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Research Overview

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My research is in emerging fast methods for the solution of nonlinear partial differential equations (PDEs) that arise in areas such as mechanics and computer vision. I am particularly interested in static solutions of Hamilton-Jacobi equations and hyperbolic conservation laws, and developing accessible scientific software for these and other problems. This includes research in numerical methods for partial differential equations, high performance scientific computing, and efficient algorithm development combining traditional discretization techniques and modern machine learning (ML) methods, for high-order solutions to nonlinear problems with steep gradients and shocks.

Current work is in utilizing traditional discretization of PDEs and ML techniques, particularly fixed-point methods and physics-informed neural networks (PINNs), for solving eikonal equations with applications in computer vision. This includes object reconstruction through solving inverse problems and object identification through multi-objective methods of reconstruction and surface recognition. Other projects include investigating the use of adaptive mesh refinement (AMR) for efficiently producing high-resolution solutions of Hamilton-Jacobi equations, particularly in high-dimensions.

Past work has been in high-order accurate methods for hyperbolic conservation laws and high-dimensional Hamilton-Jacobi equations, with a focus on improving efficiency while maintaining accuracy. The above image is an example of the solution to a shape from shading problem, a common problem in computer vision of reconstructing a 3D shape from the shading data of a 2D image by solving a nonlinear Hamilton-Jacobi equation.

Publications

Refereed Journal

  1. Z.M. Miksis and Y.-T. Zhang, Sparse-grid implementation of fixed-point fast sweeping WENO schemes for Eikonal equations, Communications on Applied Mathematics and Computation, Accepted (2022). Published online first: https://doi.org/10.1007/s42967-022-00209-x (pdf)

Conference Publications

  1. S. Abraham, J. Kinnison, Z.M. Miksis, D. Poster, S. You, J. Hauenstein, and W. Scheirer, Efficient hyperparameter optimization for ATR using homotopy parametrization, to appear at Society of Photographic Instrumentation Engineers Defense + Commercial Sensing (SPIE 2023), May 2023.

Theses

  1. Z.M. Miksis, Sparse-grid implementation of fixed-point fast sweeping WENO schemes for Eikonal equations, PhD dissertation, University of Notre Dame, 2022.
  1. Z.M. Miksis, An Accelerating Couette Flow in Nek5000: Applications in Oceanography and Magnetohydrodynamics, MS thesis, Illinois Institute of Technology, 2017.

Presentations

Invited Talks

  1. Numerical solutions to Hamilton-Jacobi equations on sparse grids and a Stokes problem variant with Nek5000, Virtual presentation, Sandia National Laboratory, Albuquerque, NM, June 2022.

Contributed Talks

  1. Parallel implementation of a sparse grid fast sweeping WENO method for Eikonal equations, Midwest Numerical Analysis Day, University of Michigan, Ann Arbor, MI, May 2022.

  2. A sparse grid fast sweeping WENO method for Eikonal equations, Midwest Numerical Analysis Day, Missouri University of Science and Technology, Rolla, MO, October 2021.

  3. An accelerating Couette flow with MHD in Nek5000, Summer Argonne Student Symposium, Argonne National Laboratory, Lemont, IL, August 2016.

  4. MHD with Nek5000, Nek5000 Users and Developers Meeting, Massachusetts Institute of Technology, Cambridge, MA, August 2016.

  5. Verification of a Stokes problem variant, Graduate Student Colloqium, Illinois Institute of Technology, Chicago, IL, March 2016.