About

Tonatiuh Sanchez-Vizuet is an Assistant Professor in the Department of Mathematics at The University of Arizona, where he is also a member of the Graduate Faculty. His role involves teaching and research within applied mathematics, contributing to the department's academic and research missions. He is associated with the Graduate Interdisciplinary Program (GIDP) and actively participates in the department's scholarly activities. His contact information includes an office at ENR2-S459, and he can be reached via email at tonatiuh@arizona.edu or by phone at 520-621-6098. The department offers a range of undergraduate and graduate programs in mathematics and statistics, and Sanchez-Vizuet's work supports the department's focus on advancing mathematical research and education.

Research topics

• Mathematics
• Mathematical analysis
• Applied mathematics
• Physics
• Statistical physics

Selected publications

• Lambert’s problem in orbital dynamics: a self-contained introduction

  The European Physical Journal Plus · 2026-02-15

  articleSenior authorCorresponding
  PublisherDOI

• Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem

  ArXiv.org · 2025-01-14

  preprintOpen accessSenior author

  We explore a hybrid technique to quantify the variability in the numerical solutions to a free boundary problem associated with magnetic equilibrium in axisymmetric fusion reactors amidst parameter uncertainties. The method aims at reducing computational costs by integrating a surrogate model into a multilevel Monte Carlo method. The resulting surrogate-enhanced multilevel Monte Carlo methods reduce the cost of simulation by factors as large as $10^4$ compared to standard Monte Carlo simulations involving direct numerical solutions of the associated Grad-Shafranov partial differential equation. Accuracy assessments also show that surrogate-based sampling closely aligns with the results of direct computation, confirming its effectiveness in capturing the behavior of plasma boundary and geometric descriptors.

  PublisherOA PDFDOI

• A symmetric boundary integral formulation for time-domain acoustic-elastic scattering

  ArXiv.org · 2025-02-07

  preprintOpen access1st authorCorresponding

  A symmetric boundary integral formulation for the transient scattering of acoustic waves off homogeneous and isotropic elastic obstacles is analyzed. Both the acoustic scattered field and the elastodynamic excited field are represented through a direct integral representation, resulting in a coupled system of interior/exterior integral equations that is symmetrized through the introduction of an auxiliary mortar variable. The analysis of each system and of its Galerkin discretization is done through the passage to the Laplace domain, which allows for the use of convolution quadrature for time discretization. Since the operators of the acustic and elastic Calderón calculus appear independently of each other, the formulation is well suited for non-intrusive numerical impementations (i.e. existing codes for acoustic and elastic problems can be used without any modification).

  PublisherOA PDFDOI

• A symmetric boundary integral formulation for time–domain acoustic-elastic scattering

  Computational Mechanics · 2025-10-24 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Boundary-Field Formulation for Transient Electromagnetic Scattering by Dielectric Scatterers and Coated Conductors

  SIAM Journal on Mathematical Analysis · 2025-03-24 · 1 citations

  article
  PublisherDOI

• A coupled HDG discretization for the interaction between acoustic and elastic waves

  ArXiv.org · 2025-05-21

  preprintOpen access

  We propose and analyze an HDG scheme for the Laplace-domain interaction between a transient acoustic wave and a bounded elastic solid embedded in an unbounded fluid medium. Two mixed variables (the stress tensor and the velocity of the acoustic wave) are included while the symmetry of the stress tensor is imposed weakly by considering the antisymmetric part of the strain tensor (the spin or vorticity tensor) as an additional unknown. Convergence of the method is demonstrated and theoretical rates are obtained; numerical results suggesting optimal order of convergence and superconvergence of the traces are presented.

  PublisherOA PDFDOI

• Well-posedness for the biharmonic scattering problem for a penetrable obstacle

  ArXiv.org · 2025-06-11

  preprintOpen accessSenior author

  We address the direct scattering problem for a penetrable obstacle in an infinite elastic two--dimensional Kirchhoff--Love plate. Under the assumption that the plate's thickness is small relative to the wavelength of the incident wave, the propagation of perturbations on the plate is governed by the two-dimensional biharmonic wave equation, which we study in the frequency domain. With the help of an operator factorization, the scattering problem is analyzed from the perspective of a coupled boundary value problem involving the Helmholtz and modified Helmholtz equations. Well-posedness and reciprocity relations for the problem are established. Numerical examples for some special cases are provided to validate the theoretical findings.

  PublisherOA PDFDOI

• A boundary integral equation formulation for transient electromagnetic transmission problems on Lipschitz domains

  Springer Link (Chiba Institute of Technology) · 2025-06-26

  article1st authorCorresponding

  We propose a boundary integral formulation for the dynamic problem of electromagnetic scattering and transmission by homogeneous dielectric obstacles. In the spirit of Costabel and Stephan, we use the transmission conditions to reduce the number of unknown densities and to formulate a system of coupled boundary integral equations describing the scattered and transmitted waves. The system is transformed into the Laplace domain where it is proven to be stable and uniquely solvable. The Laplace domain stability estimates are then used to establish the stability and unique solvability of the original time domain problem. Finally, we show how the bounds obtained in both Laplace and time domains can be used to derive error estimates for semi discrete Galerkin discretizations in space and for fully discrete numerical schemes that use Convolution Quadrature for time discretization and a conforming Galerkin method for discretization of the space variables.

  DOI

• A boundary integral equation formulation for transient electromagnetic transmission problems on Lipschitz domains

  ESAIM. Mathematical modelling and numerical analysis · 2025-05-01

  articleOpen access1st authorCorresponding

  We propose a boundary integral formulation for the dynamic problem of electromagnetic scattering and transmission by homogeneous dielectric obstacles. In the spirit of Costabel and Stephan, we use the transmission conditions to reduce the number of unknown densities and to formulate a system of coupled boundary integral equations describing the scattered and transmitted waves. The system is transformed into the Laplace domain where it is proven to be stable and uniquely solvable. The Laplace domain stability estimates are then used to establish the stability and unique solvability of the original time domain problem. Finally, we show how the bounds obtained in both Laplace and time domains can be used to derive error estimates for semi discrete Galerkin discretizations in space and for fully discrete numerical schemes that use Convolution Quadrature for time discretization and a conforming Galerkin method for discretization of the space variables.

  PublisherDOI

• An unfitted HDG discretization for a model problem in shape optimization

  ArXiv.org · 2025-09-25

  preprintOpen access

  We apply an unfitted HDG discretization to a model problem in shape optimization. The method proposed uses a fixed, shape regular, non-geometry conforming mesh and a high order transfer technique to deal with the curved boundaries arising in the optimization process. The use of this strategy avoids the need for constant remeshing and enables a highly accurate description of the domain using a coarse computational mesh. We develop a rigorous analysis of the well-posedness of the problems that arise from the optimality conditions, and provide an a priori error analysis for the resulting discrete schemes. Numerical examples with manufactured problems are provided demonstrating the convergence of the scheme and the efficiency of the transfer path method. The approach proposed yields high resolution approximations of the boundary using grids with as few as 100 times less elements than an interpolatory technique.

  PublisherOA PDFDOI

Recent grants

• LEAPS-MPS: Hybridizable discontinuous Galerkin methods for non-linear integro-differential boundary value problems in magnetic plasma confinement

  NSF · $250k · 2021–2025

Frequent coauthors

• Manuel Solano

  University of Concepción

  14 shared

• Nestor Sánchez

  Universidad Nacional Autónoma de México

  13 shared

• George C. Hsiao

  12 shared

• Jiaxing Liang

  University of Maryland, College Park

  11 shared

• Francisco‐Javier Sayas

  University of Delaware

  8 shared

• Howard C. Elman

  University of Maryland, College Park

  7 shared

• Antoine Cerfon

  New York University

  5 shared

• Wolfgang L. Wendland

  University of Stuttgart

  3 shared