5465: The first semester of our Numerical Analysis sequence emphasizes error analysis for numerical linear algebra including linear systems and eigenvalue problems.
5466: The second semeseter of Numerical Analysis emphasizes elements of approximation theory such as approximation by polynomials, quadrature, as well as nonlinear equations and optimization and an overview of numerical methods for ordinary differential equations. MATH 5465 is not a prerequisite, but students should be familiar with analysis and implementation of numerical algorithms.
My research is in the development and analysis of computational tools for optimal design and control of nonlinear PDEs (for example, active and passive control of fluid flows modeled by the Navier-Stokes equations). This research requires developments in a wide range of mathematical topics. A partial list is given below.
- Sensitivity Analysis:
- Adaptive Finite Element Calculations,
- Automatic Differentiation,
- Local Improvements in Reduced-Order Models,
- Nearby Solutions,
- Stability Analysis,
- Uncertainty Quantification
- Optimal Design Algorithms:
- Convergence Theory Based on Asymptotic Consistency,
- Forebody Simulator Design Problem,
- Parameter Estimation
- Computational Methods for Control of PDEs:
- Approximation to PDE Riccati Equations,
- Data Assimilation,
- Optimal Actuator and Sensor Placement
- Model Reduction Methods:
- Principal Interval Decomposition,
- Proper Orthogonal Decomposition,
- Extensions to Parameter Dependent Models,
- Extensions to Complex Turbulent Flows,
- Applications to Fluid Flow Control
- Applied Mathematics / Computational Science:
- Design of Energy Efficient Buildings,
- Computational Fluid Dynamics,
- Modeling for Ventilation in Mines,
- Optimal Zonation for Groundwater Models
A more complete description of my research can be found here.