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 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, including specifically also machine learning models thereof. I am also more generally interested in computational orthogonal polynomials and their applications in such methods.

Research

The following is a non-exhaustive list of things I have worked on and am currently interested in.

Dirac cones in the conduction and valence bands of graphene reconstructed from smooth interpolants by learning invariants instead of the surfaces themselves.

Machine learning conical intersections

Conical intersections play a crucial role in determining the dynamics of molecules and materials upon light excitation, molecular orbitals and material band structure. In the relevant context of high-dimensional non-smooth hyper-surfaces, common machine learning models perform poorly.

I have worked on overcoming these issues by learning globally smooth invariant quantities from which the desired conical intersections can be reconstructed accurately.

Molecular dynamics simulation of an Ala-Ala-Ala molecule (C₉H₁₇N₃O₄).

Machine learning of interaction models

Many physical and biological systems can be accurately modeled using atomistic many body interactions. Machine learning based methods allow one to infer succesful models for otherwise prohibitively large and complex systems.

I work on novel methodologies and applications of such frameworks as well as the closely related problem of analyzing their theoretical foundations.

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 derivatives and closely related objects such as Riesz potentials appear in applications featuring memory effects, e.g. photoacoustic imaging in medical physics or in the continuous limit of classical particle swarms.

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.

With appropriate orthogonal polynomials for a given domain and weight function in hand one can construct quadrature methods as well as spectral methods for solving integral and partial differential equations with sparse discretisations and spectral convergence.

Publications

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

  1. Parameterizing Intersecting Surfaces via Invariants
    arXiv   T. S. Gutleb, R. Barrett, J. Westermayr, C. Ortner.   (2024).
  2. Atomic Cluster Expansion Without Self-Interaction
    arXiv   C.H. Ho, T. S. Gutleb, C. Ortner.   (2024)
  3. A Frame Approach for Equations Involving the Fractional Laplacian
    arXiv   Ioannis P. A. Papadopoulos, T. S. Gutleb, J. A. Carrillo, S. Olver.   (2023)
  4. Explicit Fractional Laplacians and Riesz Potentials of Classical Functions
    arXiv   T. S. Gutleb, Ioannis P. A. Papadopoulos.   (2023)
  5. A Static Memory Sparse Spectral Method for Time-Fractional PDEs
    arXiv  DOI  T. S. Gutleb, J. A. Carrillo.   Journal of Computational Physics (2023).
  6. Building Hierarchies of Semiclassical Jacobi Polynomials for Spectral Methods In Annuli
    arXiv   I. P. A. Papadopoulos, T. S. Gutleb, R. M. Slevinsky, S. Olver.   (2023).
  7. Polyn. and Rational Measure Modifications of Orth. Polynomials via Inf.-Dim. Banded Matrix Factorizations
    arXiv   T. S. Gutleb, S. Olver, R. M. Slevinsky.   (2023)
  8. Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension
    arXiv  DOI   T. S. Gutleb, J. A. Carrillo, S. OlverConstructive Approximation (2022).
  9. Computing Equilibrium Measures with Power Law Kernels
    arXiv   DOI  T. S. Gutleb, J. A. Carrillo, S. OlverMathematics of Computation (2022).
  10. A Fast Sparse Spectral Method for Nonlinear Integro-differential Volterra Equations with General Kernels
    arXiv   DOI  T. S. Gutleb.   Advances in Computational Mathematics (2021).
  11. A Time Splitting Method for the Three-Dimensional Linear Pauli Equation
    arXiv   DOI   T. S. Gutleb, N. Mauser, M. Ruggeri, H. Stimming.   Comp. Methods in Appl. Math. (2023).
  12. A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle
    arXiv   DOI   T. S. Gutleb, S. Olver.   SIAM Journal on Numerical Analysis (2020).

Selected Talks

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

Teaching

  • Small Class Instructor for Integral Calculus with Applications
    Jun 2024 - Aug 2024, University of British Columbia, Canada
  • 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