arXiv subjects

Jeff Borggaard

Publications and source records attributed to Jeff Borggaard.

17 recordsLinked to original sources

arXiv2026

Control of the Fluidic Pinball using the Quadratic-Quadratic Regulator

The fluidic pinball presents a significant benchmark for nonlinear flow control, managing the complex interactions of three cylinder wakes. This study addresses the stabilization of the fluidic pinball to its unstable steady-state solution using a model-based nonlinear feedback strategy. We propose a framework that combines interpolatory model order reduction (IMOR) with the quadratic-quadratic regulator (QQR), a feedback control methodology that is specifically suited to the quadratic nonlinearity of the Navier-Stokes equations. A finite element model (FEM) of the problem coupled with IMOR is used to produce a reduced-order model (ROM) that accurately represents the input-output dynamics of the actuated wake. The performance of the QQR control is evaluated against the traditional linear feedback control for two different Reynolds numbers, $Re_D = 30$ and $Re_D = 50$. At $Re_D = 30$, the QQR controller is able to stabilize the wake and reaches the desired performance criteria 40.1\% faster than using a linear feedback controller. More significantly, at $Re_D = 50$, the QQR controller successfully stabilizes the wake, whereas the linear controller fails to overcome the nonlinearity of the flow. The QQR control effectively suppresses vortex shedding, resulting in the elimination of lift oscillations and a reduction in the drag coefficient. These results demonstrate that the IMOR-QQR framework provides an effective model-based control strategy that can manage nonlinear hydrodynamic instabilities in such complex wake flows.

math.OC↗

arXiv2026

Energy function approximations for differential algebraic polynomial systems of Stokes-type

Energy functions are generalizations of controllability and observability Gramians to nonlinear systems and as such find applications in both nonlinear balanced truncation and feedback control. These energy functions are solutions to the Hamilton-Jacobi-Bellman (HJB) equations partial differential equations defined over spatial dimensions determined by the number of state variables in the nonlinear system. Thus, they cannot be resolved with local basis functions for problems of even modest dimension. In this paper, we extend recent results that utilize Kronecker products to generate polynomial approximations to HJB equations. Specifically, we consider the addition of linear drift terms that exhibit a Stokes-type differential-algebraic equation (DAE) structure. This extension leverages the so-called strangeness framework for DAEs to create separate sets of algebraic and differential variables with differential equations that only involve the differential variables and algebraic equations that relate them both. At this point, the existing polynomial approximations using Kronecker products can be applied to find approximations to the energy functions. This approach is demonstrated for two polynomial feedback control problems. The standard transformation destroys the sparsity in the original system. Thus, we also present a formulation that preserves the original sparsity structure.

math.OC↗

arXiv2024

Sum of squares approximations to energy functions

Energy functions offer natural extensions of controllability and observability Gramians to nonlinear systems, enabling various applications such as computing reachable sets, optimizing actuator and sensor placement, performing balanced truncation, and designing feedback controllers. However, these extensions to nonlinear systems depend on solving Hamilton-Jacobi-Bellman (HJB) partial differential equations, which are infeasible for large-scale systems. Polynomial approximations are a viable alternative for modest-sized systems, but conventional polynomial approximations may yield negative values of the energy away from the origin. To address this issue, we explore polynomial approximations expressed as a sum of squares to ensure non-negative approximations. In this study, we focus on a reduced sum of squares polynomial where the coefficients are found through least-squares collocation -- minimizing the HJB residual at sample points within a desired neighborhood of the origin. We validate the accuracy of these approximations through a case study with a closed-form solution and assess their effectiveness for controlling a ring of van der Pol oscillators with a Laplacian-like coupling term and discretized Burgers equation with source terms.

math.OC↗

arXiv2024

Scalable Computation of Energy Functions for Nonlinear Balanced Truncation

Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. A computational challenges that has so far prevented its deployment on large-scale systems is that the energy functions required for characterization of controllability and observability are solutions of various high-dimensional Hamilton-Jacobi-(Bellman) equations, which are computationally intractable in high dimensions. This work proposes a unifying and scalable approach to this challenge by considering a Taylor-series-based approximation to solve a class of parametrized Hamilton-Jacobi-Bellman equations that are at the core of nonlinear balancing. The value of a formulation parameter provides either open-loop balancing or a variety of closed-loop balancing options. To solve for the coefficients of Taylor-series approximations to the energy functions, the presented method derives a linear tensor system and heavily utilizes it to numerically solve structured linear systems with billions of unknowns. The strength and scalability of the algorithm is demonstrated on two semi-discretized partial differential equations, namely the Burgers and the Kuramoto-Sivashinsky equations.

math.OC↗

arXiv2023

Nonlinear Balanced Truncation: Part 2 -- Model Reduction on Manifolds

Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. Two computational challenges have so far prevented its deployment on large-scale systems: (a) the energy functions required for characterization of controllability and observability are solutions of high-dimensional Hamilton-Jacobi-(Bellman) equations, which have been computationally intractable and (b) the transformations to construct the reduced-order models (ROMs) are potentially ill-conditioned and the resulting ROMs are difficult to simulate on the nonlinear balanced manifolds. Part~1 of this two-part article addressed challenge (a) via a scalable tensor-based method to solve for polynomial approximations of the open- and closed-loop energy functions. This article, (Part~2), addresses challenge (b) by presenting a novel and scalable method to reduce the dimensionality of the full-order model via model reduction on polynomially-nonlinear balanced manifolds. The associated nonlinear state transformation simultaneously 'diagonalizes' relevant energy functions in the new coordinates. Since this nonlinear balancing transformation can be ill-conditioned and expensive to evaluate, inspired by the linear case we develop a computationally efficient balance-and-reduce strategy, resulting in a scalable and better conditioned truncated transformation to produce balanced nonlinear ROMs. The algorithm is demonstrated on a semi-discretized partial differential equation, namely Burgers equation, which illustrates that higher-degree transformations can improve the accuracy of ROM outputs.

math.OC↗

arXiv2022

A Statistical Framework for Domain Shape Estimation in Stokes Flows

We develop and implement a Bayesian approach for the estimation of the shape of a two dimensional annular domain enclosing a Stokes flow from sparse and noisy observations of the enclosed fluid. Our setup includes the case of direct observations of the flow field as well as the measurement of concentrations of a solute passively advected by and diffusing within the flow. Adopting a statistical approach provides estimates of uncertainty in the shape due both to the non-invertibility of the forward map and to error in the measurements. When the shape represents a design problem of attempting to match desired target outcomes, this "uncertainty" can be interpreted as identifying remaining degrees of freedom available to the designer. We demonstrate the viability of our framework on three concrete test problems. These problems illustrate the promise of our framework for applications while providing a collection of test cases for recently developed Markov Chain Monte Carlo (MCMC) algorithms designed to resolve infinite dimensional statistical quantities.

math.OC↗

arXiv2022

Performance Assessment of Energy-preserving, Adaptive Time-step Variational Integrators

A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive time-step variational integrators that conserve the energy in addition to being symplectic and momentum-preserving. Their utility, however, is still an open question due to the numerical difficulties associated with solving the discrete governing equations. In this work, we investigate the numerical performance of energy-preserving, adaptive time-step variational integrators. First, we compare the time adaptation and energy performance of the energy-preserving adaptive algorithm with the adaptive variational integrator for Kepler's two-body problem. Second, we apply tools from Lagrangian backward error analysis to investigate numerical stability of the energy-preserving adaptive algorithm. Finally, we consider a simple mechanical system example to illustrate the backward stability of this energy-preserving, adaptive time-step variational integrator.

math.NA↗

arXiv2020

Quantum Distance to Uncontrollability and Quantum Speed Limits

Distance to Uncontrollability is a crucial concept in classical control theory. Here, we introduce Quantum Distance to Uncontrollability as a measure how close a universal quantum system is to a non-universal one. This allows us to provide a quantitative version of the Quantum Speed Limit, decomposing the bound into a geometric and dynamical component. We consider several physical examples including globally controlled solid state qubits and a cross-Kerr system, showing that the Quantum Distance to Uncontrollability provides a precise meaning to spectral crowding, weak interactions and other bottlenecks to universality. We suggest that this measure should be taken into consideration in the design of quantum technology.

quant-ph↗

arXiv2020

On Approximating Polynomial-Quadratic Regulator Problems

Feedback control problems involving autonomous polynomial systems are prevalent, yet there are limited algorithms and software for approximating their solution. This paper represents a step forward by considering the special case of the regulator problem where the state equation has polynomial nonlinearity, control costs are quadratic, and the feedback control is approximated by low-degree polynomials. As this represents the natural extension of the linear-quadratic regulator (LQR) and quadratic-quadratic regulator (QQR) problems, we denote this class as polynomial-quadratic regulator (PQR) problems. The present approach is amenable to feedback approximations with low degree polynomials and to problems of modest model dimension. This setting can be achieved in many problems using modern model reduction methods. The Al'Brekht algorithm, when applied to polynomial nonlinearities represented as Kronecker products leads to an elegant formulation. The terms of the feedback control lead to large linear systems that can be effectively solved with an N-way generalization of the Bartels-Stewart algorithm. We demonstrate our algorithm with numerical examples that include the Lorenz equations, a ring of van der Pol oscillators, and a discretized version of the Burgers equation. The software described here is available on Github.

math.OC↗

arXiv2019

The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems

Feedback control problems involving autonomous quadratic systems are prevalent, yet there are only a limited number of software tools available for approximating their solution due to the complexity of the problem. This paper represents a step forward in the special case where both the state equation and the control costs are quadratic. As it represents the natural extension of the linear-quadratic regulator (LQR) problem, we describe this setting as the quadratic-quadratic regulator (QQR) problem. This is significantly more challenging and holds the LQR as special case that must be solved along the way. We describe an algorithm that exploits the structure of the QQR problem that arises when implementing Al'Brekht's method. This approach is amenable to feedback laws with low degree polynomials but have a relatively modest model dimension that could be achieved by modern model reduction methods. This problem has an elegant formulation and a solution that introduces several linear systems where the structure suggests modern tensor-based linear solvers. We demonstrate this algorithm on a suite of random test problems then apply it to a distributed parameter control problem that fits the QQR framework. Comparisons to linear feedback control laws show a modest benefit using the QQR formulation.

math.OC↗

arXiv2019

On Bayesian Consistency for Flows Observed Through a Passive Scalar

We consider the statistical inverse problem of estimating a background fluid flow field $\mathbf{v}$ from the partial, noisy observations of the concentration $θ$ of a substance passively advected by the fluid, so that $θ$ is governed by the partial differential equation \[ \frac{\partial}{\partial t}θ(t,\mathbf{x}) = -\mathbf{v}(\mathbf{x}) \cdot \nabla θ(t,\mathbf{x}) + κΔθ(t,\mathbf{x}) \quad \text{ , } \quad θ(0,\mathbf{x}) = θ_0(\mathbf{x}) \] for $t \in [0,T], T>0$ and $\mathbf{x} \in \mathbb{T}=[0,1]^2$. The initial condition $θ_0$ and diffusion coefficient $κ$ are assumed to be known and the data consist of point observations of the scalar field $θ$ corrupted by additive, i.i.d. Gaussian noise. We adopt a Bayesian approach to this estimation problem and establish that the inference is consistent, i.e., that the posterior measure identifies the true background flow as the number of scalar observations grows large. Since the inverse map is ill-defined for some classes of problems even for perfect, infinite measurements of $θ$, multiple experiments (initial conditions) are required to resolve the true fluid flow. Under this assumption, suitable conditions on the observation points, and given support and tail conditions on the prior measure, we show that the posterior measure converges to a Dirac measure centered on the true flow as the number of observations goes to infinity.

math.ST↗

arXiv2019

A Bayesian Approach to Estimating Background Flows from a Passive Scalar

We consider the statistical inverse problem of estimating a background flow field (e.g., of air or water) from the partial and noisy observation of a passive scalar (e.g., the concentration of a solute), a common experimental approach to visualizing complex fluid flows. Here the unknown is a vector field that is specified by a large or infinite number of degrees of freedom. Since the inverse problem is ill-posed, i.e., there may be many or no background flows that match a given set of observations, we adopt a Bayesian approach to regularize it. In doing so, we leverage frameworks developed in recent years for infinite-dimensional Bayesian inference. The contributions in this work are threefold. First, we lay out a functional analytic and Bayesian framework for approaching this problem. Second, we define an adjoint method for efficient computation of the gradient of the log likelihood, a key ingredient in many numerical methods. Finally, we identify interesting example problems that exhibit posterior measures with simple and complex structure. We use these examples to conduct a large-scale benchmark of Markov Chain Monte Carlo methods developed in recent years for infinite-dimensional settings. Our results indicate that these methods are capable of resolving complex multimodal posteriors in high dimensions.

math.NA↗

arXiv2018

GPU-Accelerated Particle Methods for Evaluation of Sparse Observations for PDE-Constrained Inverse Problems

We consider the inverse problem of estimating parameters of a driven diffusion (e.g., the underlying fluid flow, diffusion coefficient, or source terms) from point measurements of a passive scalar (e.g., the concentration of a pollutant). We present two particle methods that leverage the structure of the inverse problem to enable efficient computation of the forward map, one for time evolution problems and one for a Dirichlet boundary-value problem. The methods scale in a natural fashion to modern computational architectures, enabling substantial speedup for applications involving sparse observations and high-dimensional unknowns. Numerical examples of applications to Bayesian inference and numerical optimization are provided.

math.NA↗

arXiv2017

Interpolatory Model Reduction of Parameterized Bilinear Dynamical Systems

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this is not the case for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function is matched tangentially by those of the corresponding reduced order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including extra vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.

math.NA↗

arXiv2016

Robust Reduced-Order Model Stabilization for Partial Differential Equations Based on Lyapunov Theory and Extremum Seeking with Application to the 3D Boussinesq Equations

We present some results on stabilization for reduced-order models (ROMs) of partial differential equations. The stabilization is achieved using Lyapunov theory to design a new closure model that is robust to parametric uncertainties. The free parameters in the proposed ROM stabilization method are optimized using a model-free multi-parametric extremum seeking (MES) algorithm. The 3D Boussinesq equations provide a challenging numerical test-problem that is used to demonstrate the advantages of the proposed method.

eess.SY↗

arXiv2014

Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations

This paper addresses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called finite noise problem in the space of functions with bounded mixed derivatives. We prove convergence of finite noise minimizers to the appropriate infinite dimensional ones, and devise a stochastic augmented Lagrangian method for locating these numerically. Lastly, we illustrate our method with three numerical examples.

math.OC↗

arXiv2011

Proper Orthogonal Decomposition Closure Models For Turbulent Flows: A Numerical Comparison

This paper puts forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a 3D turbulent flow around a circular cylinder at Re = 1,000. Two criteria are used in judging the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models perform best.

physics.comp-ph↗