Akil Narayan

Research Interests


My broad research interests are numerical analysis and scientific computing. Specifically, I am interested in high-order numerical methods for differential equations, applications in the field of uncertainty quantification, spectral methods, and orthogonal polynomials.

My current and past projects include:
  • spectral expansions on infinite intervals
  • theory and computational methods for univariate orthogonal polynomials
  • uncertainty quantification (UQ) for stochastic systems
    • generalized polynomial chaos methods
    • sensitivity of stochastic systems with respect to probability distribution
    • assimilation of multiple models and sources of data
  • polynomial interpolation in high dimensions on unstructured grids with applications to UQ


Publications

  • M. Feiszli and A. Narayan. Numerical computation of Weil-Peterson geodesics on the Universal Teichmüller Space. In Preparation.
  • S. Kushnarev and A. Narayan. Teichon solutions for geodesics on the Universal Teichmüller Space. In Preparation.
  • J. Jakeman, A. Narayan, and D. Xiu. Minimal element stochastic collocation for uncertainty quantification of discontinuous functions. Submitted.
  • A. Narayan and J. S. Hesthaven. A Generalization of the Wiener Rational Basis Functions on Infinite Intervals: Part II -- Numerical Examples. Submitted.
  • A. Narayan, Y. Marzouk and D. Xiu. Sequential assimilation of multiple models. To appear: Journal of Computational Physics
  • A. Narayan and D. Xiu. Stochastic collocation methods on unstructured grids in high dimensions via interpolation. To appear: SIAM Journal on Scientific Computing
  • A. Narayan and J. S. Hesthaven. Computation of connection coefficients and measure modifications for orthogonal polynomials. BIT Numerical Mathematics, 2011.
  • A. Narayan and D. Xiu. Distributional Sensitivity for Uncertainty Quantification. Communications in Computational Physics 10:140-160, 2011.
  • A. Narayan and J. S. Hesthaven A Generalization of the Wiener Rational Basis Functions on Infinite Intervals: Part I -- Derivation and Theory. Mathematics of Computation 80:1557-1583, 2010.

Theses and Technical Reports

  • A. Narayan and A. Klöckner. Deterministic Methods for the Boltzmann Equation. Brown University Technical Report, 2009
  • A. Narayan. Generalized Wiener rational functions for spectral expansions on an infinite interval. Ph.D. thesis, Brown University, 2009
  • A. Narayan. A study of spatial solitons. Undergraduate thesis, Northwestern University, 2003.
  • A. Narayan. Modelling fiber-optic generated continuum. Undergraduate thesis, Northwestern University, 2003.