Computation for science

Numerical methods for PDEs in computational biophysics, and compartmental models of transmission and control.

Focus Areas

I develop and implement numerical methods for multidimensional partial differential equations: finite difference schemes, Matched Interface and Boundary (MIB) methods, and finite element methods, run on high-performance computing platforms. The applications are in drug design and protein engineering.

A separate part of my work uses compartmental dynamical systems, where the questions are about thresholds and control: what the reproduction number is, whether the endemic equilibrium is stable, and which intervention strategy is worth what it costs.

Mathematical Biology Implicit solvation models, biomolecular solvation analysis, ensemble averaged electrostatics, PDE modeling of molecular surfaces.
Scientific Computing Electrostatic analysis, material interface problems, biomolecular simulations on HPC platforms.
Numerical Methods for PDEs Matched Interface and Boundary (MIB) method, Alternating Direction Implicit (ADI) methods.
Mathematical Epidemiology Compartmental transmission models, reproduction numbers via the next-generation matrix, stability and bifurcation analysis, optimal control and cost-effectiveness of intervention strategies.

Current Projects

Biomolecular Solvation

Developing a Variational Implicit Solvation Model incorporating Size-Modified Poisson-Boltzmann (SMPB) theory, diffuse interface approaches, and ensemble averaged electrostatics for accurate free energy calculations in drug design and protein engineering.

Poisson-Boltzmann Free Energy Protein Engineering

Related manuscript: On calculating polar solvation energy of nonrigid proteins in the Poisson-Boltzmann theory (submitted) →

High-Order Numerical Schemes

Developing robust solvers for interface problems and stiff systems, with focus on three-dimensional p-Laplace equations using finite difference methods, efficient ADI time-stepping for parabolic equations, and MIB methods for complex geometry in solvation models.

p-Laplace ADI Methods MIB Method

Related paper: Numerical solution to the three-dimensional p-Laplace equation (JCAM, 2026) →

Mathematical Epidemiology & Transmission Dynamics

I build and analyze nonlinear compartmental models of spread through a structured population. The analysis follows a familiar course: positivity and boundedness of solutions, the reproduction number from the next-generation matrix, then the disease-free and endemic equilibria and their local and global stability, usually by Lyapunov arguments. I also check for backward bifurcation, since where it occurs, pushing the reproduction number below one is no longer enough to eliminate the disease. Optimal control and cost-effectiveness analysis come last, to rank the candidate interventions.

Methods

Compartmental ODE Models Next-Generation Matrix Reproduction Number Equilibrium & Stability Analysis Backward Bifurcation Optimal Control Cost-Effectiveness Analysis Sensitivity Analysis

Related work

Computational Tools

FORTRAN MATLAB Python R APBS DelPhi VMD HPC / Slurm Finite Difference Methods Variational Implicit Solvent Models Matched Interface and Boundary

Selected Visualizations

Electrostatic potential of protein 451c
Electrostatic Analysis Electrostatic surface potential of protein 451c (PDB ID) mapped onto surface generated after solving the p-Laplace equation (p = 1.2). Visualized in VMD.
Diatom molecular surfaces generated with p-Laplace exponents 2.00, 1.33, 1.20 and 1.00
Molecular Surfaces from the p-Laplacian Isosurfaces of a diatom system as the exponent pm goes from 2 to 1. As pm approaches 1, surface area and enclosed volume converge to the reference values. View paper →
CPU time against number of atoms for Picard iteration, ADI and Crank-Nicolson
Numerical Scheme Efficiency CPU time for ADI, Crank-Nicolson, and Picard iteration on the p-Laplace equation across 23 proteins. ADI is the fastest of the three at every problem size. View paper →
Flow diagram of the crime dynamics model
Compartmental Transmission Model Flow diagram for the crime dynamics model, showing how intervention strategies change the transfer rates between compartments. View paper →
Contour plot of the effective reproduction number against intervention efficacy and compliance rate
Reproduction Number Under Control The effective reproduction number RE as a function of intervention efficacy (ε) and compliance rate (η). Both have to be high before RE falls substantially. View paper →
Bar chart of partial rank correlation coefficients for the crime model parameters
Global Sensitivity Analysis Partial rank correlation coefficients (PRCCs) from Latin Hypercube Sampling, ranking how strongly each parameter drives the criminal population. View paper →