arXiv subjects

Michael Holst

Publications and source records attributed to Michael Holst.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.

math.OC↗

arXiv2025

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

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.

math.OC↗

arXiv2023

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

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.

math.AP↗

arXiv2022

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

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.

physics.comp-ph↗

arXiv2021

Symmetry Breaking in Density Functional Theory due to Dirac Exchange for a Hydrogen Molecule

We study symmetry breaking in the mean field solutions to the 2 electron hydrogen molecule within Kohn Sham (KS) local spin density function theory with Dirac exchange (the XLDA model). This simplified model shows behavior related to that of the (KS) spin density functional theory (SDFT) predictions in condensed and molecular systems. The Kohn Sham solutions to the constrained SDFT variation problem undergo spontaneous symmetry breaking as the relative strength of the non-convex exchange term increases. This results in the change of the molecular ground state from a paramagnetic state to an antiferromagnetic ground states and a stationary symmetric delocalized 1st excited state. We further characterize the limiting behavior of the minimizer when the strength of the exchange term goes to infinity. This leads to further bifurcations and highly localized states with varying character. The stability of the various solution classes is demonstrated by Hessian analysis. Finite element numerical results provide support for the formal conjectures.

math-ph↗

arXiv2021

A Note On Determining Projections for Non-Homogeneous Incompressible Fluids

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.

math.AP↗

arXiv2019

Applications and Challenges of Machine Learning to Enable Realistic Cellular Simulations

In this perspective, we examine three key aspects of an end-to-end pipeline for realistic cellular simulations: reconstruction and segmentation of cellular structures; generation of cellular structures; and mesh generation, simulation, and data analysis. We highlight some of the relevant prior work in these distinct but overlapping areas, with a particular emphasis on current use of machine learning technologies, as well as on future opportunities.

physics.bio-ph↗

arXiv2018

Numerical Bifurcation Analysis of the Conformal Method

The conformal formulation of the Einstein constraint equations has been studied intensively since the modern version of the conformal method was first pub- lished in the early 1970s. Proofs of existence and uniqueness of solutions were limited to the constant mean curvature (CMC) case through the early 90s, with analogous results for the near-CMC case beginning to appear thereafter. In the last decade, there has been some limited progress towards understanding the properties of the conformal method for far-from-CMC solutions as well. Although it was initially conceivable that that these far-from-CMC results would lead to a solution theory for the non-CMC case that would mirror the good properties of the CMC and near-CMC cases, examples of bifurcations and of nonexistence of solutions have been since discovered. Nevertheless, the general properties of the conformal method for far-from-CMC data remain unknown. In this article we apply analytic and numerical continuation techniques to the study of the con- formal method, in an attempt to give some insight into what the solution behavior is in the far-from-CMC case in various scenarios.

gr-qc↗

arXiv2017

When Do Spacetimes Have Constant Mean Curvature Slices?

Many results in mathematical relativity, including results for both the initial data problem and for the evolution problem, rely on the existence of a constant mean curvature (CMC) Cauchy surface in the underlying spacetime. However, it is known that some spacetimes have no CMC Cauchy surfaces (slices). This is an obstacle for many results and constructions with these types of spacetimes, and is particularly worrisome since it is not known whether spacetimes that do have CMC slices are in any sense generic. In this expository paper, we will discuss the known results about the existence (and non-existence) of CMC slices, examine the evidence for cases which are unknown, and make several conjectures concerning the existence of CMC slices and their generality.

gr-qc↗

arXiv2016

The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest, and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines, and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitational-field equations that are nevertheless realized in Nature, and that provided the first observed signals.

gr-qc↗

arXiv2015

Finite Element Exterior Calculus for Parabolic Evolution Problems On Riemannian Hypersurfaces

Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended the FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented.

math.NA↗

arXiv2014

Convergence and Optimality of Adaptive Mixed Methods on Surfaces

In a 1988 article, Dziuk introduced a nodal finite element method for the Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear triangulation, initiating a line of research into surface finite element methods (SFEM). Demlow and Dziuk built on the original results, introducing an adaptive method for problems on 2-surfaces, and Demlow later extended the a priori theory to 3-surfaces and higher order elements. In a separate line of research, the Finite Element Exterior Calculus (FEEC) framework has been developed over the last decade by Arnold, Falk and Winther and others as a way to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can be obtained by solving finite dimensional subproblems. In 2011, Holst and Stern merged these two lines of research by developing a framework for variational crimes in abstract Hilbert complexes, allowing for application of the FEEC framework to problems that violate the subcomplex assumption of Arnold, Falk and Winther. When applied to Euclidean hypersurfaces, this new framework recovers the original a priori results and extends the theory to problems posed on surfaces of arbitrary dimensions. In yet another seemingly distinct line of research, Holst, Mihalik and Szypowski developed a convergence theory for a specific class of adaptive problems in the FEEC framework. Here, we bring these ideas together, showing convergence and optimality of an adaptive finite element method for the mixed formulation of the Hodge Laplacian on hypersurfaces.

math.NA↗

arXiv2014

Non-CMC Solutions to the Einstein Constraint Equations on Asymptotically Euclidean Manifolds with Apparent Horizon Boundaries

In this article we further develop the solution theory for the Einstein constraint equations on an n-dimensional, asymptotically Euclidean manifold M with interior boundary S. Building on recent results for both the asymptotically Euclidean and compact with boundary settings, we show existence of far-from-CMC and near-CMC solutions to the conformal formulation of the Einstein constraints when nonlinear Robin boundary conditions are imposed on S, similar to those analyzed previously by Dain (2004), by Maxwell (2004, 2005), and by Holst and Tsogtgerel (2013) as a model of black holes in various CMC settings, and by Holst, Meier, and Tsogtgerel (2013) in the setting of far-from-CMC solutions on compact manifolds with boundary. These "marginally trapped surface" Robin conditions ensure that the expansion scalars along null geodesics perpendicular to the boundary region S are non-positive, which is considered the correct mathematical model for black holes in the context of the Einstein constraint equations. Assuming a suitable form of weak cosmic censorship, the results presented in this article guarantee the existence of initial data that will evolve into a space-time containing an arbitrary number of black holes. A particularly important feature of our results are the minimal restrictions we place on the mean curvature, giving both near- and far-from-CMC results that are new.

gr-qc↗

arXiv2013

The Lichnerowicz equation on compact manifolds with boundary

In this article we initiate a systematic study of the well-posedness theory of the Einstein constraint equations on compact manifolds with boundary. This is an important problem in general relativity, and it is particularly important in numerical relativity, as it arises in models of Cauchy surfaces containing asymptotically flat ends and/or trapped surfaces. Moreover, a number of technical obstacles that appear when developing the solution theory for open, asymptotically Euclidean manifolds have analogues on compact manifolds with boundary. As a first step, here we restrict ourselves to the Lichnerowicz equation, also called the Hamiltonian constraint equation, which is the main source of nonlinearity in the constraint system. The focus is on low regularity data and on the interaction between different types of boundary conditions, which has not been carefully analyzed before. In order to develop a well-posedness theory that mirrors the existing theory for the case of closed manifolds, we first generalize the Yamabe classification to nonsmooth metrics on compact manifolds with boundary. We then extend a result on conformal invariance to manifolds with boundary, and prove a uniqueness theorem. Finally, by using the method of sub- and super-solutions (order-preserving map iteration), we establish several existence results for a large class of problems covering a broad parameter regime, which includes most of the cases relevant in practice.

gr-qc↗

arXiv2013

Convergence and Optimality of Adaptive Methods for Poisson's Equation in the FEEC Framework

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in two dimensions, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.

math.NA↗

arXiv2013

An Alternative Between Non-unique and Negative Yamabe Solutions to the Conformal Formulation of the Einstein Constraint Equations

The conformal method has been effective for parametrizing solutions to the Einstein constraint equations on closed 3-manifolds. However, it is still not well-understood; for example, existence of solutions to the conformal equations for zero or negative Yamabe metrics is still unknown without the so-called ``CMC'' or ``near-CMC'' assumptions. The first existence results without such assumptions, termed the ``far-from-CMC'' case, were obtained by Holst, Nagy, and Tsogtgerel in 2008 for positive Yamabe metrics. However, their results are based on topological arguments, and as a result solution uniqueness is not known. Indeed, Maxwell gave evidence in 2011 that far-from-CMC solutions are not unique in certain cases. In this article, we provide further insight by establishing a type of alternative theorem for general far-from-CMC solutions. For a given manifold M that admits a metric of positive scalar curvature and scalar flat metric g(0) with no conformal Killing fields, we first prove existence of an analytic, one-parameter family of metrics g(z) through g(0) such that R(g(z)) = z. Using this family of metrics and given data (tau,sigma,rho,j), we form a one-parameter family of operators F((phi,w),z) whose zeros satisfy the conformal equations. Applying Liapnuov-Schmidt reduction, we determine an analytic solution curve for F((phi,w),z) = 0 through a critical point where the linearization of F((phi,w),z) vanishes. The regularity of this curve, the definition of F((phi,w),z), and the earlier far-from-CMC results of Holst et al. allow us to then prove the following alternative theorem for far-from-CMC solutions: either (1) there exists a z_1 >0 such that (positive Yamabe) solutions to the z_1-parameterized conformal equations are non-unique; or (2) there exists z_2 < 0 such that (negative Yamabe) solutions to the z_2-parameterized conformal equations exist.

gr-qc↗

arXiv2013

Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semilinear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori pointwise bounds and sub- and super-solutions and then apply a fixed-point argument for order-preserving maps. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and by utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, "approximating" problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed nonlinear problems with irregular data.

math.AP↗

arXiv2012

Non-uniqueness of Solutions to the Conformal Formulation

It is well-known that solutions to the conformal formulation of the Einstein constraint equations are unique in the cases of constant mean curvature (CMC) and near constant mean curvature (near-CMC). However, the new far-from-constant mean curvature (far-from-CMC) existence results due to Holst, Nagy, and Tsogtgerel in 2008, to Maxwell in 2009, and to Dahl, Gicquaud and Humbert in 2010, are based on degree theory rather than on the (uniqueness-providing) contraction arguments that had been used for all non-CMC existence results prior to 2008. In fact, Maxwell demonstrated in 2011 that solutions are non-unique in the far-from-CMC case for certain types of low-regularity mean curvature. In this article, we investigate uniqueness properties of solutions to the Einstein constraint equations on closed manifolds using tools from bifurcation theory. For positive, constant scalar curvature and constant mean curvature, we first demonstrate existence of a critical energy density for the Hamiltonian constraint with unscaled matter sources. We then show that for this choice of energy density, the linearization of the elliptic system develops a one-dimensional kernel in both the CMC and non-CMC (near and far) cases. Using Liapunov-Schmidt reduction and standard tools from nonlinear analysis, we demonstrate that solutions to the conformal formulation with unscaled data are non-unique by determining an explicit solution curve, and by analyzing its behavior in the neighborhood of a particular solution.

gr-qc↗