My general research areas are applied mathematics, scientific computation, and numerical analysis. Brief descriptions of my specific research interests are given below.
Prospective or current graduate students – feel free to email me if you are interested in talking about research or about graduate studies in applied/computational mathematics in our department.
Data-driven model order reduction of PDEs
Many mathematical models are very high dimensional, or they are infinite dimensional in the case of partial differential equations (PDEs) and other problems. Therefore, simulation, optimization, control design, and many other tasks are extremely computationally costly (or impossible) for many models. Model order reduction is a process of taking a complex model and creating a model of drastically lower dimension. The hope is that the reduced order model is still useful for simulation, optimization, control design, etc.
My research in this area has focused on data-driven model order reduction methods that use simulation or experimental data to produce the reduced order model. My recent research in this area has focused on proper orthogonal decomposition. I have also worked on balanced truncation model order reduction.
Proper orthogonal decomposition
I have worked on various aspects of model order reduction involving proper orthogonal decomposition (POD). My research focused on multiple different directions: incremental POD algorithms that reduce the computational complexity and data storage for POD computations, new methods for POD reduced order models of PDEs, proving error bounds and convergence results, as well as constructing POD-based reduced order models for certain problems that speed up the simulation of the reduced order model.
Balanced truncation
Balanced truncation is a model order reduction technique for linear differential equation and PDE systems with inputs (e.g., control inputs) and outputs (e.g., sensor measurements). The technique is very popular since there is a guaranteed error bound on the input-to-output error (in terms of the transfer functions of the original and reduced models). However, the balanced truncated reduced order model can be difficult to compute for complex systems.
My research has focused on data-driven (or snapshot) methods for balanced truncation, such as balanced POD, and related topics. My research has focused on the application of balanced POD to PDE systems without extracting approximating system matrices from existing simulation codes, proving convergence results for the algorithm applied to PDE systems, balanced POD for data reconstruction, and balanced truncation applied to nonlinear PDE systems.
Computational feedback and optimal control of PDEs
There are many systems governed by partial differential equations we want to control. For example, we would like to control air flows in order to save on fuel costs, enable stable flight of micro air vehicles, and reduce noise emissions. For another example, we would like to control heat and air flow in large commercial buildings to reduce the very large energy consumption of such buildings.
Feedback control is a method that uses a mathematical model of the system in combination with current sensor measurements of the system to produce the control input to the system. Feedback control methods are well known to be robust to uncertainties in the mathematical model – this is important for many PDE systems we would like to control. However, feedback control laws are difficult to compute and implement when the mathematical model of the system is high dimensional (or infinite dimensional, as with PDEs). This topic ties into my interest in reduced order modeling – implementation of the feedback control laws in real time necessitate accurate reduced order models.
My research in this area has focused on algorithms for Lyapunov and Riccati equations, applications of PDE feedback control, and also optimal control of PDEs.
Lyapunov and Riccati equations
Lyapunov and Riccati equations often must be solved in order to compute feedback control laws. When the mathematical model is an ordinary differential equation system, the unknowns to these equations are matrices whose dimensions increase with the dimension of the system; for PDE systems, the unknowns are infinite dimensional operators which must be approximated. My research in this area has been the development and analysis of snapshot methods to approximate the infinite dimensional Lyapunov and Riccati solution operators arising from PDE systems. The methods use snapshots of large scale simulation data to construct the approximations. This relates to my research in model reduction methods, in which I also have investigated snapshot methods.
Applications of PDE feedback control
I have also been involved in applications of feedback control for structural PDE systems. Feedback control of these specific systems has potential for flexible wing control in micro air vehicles and control of large flexible space structures.
Optimal control of PDEs
I have recently worked on computing solutions of optimal control problems for PDEs using finite element and hybridizable discontinuous Galerkin methods.
Applications in control theory, engineering, and mathematics
I have also worked on various applications in iterative learning control, forcing estimation in structural systems, sensitivity analysis, transition to turbulence, biologically inspired hair sensors, and numerical methods for PDEs.
Iterative learning control
I worked with Douglas Bristow in Mechanical and Aerospace Engineering at Missouri S&T on applications in iterative learning control (ILC), which is used to improve the performance of a process that is repeated many times. In ILC, transient behavior of the learning process is crucial, and transient growth can cause the learning process to be terminated – even if the process is stable. However, allowing an acceptable amount of transient growth can lead to gains in performance. We used and extended the concept of pseudospectra (the set of approximate eigenvalues of a matrix) in order to treat transient growth in ILC systems.
Forcing estimation in structural systems
In my research on forcing estimation in structural systems, I derived and analyzed an algorithm to estimate a load on a membrane given a finite number of sensor measurements. It may be possible to use this estimated forcing information to design control laws for micro air vehicles.
Sensitivity analysis
Sensitivity analysis is the study of how solutions to a mathematical model change as the parameters in the model change. Sensitivity analysis has many applications including optimization, parameter identification, and analyzing uncertainty in mathematical models.
A sensitivity is the derivative of a solution to the model with respect to a parameter in the model. In my research, I have used the continuous sensitivity equation method (CSEM) to compute/analyze sensitivities in PDE models. In the CSEM, one derives an equation for the sensitivity by implicitly differentiating through the model with respect to the parameter of interest.
My research in this area has focused on computational methods, theoretical analysis, and transition to turbulence in fluid flows.
Transition to turbulence
In my research in transition to turbulence in fluid flows, I examined the effect of small disturbances on transition. In low order model problems, very small disturbances can trigger transition even if the problem is stable. In fact, the small disturbances can trigger a bifurcation of equilibrium points which can destroy the stable equilibrium. In these model problems the linearized problem is crucial to transition, and linear feedback control can delay transition and also restabilize a chaotic, unstable flow.
I also showed that sensitivity analysis can be used to predict the effects of small disturbances without solving the full disturbed nonlinear flow problem. I also used sensitivity analysis to show that fluctuations about a stable laminar flow can be extremely large even for very small disturbances – therefore, very small disturbances can trigger transition to turbulence in a stable, laminar flow. The analysis shows that the linearized problem is crucial to transition – therefore linear feedback control has great potential to delay transition to turbulence in fluid flows. This links to my interest in feedback control of PDEs.
Biologically inspired hair sensors
I have been involved with modeling and simulation of biologically inspired hair sensors. There is evidence that hairs on bat wings are used to feedback airflow information for flight control. It is hoped that artificial hair sensors can be used in flight control applications for small unmanned air vehicles.
Numerical methods for PDEs
A large amount of my research involves the use of numerical methods for PDEs for applications, such as model order reduction and control. I have performed some research on analyzing the performance of various numerical methods for PDEs. Recently, I worked on various hybridizable discontinuous Galerkin methods for nonlinear PDEs. I have also studied a numerical method for an example PDE (a 1D Burgers’ equation) that is known to produce false numerical steady state solutions.