## Research

### A posteriori Error Estimation for Multi-scale, Multi-physics Systems

Multi-physics systems involve simulation of multiple physical models, often operating at different scales, and are of widespread importance in science and engineering.
In this regard, I have focused on analysis of three challenging problems: multi-rate integration for multi-physics systems, evolution solvers for non-autonomous problems that employ truncated iteration to solve the implicit system, and Implicit-Explicit (IMEX) methods.

These methods are widely employed in simulation of important real-world phenomenon, however, forming reliable error estimates for such methods is a challenging task because of the complex nature of the solution techniques. In my research I address the problem of analysis of these methods by employing a posteriori error estimation. The estimates derived in this research not only quantify the error in a quantity of interest (QoI), they also give guidance as to choice of numerical parameters needed to obtain a desired accuracy.

### Quantity-of-Interest based Least-Squares Finite Element Method

This research aims to incorporate quantity of interest in the least-squares based finite element method (LSFEM). LSFEM is an important class of robust and efficient numerical schemes. In this research, a new LSFEM method is designed for computing a QoI and analyzing error in the resulting numerical solution. The resulting framework is shown to have robust theoretical properties as well as possessing efficiency to arrive at accurate computations of the QoI.

### Adaptive Refinement and A posteriori Analysis of Parallel-in-time Algorithms

We use adjoint based analysis to identify the stability and accumulation of the solution errors. This helps guides adaptive refinement for a subsequent fine-level simulation. In particular, our analysis allows for error estimation of the Parareal algorithm, identifies sources of error and suggests appropriate strategies to accelerate convergence.

### Reduced Order Modeling

I am working on aspects of reduced order modeling, in particular developing techniques for parameter space partitioning and obtaining cheap error estimates.

### Design and Analysis of Finite Element Methods for Implicit Solvent Models

My graduate research centered on designing and analyzing numerical methods for
problems in biomolecular systems represented using implicit solvent models. The implicit models treat the solvent as a continuum, instead of representing each ion explicitly. Hence these methods are computationally efficient as compared to the traditional molecular dynamics based methods, and have seen widespread use in recent years.
I have developed novel numerical schemes based on LSFEM and analyzed the solution to non-linear equations modeling biomolecular systems. Furthermore, I developed a stabilized numerical method to account for the high electrostatic forces present in real world biomolecular systems.