The overall direction of my research is in the area of optimal design and control of distributed parameter systems. This is a multidisciplinary area and I perform a majority of my research at the Interdisciplinary Center for Applied Mathematics (ICAM). My main area of study is in system level and computational aspects of these problems covering research areas:

computational science/scientific computing, control theory, numerical analysis, numerical methods for PDEs, optimization, reduced-order modeling, sensitivity analysis, and verification and validation of software.

More details from some of these research areas are included below.
Sensitivity Analysis

My interest in sensitivity analysis goes back to my Masters research. The current line of research was initially motivated by the role of sensitivity analysis in computing derivatives for optimal design of a forebody simulator. Research continues in both development of efficient algorithms for computing sensitivity derivatives as well as their application to optimal design, parameter estimation, reduced-order modeling and uncertainty quantification.

An overview of sensitivity analysis for partial differential equations is provided here.

The general theme is that a mathematical viewpoint of the problem can inform good numerical methods for sensitivity analysis.

Examples the illuminate the utility of sensitivities are provided here.

Main Collaborators: Akhtar, Burns, Cliff, Gunzburger, Hay, Leite Nunes, Pelletier, Verma



Optimal Design and Parameter Estimation Algorithms

My primary interest is in optimization with partial differential equation constraints (frequently models of fluid behavior). The optimization parameters can be design variables, shape variables, and PDE coefficients. Earlier work considered the interplay between discretization and optimization, including the impact of using a variety of discretization methods to compute derivatives as well as other numerical noise. In recent work, I am more focused on applications, and optimization problems with distributed and/or random parameters.

  • Convergence Theory Based on Asymptotic Consistency,
  • Forebody Simulator Design Problem,
  • Optimization Algorithms for Problems with Noisy Objective Functions,
  • Optimal Zonation in Groundwater Modeling,
  • Parameter Estimation of Random Material Coefficients,
  • Trust-Region Optimization Algorithms

Main Collaborators: Burbey, Burns, Cliff, Herdman, Leite Nunes, van Wyk, Vugrin, Zietsman



Computational Methods for Control of PDEs

My main motivation is control of fluid flows, also known as flow control. Applying linear quadratic regulator theory leads to large scale Riccati equations (which can also be obtained by discretizing the Riccati partial differential equations). In some cases, we are only interested in the feedback gains (the kernel of the linear feedback operator) which has applications to actuator and sensor placement. While in other cases, we are interested in the influence of various operator discretizations on the convergence of the Riccati solutions. Applications include stabilization of flows, designing the configuration of control systems (actuator and sensor placement), and state estimation.

  • Approximation to PDE Riccati Equations,
  • Chandrasekhar Equations,
  • Data Assimilation,
  • Optimal Actuator and Sensor Placement.

Main Collaborators: Akhtar, Burns, Demetriou, Iliescu, Stoyanov, Zietsman



Model Reduction Methods

Reduced-order models seek to capture the behavior of high-dimensional, or infinite dimensional systems using low-dimensional systems. A common approach uses projection of model equations onto a reduced basis. For fluids, this typically is performed using the proper orthogonal decomposition (POD, aka Karhunen-Loeve, principle component analysis, singular value decomposition). My research involves finding better bases (e.g. using sensitivity analysis to extrapolate/interpolate bases to different parameter values, using optimization to obtained better long-time model behavior, using the principle interval decomposition to find good time windows over which to apply the POD). After projection, additional modeling is often required to capture the influence of the neglected basis functions. In many cases, this can be achieved using multiscale methods: artificial dissipation, closure models, etc.

  • Extension to Parameter Dependent Models
  • Extension to Complex Turbulent Flows
  • Applications to PDE Control Problems
  • Applications to Optimal Design Problems

More on our group's research in model reduction methods for complex flows can be found here.

Main Collaborators: Akhtar, Gugercin, Hay, Iliescu, Wang, Zietsman



Numerical Methods for PDEs

  • Extension to Parameter Dependent Models
  • Extension to Complex Turbulent Flows
  • Applications to PDE Control Problems
  • Applications to Optimal Design Problems
  • ViTLES: Virginia Tech Large Eddy Simulator

Main Collaborators: Iliescu, Pelletier, Roop



Applications

  • Model of Nitrogen Transport
  • Modeling Thermal Energy in Buildings
  • Optimal Sensor/Actuator Placement in Building HVAC Design.
  • Parameter Estimation in Groundwater Flow Models

Main Collaborators: Ahuja, Burbey, Burns, Cliff, Nunes, Shah, Surana, Wolfe, Zietsman


Funded Research

I am very grateful to have received support for this research from the Air Force Office of Scientific Research (AFOSR), the National Research Council (NRC), and the National Science Foundation (NSF):

  • Control and Optimization Tools for Systems Governed by Nonlinear Partial Differential Equations, AFOSR, 2000-2005.
  • Computational Methods for Design, Control and Optimization of Micro Air Vehicles, AFOSR, 2003-2006. (with J. Burns, E. Cliff and T. Iliescu)
  • Summer Faculty Fellowship Program, NRC, 2003.
  • Scientific Computing Research Environments in Mathematical Sciences, NSF, 2003-2005. (with T. Iliescu)
  • Computation and Analysis of Reduced-Order Models for Distributed Parameter Systems, NSF, 2005-2008. (with C. Beattie, S. Gugercin and T. Iliescu)
  • High Performance Parallel Algorithms for Improved Reduced-Order Modeling, AFOSR, 2005-2008. (with C. Beattie, S. Gugercin and T. Iliescu)
  • Summer Faculty Fellowship Program, NRC, 2007.
  • Reduced-Order Modeling for Optimization and Control of Complex Flows, AFOSR, 2007-2010. (with T. Iliescu)
  • Improved Parametrization of Groundwater Flow Models Using Interferograms and Adjoint Sensitivity Analysis, NSF, 2010-2012. (with T. Burbey and S.L. Sharpe)
  • Advanced Computer Design Tools for Modeling, Design, Control, Optimization and Sensitivity Analysis of Integrated Whole Building Systems, DOE HUB, 2010-2015. (with J. Burns, E. Cliff, S. Gugergin, T. Herdman, T. Iliescu, M. Marathe and L. Zietsman at VT, see GPIC HUB)
  • Transcending POD: Model Reduction for Complex Fluid Flows, NSF, 2010-2013. (with T. Iliescu and J.P. Roop)

I was fortunate to spend extended research stays at the Air Vehicles Directorate at AFRL, the Department of Scientific Computing at Florida State University, and Genie Mecanique at Ecole Polytechnique de Montreal in recent years.