About

Michael Holst is a professor in the Department of Mathematics at the University of California, San Diego. He holds a Ph.D. in Numerical Analysis from the University of Illinois at Urbana-Champaign, obtained in 1993. His research areas include Numerical Differential Equations, Mathematical Modeling and Applied Analysis, Mathematical Physics, and Mathematical Biology. Dr. Holst has been recognized as a Fellow of the Society for Industrial and Applied Mathematics (SIAM) and has received several awards, including the Department of Energy Office of Science Award, NSF CAREER Award, and the Hellman Fellowship. His professional focus encompasses advancing computational methods and mathematical frameworks in applied sciences, contributing to both theoretical developments and practical applications.

Research topics

• Computer Science
• Computer graphics (images)
• Physics
• Computational science
• Mathematical analysis
• Mathematics
• Discrete mathematics
• Pure mathematics
• Combinatorics
• Algorithm
• Geometry

Selected publications

• Towards Real Time Control of Water Engineering with Nonlinear Hyperbolic Partial Differential Equations

  ArXiv.org · 2025-12-16

  preprintOpen access

  This paper examines aspirational requirements for software addressing mixed-integer optimization problems constrained by the nonlinear Shallow Water partial differential equations (PDEs), motivated by applications such as river-flow management in hydropower cascades. Realistic deployment of such software would require the simultaneous treatment of nonlinear and potentially non-smooth PDE dynamics, limited theoretical guarantees on the existence and regularity of control-to-state mappings under varying boundary conditions, and computational performance compatible with operational decision-making. In addition, practical settings motivate consideration of uncertainty arising from forecasts of demand, inflows, and environmental conditions. At present, the theoretical foundations, numerical optimization methods, and large-scale scientific computing tools required to address these challenges in a unified and tractable manner remain the subject of ongoing research across the associated research communities. Rather than proposing a complete solution, this work uses the problem as a case study to identify and organize the mathematical, algorithmic, and computational components that would be necessary for its realization. The resulting framework highlights open challenges and intermediate research directions, and may inform both more circumscribed related problems and the design of future large-scale collaborative efforts aimed at addressing such objectives.

  PublisherOA PDFDOI

• Geometric transformation of finite element methods: Theory and applications

  Applied Numerical Mathematics · 2023-07-10 · 8 citations

  articleOpen access1st author

  We present a new analysis of finite element methods for partial differential equations over curved domains. In many applications, a change of variables translates a physical Poisson problem over a curved physical domain into a parametric Poisson problem over a polytopal parametric domain. Whilst this change of variables greatly simplifies the geometry and numerical implementations, the coordinate transformation typically features only low regularity. In the parametric Poisson problem, this manifests as rough coefficients and data, which diminish the elliptic regularity, and as roughness of the parametric solution. Our main result addresses how to nevertheless recover high-order finite element convergence rates, the key component being a recently developed broken Bramble-Hilbert lemma. This analysis has numerous applications, where the geometric transformation is either computable or merely a theoretical tool. We propose a simplified technique as easier, more broadly applicable, yet just as powerful as previous isoparametric methods. In particular, we reassess the error analysis of isoparametric finite element methods and prove high-order error estimates for isoparametric FEM even when the physical solution is not continuous. Numerical examples confirm our theoretical predictions.

  PublisherDOI

• A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients

  arXiv (Cornell University) · 2023-06-28

  preprintOpen access1st authorCorresponding

  We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique using rescaling estimates and apply to operators having coefficients with low regularity. For each class of function space for an operator's coefficients, we exhibit a natural associated range of function spaces of the same type for the domain of the operator and we provide regularity inference along with interior estimates. Additionally, we present a unified set of multiplication results for the function spaces we consider.

  PublisherOA PDFDOI

• Conformal fields and the structure of the space of solutions of the Einstein constraint equations

  Advances in Theoretical and Mathematical Physics · 2022-01-01 · 5 citations

  article1st authorCorresponding

  The drift method, introduced by the second author, provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly greater analytic complexity over, say, the conformal or conformal thin sandwich methods. We review this theory here and apply it to the study of solutions of the constraint equations with non-constant mean curvature. We show that this method reproduces previously known existence results obtained by other methods, and does better in one important regard. Namely, it can be applied even when the underlying metric admits conformal Killing (but not true Killing) vector fields. We also prove that the absence of true Killing fields holds generically.

  PublisherDOI

• Finite Element Methods for Linear Maxwell's Equations in Bianisotropic Media Permitting Polarization Fields and Magnetic Currents

  arXiv (Cornell University) · 2022-12-22

  preprintOpen accessSenior author

  We review Maxwell's equations and constitutive relations for 3D bianisotropic media in a generalized form: we consider all four variables and allow for nonzero polarization or magnetization, and also nonzero nonzero magnetic charge or current. After a discussion of general boundary conditions, we introduce a time-harmonic variational formulation of linear Maxwell's equations within 3D bianisotropic media in terms of the electric and magnetic fields. We showcase a finite element approximation of our variational formulation, using curl-conforming Nédélec edge elements of the first kind. Numerical examples illustrate the convergence of the method.

  PublisherOA PDFDOI

• Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

  Mathematics · 2022-02-07 · 6 citations

  articleOpen accessSenior author

  In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<t<1, and 1<p<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.

  PublisherOA PDFDOI

• Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory Due to Dirac Exchange for a Hydrogen Molecule

  Journal of Nonlinear Science · 2022-09-19

  article1st author
  PublisherDOI

• On the Space of Locally Sobolev-Slobodeckij Functions

  Journal of Function Spaces · 2022-07-18 · 2 citations

  articleOpen accessSenior author

  The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msup> <mrow> <mi>W</mi> </mrow> <mrow> <mi>m</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math> or Bessel potential spaces <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msup> </math> . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msup> <mrow> <mi>W</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math> where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>s</mi> </math> does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature.

  PublisherOA PDFDOI

• On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics

  Contemporary Mathematics · 2022-04-19 · 8 citations

  articleOpen accessSenior author

  The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.

  PublisherOA PDFDOI

• A Note On Determining Projections for Non-Homogeneous Incompressible Fluids

  arXiv (Cornell University) · 2021-02-09

  preprintOpen accessSenior author

  In this note, we consider a viscous incompressible fluid in a finite domain in both two and three dimensions, and examine the question of determining degrees of freedom (projections, functionals, and nodes). Our particular interest is the case of non-constant viscosity, representing either a fluid with viscosity that changes over time (such as an oil that loses viscosity as it degrades), or a fluid with viscosity varying spatially (as in the case of two-phase or multi-phase fluid models). Our goal is to apply the determining projection framework developed by the second author in previous work for weak solutions to the Navier-Stokes equations, in order to establish bounds on the number of determining functionals for this case, or equivalently, the dimension of a determining set, based on the approximation properties of an underlying determining projection. The results for the case of time-varying viscosity mirror those for weak solutions established in earlier work for constant viscosity. The case of space-varying viscosity, treated within a single-fluid Navier-Stokes model, is quite challenging to analyze, but we explore some preliminary ideas for understanding this case.

  PublisherOA PDFDOI

Recent grants

• Collaborative Research: Adaptive Methods and Finite Element Exterior Calculus for Nonlinear Geometric PDE

  NSF · $145k · 2012–2016

• Numerical Methods for Geometric Partial Differential Equations with Applications in Numerical Relativity

  NSF · $450k · 2020–2025

• FRG: Collaborative Research: Error Quantification and Control for Gravitational Waveform Simulation

  NSF · $455k · 2011–2015

• Collaborative Research: Numerical Methods for Nonlinear Diffusion Problems

  NSF · $239k · 2004–2007

• FRG: Collaborative Research: Analysis of the Einstein Constraint Equations

  NSF · $251k · 2013–2019

Frequent coauthors

• J. Andrew McCammon

  University of California, San Diego

  71 shared

• Yuhui Cheng

  State Key Laboratory of Biotherapy

  19 shared

• Padmini Rangamani

  University of California, San Diego

  19 shared

• Yunrong Zhu

  17 shared

• Christopher T. Lee

  University of California, San Diego

  15 shared

• Stephen Bond

  Sandia National Laboratories

  15 shared

• Zeyun Yu

  University of Wisconsin–Milwaukee

  15 shared

• Y. C. Zhou

  Nantong University

  15 shared

Awards & honors

• Fellow of the Society for Industrial and Applied Mathematics…
• Department of Energy Office of Science Award
• NSF CAREER Award
• Hellman Fellowship