Broadly speaking, my research falls into two categories: probability (mainly, but not exclusively, uncertainty quantification) and nonlinear dynamics, with some crossover between the two. I also have some interests in theoretical computer science, stimulated by the kinds of optimizations problems that have arisen in my uncertainty quantification research.
Since 2009, much of my reasearch has been into uncertainty quantification (UQ). Along with collaborators at Caltech and Los Alamos National Laboratory, I have developed a point of view on UQ as the intersection of probability theory with optimization. More precisely, for some quantity of interest that depends upon probability measures, functions and other objects about which one has partial information, one seeks rigorous optimal lower and upper bounds on the quantity of interest with respect to that information. This general framework offers a sharp quantitative understanding of the relationships among input and output uncertainties in complex physical systems.
|Joint research with Mike McKerns, Dominik Meyer, Michael Ortiz, Houman Owhadi, Clint Scovel, Florian Theil.|
I am also interested in using the Optimal UQ framework to provide a robust Bayesian perspective on inverse problems for (stochastic) partial differential equations, including elliptic PDEs such as the pressure equation, and non-linear PDEs such as the Navier–Stokes and MHD equations.
|Joint research with Houman Owhadi and Andrew Stuart.|
Massively Parallel Optimization
I am interested in the implications of massive parallelism for the way that large calculations, especially numerical optimization algorithms, are performed. In the presence of massive parallelism, it becomes not just desirable but imperative to break the iterative blocking inherent in serial algorithms. In this area, I have contributed to the development of the mystic optimization framework, particularly its probabilistic and UQ components.
|Joint research with Michael Aivazis and Mike McKerns.|
Gradient Descents in Random Environments
A gradient descent in a space X is a process z: [0, T] → X along which a prescribed energetic potential E(t, z(t)) decreases “as quickly possible”, as allowed by a dissipation potential Ψ. The classic example is the gradient descent in ℝn described by the ordinary differential equation
ż(t) = −∇E(t, z(t)),
in which case the dissipation potential is Ψ(v) = ½∥v∥2. I am interested in the behaviour of such processes when the energetic potential E is random in some way, and especially in limits of families of such processes. Part of my PhD thesis work was to show that, in the case X = ℝn, the solutions zε to
żε(t) = −ε−1 ∇Eε(t, zε(t)),
where Eε is a suitably nice potential E randomly “dented” by a Poisson point process, converge in probability to the rate-independent process z that satisfies
∂Ψ(ż) ∋ −∇E(t, z),
where Ψ is homogeneous of degree one. That is, as ε → 0, the random “wiggles” in the energetic potential E “average out” to give a qualitative change in the dissipation potential, from 2-homogeneity to 1-homogeneity. This limit passage justifies the use of rate-independent models in the limit of vanishing inertia and relaxation time.
|Joint research with Florian Theil.|
Rate-independent dyanmical systems are evolutionary processes that have no intrinsic time-scale: in some sense, they vary only as quickly as their time-dependent inputs. More precisely, a rate-independent system is one for which the solution operator commutes with (smooth) strictly increasing reparametrizations of time. Such systems have a strongly geometric character, and arise often in the study of physical processes such as plasticity and electromagnetism.
Rate-independent systems are often described using differential inclusions such as
∂Ψ(ż) ∋ −∇E(t, z).
The assumption of rate independence is typically made in the limit of vanishing inertia, relaxation times and thermal effects. In this paper and my PhD thesis, I examined the effect of coupling a rate-independent process to a heat bath. In discrete time, the heat bath generates a non-trivial stochastic process; in the limit as the time step vanishes, though, the randomness disappears, and leaves a non-trivial deterministic limit process which can be characterized as a nonlinear gradient descent in E with respect to a suitable integral transformation of Ψ.
|Joint research with Marisol Koslowski, Michael Ortiz and Florian Theil.|
ü(t, x) = c2 Δu(t, x),
u: [0, ∞) × ℤn → ℝ,
with suitable initial conditions, where Δ denotes the discrete Laplacian on ℤn. It turns out that this system is similar to the continuum wave equation on ℝn in some ways, but very different in others; also, it has some properties that only make sense in the spatially discrete setting. My dissertation partly inspired this more in-depth analysis by my PhD advisor Florian Theil, his student Lisa Harris, and others.