PDE Constrained Optimization: My principle area of research is in Partial Differential Equations (PDE) constrained optimization. Optimal design, optimal control, and parameter estimation of systems goverened by partial differential equations (PDE) give rise to a class of problems known as PDE constrained optimization. The size and complexity of the PDEs often pose significant challenges for contemporary optimization methods. An example is the localization of airborne contaminant releases in regional atmospheric transport models from sparse observations. Given measurements of the contaminant over an observation window at a small number of points in space, and a velocity field as predicted for example by a mesoscopic weather model, an estimate of the state of the contaminant at the begining of the observation interval is sought that minimizes the least squares misfit between measured and predicted contaminant field, subject to the convection-diffusion equation for the contaminant. Once the initial conditions are estimated by solution of the inverse problem, predictions can be issued of the evolution of the contaminant, the observation window is advanced in time, and the process repeated to issue a new prediction, in the style of 4D-Var. In collaboration with CMU, Rice, and UT Austin, we have investigated special preconditioning methods and fast algorithms to solve the inversion problem for internal facilities, water distribution systems, and regional models.

Reduced Order Modeling : In the case of a contamination scenario, a real time response is critical to efficient communicate evacuation procedures and support the mitigation process. Even though PDE constrained optimization methods are very efficient and state-of-the-art parallel linear solvers can be leveraged with the largest computational resources, a field implementation requires modest computational platforms and real time response. For the forward model, Proper Orthogional Decomposition (POD) can be used, provided the system is linear. However, POD is not as effective with highly nonlinear systems, nor is it applicable to solving optimization problems. My research in collaboration with MIT and UT Austin is focussed on goal-oriented approaches in which ony a few observation points are considered at the cost of large errors everywhere else. We have leveraged PDE constrained optimization methods to calculate optimal basis that are sensitive to any initial coditions for the specific purpose of inverting in real time.

Statistical Inverse problems and Uncertainty quantification : Many classes of problems in simulation-based science and engineering are characterized by a cycle of observation, parameter/state estimation, prediction, and decision-making. The critical steps in this process involve: (1) assimilating observational data into large-scale simulations to estimate uncertainties in input parameters, (2) propagation of those uncertainties through the simulation to predict output quantities of interest, and (3) determination of an optimal control or decision-making strategy taking into account the uncertain outputs. For many problems, the input parameters cannot be measured directly; instead they must be inferred from observations of simulation outputs. The estimation of input parameters and associated uncertainties from observations and from a computational model linking inputs to outputs constitutes a statistical inverse problem. The uncertainties in the input parameters result from observational errors, inadequate mathematical/computational models, and uncertain prior models of the inputs. Characterization of the uncertainties in the inputs for high-dimensional parameter spaces and expensive forward simulations remains a tremendous challenge for many problems today. Yet despite their difficulties, there is a crucial unmet need for the development of scalable numerical algorithms for the solution of large-scale statistical inverse problems: uncertainty estimation in model inputs is a important precursor to the quantification of uncertainties underpinning prediction and decision-making. While complete quantification of uncertainty in inverse problems for very large scale nonlinear systems has been often intractable, several recent developments are making it viable: (1) the maturing state of algorithms and software for forward simulation for many classes of problems; (2) the imminent arrival of the petascale computing era; and (3) the explosion of available observational data in many scientific areas. I am interested in investigating Bayesian framework type methods, in addition to leveraging computationally efficient deterministic methods such as PDE constrained optimization. Reduced order models are considered to enable tractable Monte Carlo strategies in addition to the use of Polynomial Chaos to pseudo discretize the stochastic space.

Software and Applications : The computational science environment is continuously changing, as exhibited by evolving hardware architectures, maturing simulation technologies and advancing analysis algorithms. New software capabilities are emerging and in particular the process of developing simulators is transforming. Historically, simulation development consisted of implementing algorithms from scratch requiring considerable effort. The changes in computational science however are driving the need for more flexible and efficient methods, not only isolating users from the low level programming tasks but also shifting the focus to the use of complex analysis and design algorithms. The purpose of this research is to develop near-real time simulation development capabilities fully enabled with sophisticated and computationally efficient analysis capabilities. Furthermore, a target tool set is to be endowed with interfaces that support a modular design so that modifications and extensions can be applied seamlessly. I am currently working on a new toolset called NIHILO that makes use of Sundance which is a powerful prototyping capability.