About

I am a PIMS-Simons Postdoctoral Fellow working with the group of Christoph Ortner at the University of British Columbia. Previously I held a postdoctoral research associate position at the University of Oxford after having completed a PhD in applied mathematics at the Imperial College London.

My research focus is on numerical analysis and the development of computational tools for PDEs, integral equations and fractional differential equations with applications in the natural sciences such as quantum physics and collective behavior in biology. I am also more generally interested in computational orthogonal polynomials and their applications in the development of multivariate spectral methods.

Publications

See also the automated listings at Google Scholar or Semantic Scholar.

Wave equation on a disk dampened by a time-fractional Caputo derivative term.

Fractional Differential Equations

Fractional differential equations generalize ODEs and PDEs to include derivative operators of non-integer order. The resulting operators are generally nonlocal and thus pose several challenges to conventional numerical approaches.

Fractional PDEs appear in natural science applications featuring memory effects, e.g. photoacoustic imaging in medical physics.

  1. A frame approach for equations involving the fractional Laplacian
    Ioannis P. A. Papadopoulos, T. S. Gutleb, J. A. Carrillo, S. Olver.  arXiv
  2. Explicit fractional Laplacians and Riesz potentials of classical functions
    T. S. Gutleb, Ioannis P. A. Papadopoulos.  arXiv
  3. A static memory sparse spectral method for time-fractional PDEs
    T. S. Gutleb, J. A. Carrillo. Journal of Computational Physics (2023).   arXiv  DOI
Numerical solution to an inhomogeneous Helmholtz equation with k = 80 and right-hand side f(x,y) = sin(100x) on an annular domain with inner radius 0.5.

Orthogonal Polynomials and Spectral Methods

Orthogonal polynomials can be used to accurately and efficiently approximate functions in one and higher dimensions.

Alongside their use in quadrature methods they can be used to construct spectral methods for solving integral and partial differential equations with sparse discretisations and spectral convergence.

  1. Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli
    I. P. A. Papadopoulos, T. S. Gutleb, R. M. Slevinsky, S. Olver.  arXiv
  2. Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations
    T. S. Gutleb, S. Olver, R. M. Slevinsky.  arXiv
  3. A Fast Sparse Spectral Method for Nonlinear Integro-differential Volterra Equations with General Kernels
    T. S. Gutleb. Advances in Computational Mathematics (2021).   arXiv   DOI
  4. A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle
    T. S. Gutleb, S. Olver. SIAM Journal on Numerical Analysis (2020).   arXiv   DOI
Convergence of histograms associated with attractive-repulsive particle simulation to the equilibrium measure.

Computing Equilibrium Measures

Equilibrium measure problems arise in the continuous mean-field limit of attractive-repulsive particle systems and have diverse applications ranging from swarming behavior in animals to self-assembly properties of particle systems.

As power law interactions involve Riesz potentials, this special case also relates to fractional differential equations.

  1. Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension
    T. S. Gutleb, J. A. Carrillo, S. Olver. Constructive Approximation (2022).   arXiv  DOI
  2. Computing Equilibrium Measures with Power Law Kernels
    T. S. Gutleb, J. A. Carrillo, S. Olver. Mathematics of Computation (2022).   arXiv   DOI
Isosurface evolution of spin up (red) and spin down (blue) states as solutions of the Pauli equation coupled to an external magnetic field.

Numerics for magnetic Schrödinger equations

Strang splitting methods have been successfully used for the numerical treatment of non-relativistic magnetic Schrödinger equations in their appropriate 3D context.

I have worked towards extending these results to also work for relativistic equations which account for quantum mechanical spin.

  1. A time splitting method for the three-dimensional linear Pauli equation
    T. S. Gutleb, N. J. Mauser, M. Ruggeri, H. P. Stimming. Computational Methods in Applied Mathematics (2023). arXiv   DOI

Selected Talks

Below is a selection of my conference talks. I'll update the website with a more complete list soon.

Teaching

  • Intercollegiate Tutor for Numerical Linear Algebra
    Oct 2022 - Jan 2023, University of Oxford, UK
  • Senior GTA for Numerical Analysis
    Jan 2022 - Apr 2022, Imperial College London, UK
  • GTA and student project mentor for Mathematical Modeling
    Mar 2021 - Jun 2021 & May 2022 - August 2022, Wolfgang Pauli Institute, University of Vienna, Austria
  • Graduate Teaching Assistant (GTA) in Analysis 1 and 2 for Physicists
    Oct 2015 - Jul 2018 , University of Vienna, Austria