-coded special-purpose simulators. Our approach is the formulation of programming tasks as mathematical problems whose solutions are then automated. Central to this is a theorem establishing Frechet differentiation as a ``bridge'' between high-level symbolic programming and high-performance numerical computing. Our previous work has laid the foundations for this; the proposed project will extend that work to other aspects of PDE simulation as well as to non-PDE paradigms such as density functional theory (DFT), and investigate intrusive preconditioners for coupled multiphysics problems.

Inverse problems

Large-scale PDE-constrained inverse problems are ubiquitous in science. In the context of fundamental science, they provide a meeting point between theory and experiment; in the context of engineering, they are used in optimal design, planning, and nondestructive evaluation. In one way or another, I've been working with computational inverse problems for most of my career: for example, my undergraduate internship at NBS was on non-destructive inversion of conductivity profiles in metals through eddy current measurements; later, part of my thesis work was on methods for exploiting symmetry to deproject accurate non-circular velocity fields of barred galaxies from observations. At Sandia, I've worked on inverse problems in a wide range of settings including various problems in source location and characterization for WMD defense, sensor placement for WMD defense, and shape optimization for the design of microfluidic instruments.

My recent work in inverse problems has been on applications of, and algorithms and software for, PDE-constrained optimization. An exciting development in inverse problems is the potential for combination with new methods in spectral UQ to provide information on uncertainties in inference due to errors in data, model, and simulation. One of the drawbacks to the use of simulation in science is that it is difficult to understand and quantify the combined effect of modeling and discretization errors when combining simulations using disparate domains of physics and multiple scales. In computational physics, we often compared simulations to experiment dutifully plotting error bars on the experimental data but knowing little about the errors inherent in the simulation. Recent work in spectral UQ has the potential to greatly improve our ability to understand simulation error, thus making simulation a much more powerful tool for making tests of theoretical predictions in physics, and a much more reliable tool for making engineering decisions involving millions of dollars or human lives.

Advanced methods in inversion and UQ have been adopted very slowly in part because of the difficulty in programming them or retrofitting old simulation codes to use them. To help bring these methods into the mainstream of computational science and engineering, I have pioneered the design and development of high-level, high-performance general-purpose simulation software that has capabilities for advanced optimization and spectral UQ built in at an architectural level.

Metanumerical people

Some of my other collaborators

Research areas

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