Carlos N. Rautenberg

Associate Professor  ·  Department of Mathematical Sciences  ·  George Mason University

math.gmu.edu ↗

Welcome. I am an Associate Professor at George Mason University in the Department of Mathematical Sciences.

My research spans several areas of analysis and applied mathematics: optimal control, variational and quasi-variational inequalities, optimal filtering, parameter estimation, sensor placement, Riccati equations, and mathematical imaging.

I am a former ECMath Junior Research Group Leader at the Department of Mathematics of Humboldt-Universität zu Berlin, and former deputy head of Research Group 8 at the Weierstrass Institute for Applied Analysis and Stochastics, Berlin.

Quasi-variational inequalities Sensor placement Density in function spaces Image processing Optimal control Thermal fluids
Carlos N. Rautenberg

Short CV

Education & Research Positions

2022–present Associate Professor Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA.
2019–2022 Assistant Professor Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA.
2016–2019 Researcher Weierstrass Institute, Berlin, Germany.
2014–2019 Junior Research Group Leader ECMath-Nachwuchsgruppenleiter, Department of Mathematics, Humboldt-Universität zu Berlin, Germany.
2010–2014 Postdoctoral Researcher Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Austria.
Supervised by Michael Hintermüller.
2006–2010 Ph.D., Mathematics Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg VA, USA.
Supervised by John A. Burns.

For the full CV, please write to crautenb@gmu.edu.

Quasi-variational Inequalities & Optimal Control

Research on this topic is coming soon. In the meantime, see the publications page for relevant papers, or visit the projects page for funded research in this area (NSF DMS-2012391, DFG SPP 1962).

Optimal & Robust Sensor Placement

The optimal placement of sensors (such as thermostats) is a complex multiscale problem where multiple nonlinear processes are coupled. Let \(\Omega\) be a room or building with a ventilation system, and let \(u:\Omega\to\mathbb{R}\) describe the temperature field advected by an air velocity profile \(\vec{v}:\Omega\to\mathbb{R}^n\).

The convection-diffusion process is perturbed by a stochastic (Wiener) process \(\eta\). A sensor at location \(x\) measures \(u\) by a weighted average within its effective range, yielding output \(h_x\) perturbed by noise \(\nu\): \[u' = \alpha\Delta u + v\cdot\nabla u + \eta, \qquad h_x = C(x)u + \nu,\] where \(C(x)w = \int_\Omega K(y-x)\,w(y)\,dy\) and \(K\) encodes the sensor's properties.

A natural optimality criterion minimizes the expected state estimation error \(\mathbb{E}\|u - \tilde{u}_x\|_{L^2}^2\), where \(\tilde{u}_x\) is the output of the Kalman–Bucy filter. The objective \[x \mapsto J(x) := \int_0^1 \mathbb{E}\|u - \tilde{u}_x\|_{L^2}^2\,dt\] exhibits convex-like behaviour for simple velocity fields, but becomes non-convex when \(\vec{v}\) is the solution to the Navier–Stokes equations on \(\Omega\).

Relevant publications

Density of Smooth Functions in Convex Sets

In function spaces \(X\) (e.g. \(L^p(\Omega)\), \(W^{1,p}(\Omega)\)) we often encounter constrained sets of the form \[\mathbf{K}(X) = \{v\in X : |v(x)| \leq \alpha(x) \text{ a.e. in } \Omega\}\] for some given \(\alpha:\Omega\to\mathbb{R}\). For smooth enough \(\partial\Omega\), one has the standard density results \[\overline{C^\infty(\Omega)}^{L^p} = L^p,\qquad \overline{C_c^\infty(\Omega)}^{W_0^{1,p}} = W_0^{1,p},\qquad \overline{C^\infty(\bar\Omega)}^{W^{1,p}} = W^{1,p}.\] The constrained analogue \(\overline{\mathbf{K}(W^{1,p})\cap C^\infty(\bar\Omega)}^{W^{1,p}} = \mathbf{K}(W^{1,p})\) requires additional regularity on \(\alpha\). Irregular enough \(\alpha\) can make this fail (Hintermüller–Rautenberg–Rösel 2017), and in general a dense embedding \(Y\hookrightarrow X\) alone does not guarantee \(\overline{\mathbf{K}\cap Y}^X = \mathbf{K}\).

Connection to regularization

Sets \(\mathbf{K}\) arise as constraints in optimization problems. Regularization replaces \(\min_{u\in\mathbf{K}} J(u)\) by \(\min_{u\in Y} J(u) + \gamma R(u)\), and convergence of minimizers as \(\gamma\to 0\) hinges precisely on the density condition \(\overline{\mathbf{K}\cap Y}^X = \mathbf{K}\).

Relevant publications

Selected Publications by Year

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Selected Talks by Year

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Teaching

George Mason University

Advanced Linear Algebra MATH 322
Elementary Differential Equations MATH 214
Nonlinear Functional Analysis MATH 784
Advanced Calculus I MATH 315
Partial Differential Equations MATH 678
Modern Applied Mathematics II MATH 414
Ordinary Differential Equations MATH 677
Analytic Geometry and Calculus III MATH 213
Linear Analysis MATH 675
Operator Theory and Applications (created course) MATH 689
Introduction to Partial Differential Equations with Numerical Methods MATH 478
Advanced Nonlinear Partial Differential Equations MATH 697
Readings in Obstacle Problems MATH 697
Polar Bear Population Dynamics MATH 697
Nonlinear Partial Differential Equations MATH 697
Convex Analysis MATH 697
Numerics for Measure Spaces MATH 697
Mathematical Models of Soap Bubbles MATH 697
Elliptic Variational Inequalities MATH 697
Minimization on Measure Spaces MATH 697
Optimization with Partial Differential Equations MATH 697
Variational Analysis MATH 697
Partial Differential Equations with Measure Data MATH 697

Humboldt-Universität zu Berlin

Optimal Control with Partial Differential Equations BMS Advanced Course

Assignments available here.

Research Projects

Current

NSF DMS-2012391  ·  PI

Optimization Problems with Quasi-Equilibrium Constraints: Control, Identification, and Design

A wide range of problems in applied sciences involve constraints on variables of interest arising from complex physical phenomena, hierarchy, or competition. This project develops theory and numerical methods for implicitly constrained problems — such as an elastic membrane on a deformable obstacle — with applications to composite materials design, OLED fabrication, and structural crack detection.

Past

Math+ Project AA4-3  ·  Co-PI

Equilibria for Energy Markets with Transport

Motivated by efficient energy distribution, this project develops theory and algorithms for a new class of generalized Nash equilibrium problems (GNEPs) in game-theoretic formulations of energy markets, with model predictive control strategies and stochastic perturbations.

ECMath Project C-SE15/19  ·  PI

Optimal Network Sensor Placement for Energy Efficiency

Optimal sensor placement for estimating temperature distributions in buildings: a highly nonlinear, multiscale problem with stochastic perturbations, aimed at reducing the energy consumption of feedback control systems.

DFG SPP 1962 – P11  ·  Co-PI

Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities

Analysis and numerical methods for optimal control problems where quasi-variational inequalities form non-smooth, nonlinear constraints arising from state-dependent constraint sets.

DFG SPP 1962 – P10  ·  Co-PI

Optimal Uncertainty Quantification and Risk-Averse Optimization with QVI Constraints

Establishing a rigorous mathematical framework for uncertainty quantification and risk-averse optimization suited to quasi-variational inequalities, where the constraint set is not known a priori.

Math+ Transition Project  ·  Co-PI

Optimization and Control of Electrowetting on Dielectric for Digital Microfluidics

Deriving feedback laws for nonsmooth optimal control problems with moving boundaries in a sharp interface setting, enabling automation with dynamic integration of measurements.

Contact

Office
Department of Mathematical Sciences
George Mason University
4400 University Drive, MS: 3F2
Exploratory Hall, Room 4203
Fairfax, Virginia 22030, USA
Email
crautenb@gmu.edu
Phone
+1 (703) 993-1476