Orthogonal Polynomials and Biomedical Uncertainty Quantification

My Ph.D. research at the University of Utah focused on scientific computing and uncertainty quantification, advised by Professor Akil Narayan. The core mathematical problem was how to construct stable orthogonal polynomial bases for nonclassical measures, then use them in polynomial chaos expansion for expensive biomedical simulations.

The work connects three layers:

• Univariate recurrence algorithms. Compute three-term recurrence coefficients for generalized orthogonal polynomial families when closed-form classical formulas are unavailable.
• Multivariate orthogonal polynomials. Develop recurrence-matrix algorithms that make stable multivariate polynomial bases practical for approximation and quadrature.
• Biomedical uncertainty quantification. Package the mathematics into noninvasive UQ workflows through UncertainSCI and apply it to cardiac simulation, brain stimulation, and coronary biomechanics.

Representative outcomes:

• Univariate recurrence examples
• Multivariate recurrence examples
• UncertainSCI

This project is the foundation of my quantitative background: numerical analysis, approximation theory, uncertainty quantification, and mathematical software for reliable biomedical modeling.