Christopher A. Beattie

• PhD (Johns Hopkins University)
• Professor (Full) at Virginia Tech

New observations of ocean surface topography obtained by wide-swath satellite interferometry require new capabilities to process spatially correlated errors in order to assimilate these data into numerical models. The sea surface height (SSH) variations have to be weighted against other types of assimilated data using information on their precision...

We present a novel data-driven reformulation of the iterative SVD-rational Krylov algorithm (ISRK), in its original formulation a Petrov-Galerkin (two-sided) projection-based iterative method for model reduction combining rational Krylov subspaces (on one side) with Gramian/SVD based subspaces (on the other side). We show that in each step of ISRK,...

The recently deployed Surface Water and Ocean Topography (SWOT) mission for the first time has observed the ocean surface at a spatial resolution of 1 km, thus giving an opportunity to directly monitor submesoscale sea surface height (SSH) variations that have a typical magnitude of a few centimeters. This progress comes at the expense of the neces...

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed point iteration developed to minimize the ${\mathcal {H}}_{2}$ model order reduction error. In this work, IRKA is recast as a Riemannian gradient descent method with a fixed step size over the manifold of rational functions having fixed degree. This interpretation motivates t...

The upcoming technology of wide-swath altimetry from space will enable monitoring the ocean surface at 4–5 times better spatial resolution and 2–3 times better accuracy than traditional nadir altimeters. This development will provide a chance to directly observe submesoscale sea surface height (SSH) variations that have a typical magnitude of a few...

The surge of interest in machine learning has led to increased emphasis on the value of accurate, efficient surrogate models. The book is divided into four parts. Part 1, consisting of Chapters 1 and 2, introduces the reader to the basic ideas underlying general model reduction and provides the relevant system-theoretic background needed for the re...

We examine interpolatory model reduction methods that are particularly well-suited for treating large-scale port-Hamiltonian differential-algebraic systems. We are able to take advantage of underlying structural features of the system in a way that preserves them in the reduced model, using approaches that incorporate regularization and a prudent s...

We formulate here an approach to model reduction that is well-suited for linear time-invariant control systems that are stabilizable and detectable but may otherwise be unstable. We introduce a modified H2-error metric, the H2-gap, that provides an effective measure of model fidelity in this setting. While the direct evaluation of the H2-gap requir...

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribut...

The question when a general linear time invariant control system is equivalent to a port-Hamiltonian system is answered. Several equivalent characterizations are derived which extend the characterizations of Willems to the general non-minimal case and to the case where the feedthrough term does not have an invertible symmetric part. An explicit con...

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribut...

The ongoing transition to coupled data assimilation (DA) systems encounters substantial technical difficulties associated with the need to merge together different elements of atmospheric and ocean DA systems that typically have had independent development paths for decades. In this study, we consider the incorporation of strong coupling in the obs...

We present a novel reformulation of balanced truncation, a classical model reduction method. The principal innovation that we introduce comes through the use of system response data that has been either measured or computed, without reference to any prescribed realization of the original model. Data are represented by sampled values of the transfer...

We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the pri...

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order \({\mathscr{H}}_{2}\)-approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in...

We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that differ from one another by a low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to isolate representative models on which to focus model reductio...

The iterative rational Krylov algorithm (\textsf{IRKA}) is a popular approach for producing locally optimal reduced-order $\mathcal{H}_2$-approximations to linear time-invariant (LTI) dynamical systems. Overall, \textsf{IRKA} has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully...

We examine interpolatory model reduction methods that are well-suited for treating large scale port-Hamiltonian differential-algebraic systems in a way that is able to preserve and indeed, take advantage of the underlying structural features of the system. We introduce approaches that incorporate regularization together with prudent selection of in...

We formulate here an approach to model reduction that is well-suited for linear time-invariant control systems that are stabilizable and detectable but may otherwise be unstable. We introduce a modified $\mathcal{H}_2$-error metric, the $\mathcal{H}_2$-gap, that provides an effective measure of model fidelity in this setting. While the direct evalu...

We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. The objective function is non-convex and the associated optimi...

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also...

Given a set of response observations for a parametrized dynamical system, we seek a parametrized dynamical model that will yield uniformly small response error over a range of parameter values yet has low order. Frequently, access to internal system dynamics or equivalently, to realizations of the original system is either not possible or not pract...

Formulations of open physical systems within the framework of Non-Equilibrium Reversible/Irreversible Coupling (associated with the acronym "GENERIC") is related in this work with state-space realizations that are given as boundary port-Hamiltonian systems. This reformulation is carried out explicitly by splitting the dynamics of the system into a...

We discuss the problem of robust representations of stable and passive transfer functions in particular coordinate systems, and focus in particular on the so-called port-Hamiltonian representations. Such representations are typically far from unique and the degrees of freedom are related to the solution set of the so-called Kalman-Yakubovich-Popov...

We discuss the problem of robust representations of stable and passive transfer functions in particular coordinate systems, and focus in particular on the so-called port-Hamiltonian representations. Such representations are typically far from unique and the degrees of freedom are related to the solution set of the so-called Kalman-Yakubovich-Popov...

The ongoing trend towards parallelization in computer technologies propels ensemble methods toward the forefront of data assimilation studies in geophysics. Of particular interest are ensemble techniques which do not require the development of tangent linear numerical models and their adjoints for optimization . These “adjoint-free” methods detect...

We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form C˜(∑k=1Khk(s)A˜k)−1B˜ where h1,h2,...,hK are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements o...

Development and maintenance of the linearized and adjoint code for advanced circulation models is a challenging issue, requiring a significant proportion of total effort in operational data assimilation (DA). The ensemble-based DA techniques provide a derivative-free alternative, which appears to be competitive with variational methods in many prac...

We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. The objective function is non-convex and the associated optimi...

Linear time-periodic (LTP) dynamical systems frequently appear in the modeling of phenomena related to fluid dynamics, electronic circuits, and structural mechanics via linearization centered around known periodic orbits of nonlinear models. Such LTP systems can reach orders that make repeated simulation or other necessary analysis prohibitive, mot...

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also...

We consider the model reduction problem for linear time-invariant dynamical systems having nonzero (but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a...

Very large-scale dynamical systems, even linear time-invariant systems, can present significant computational difficulties when used in numerical simulation. Model reduction is one response to this challenge but standard methods often are restricted to systems that are presented as standard first-order realizations; in the frequency domain such sys...

In this note, we first attempt to generalize the H2 optimal interpolation conditions for more general reduced order models. To this aim, we first expose the necessary optimality conditions in the case where the reduced system is of dimension one and have a single state delay structure. This can be viewed as a first step toward the H2 optimal model...

We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form $C(\sum_{k=1}^K h_k(s)A_k)^{-1}B$ where $h_1,h_2,\ldots,h_K$ are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely...

We consider the model reduction problem for linear time-invariant dynamical systems having nonzero (but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a...

We develop here a computationally effective approach for producing high-quality $\mathcal{H}_\infty$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg, Beattie, and Gugercin for the single-input/single-output (...

We develop here a computationally effective approach for producing high-quality $\mathcal{H}_\infty$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg, Beattie, and Gugercin for the single-input/single-output (...

This paper presents a structure-preserving model reduction approach applicable to large-scale, nonlinear port-Hamiltonian systems. Structure preservation in the reduction step ensures the retention of port-Hamiltonian structure which, in turn, ensures the stability and passivity of the reduced model. Our analysis provides a priori error bounds for...

Iterative Rational Krylov Algorithm (irka) of Gugercin et al. [2008] is an effective tool for optimal H2 rational approximation. Beattie and Gugercin [2012] has recently developed a new formulation of irka that only uses transfer function evaluations, without requiring any particular realization; thus extending irka to H2 approximation of irrationa...

Performance of the adjoint and adjoint-free 4-dimensional variational (4dVar) data assimilation techniques is compared in application to the hydrographic surveys and velocity observations collected in the Adriatic Sea in 2006. Assimilating the data into the Navy Coastal Ocean Model (NCOM) has shown that both methods deliver similar reduction of the...

Vector Fitting (VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and established a connection with optimal $H_2$ approximation. We build on this work and extend the previous framew...

Vector fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the pe...

Linear dynamical systems with an affine parameter dependence producing low-rank variation in the state matrix can be recast as a nonparameterized system operating in parallel with a parameterized feed forward term operating under output constraints. This mapping permits the application of standard (nonparametric) model reduction strategies to solve...

The last two decades have seen major developments in interpolatory methods for model reduction of large-scale linear dynamical systems. Advances of note include the ability to produce (locally) optimal reduced models at modest cost; refined methods for deriving interpolatory reduced models directly from input/output measurements; and extensions for...

Vector Fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the pe...

Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of large-scale discretized, parametrized, partial differential equations...

This paper extends an interpolatory framework for weighted-H2 model reduction to MIMO dynamical systems. A new representation of the weighted-H2 inner product in MIMO settings is presented together with associated first-order necessary conditions for an optimal weighted-H2 reduced-order model. Equivalence of these conditions with necessary conditio...

We introduce an approach to H∞H∞ model reduction that is founded on ideas originating in realization theory, interpolatory H2H2-optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves...

The Iterative Rational Krylov Algorithm (IRKA) of [9] is an effective tool for approaching the H2-optimal model reduction problem. However, it has relied on the availability of a standard first-order state-space realization of the model-to-be-reduced. In this paper, we employ a Loewner-matrix approach for interpolation, and develop a new formulatio...

This paper introduces an interpolation framework for the weighted-H2 model reduction problem. We obtain a new representation of the weighted-H2 norm of SISO systems that provides new interpolatory first order necessary conditions for an optimal reduced-order model. The H2 norm representation also provides an error expression that motivates a new we...

In this article, we propose a numerical algorithm for efficient and robust solution of a sequence of shifted Hessenberg linear systems. In particular, we show how the frequency response &calG;(σ) = d-C(A-σ I)-1b in the single input case can be computed more efficiently than with other state-of-the-art methods. We also provide a backward stability a...

Port-Hamiltonian systems result from port-based network modeling of physical systems and constitute an important class of passive nonlinear state-space systems. In this paper, we develop a framework for model reduction of large-scale multi-input/multi-output nonlinear port-Hamiltonian systems that retains the port-Hamiltonian structure in the reduc...

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order...

Weighted model reduction problems appear in many important applications such as controller reduction. The most common approach to this problem is the weighted balanced truncation method. Interpolatory approaches to weighted model reduction have been mostly limited to simply choosing interpolation points in the regions where the weights are dominant...

The Iterative Rational Krylov Algorithm (IRKA) of [8] is an interpolatory model reduction approach to the optimal $\mathcal{H}_2$ approximation problem. Even though the method has been illustrated to show rapid convergence in various examples, a proof of convergence has not been provided yet. In this note, we show that in the case of state-space sy...

Port-Hamiltonian systems result from port-based network modeling of physical systems and are an important example of passive state-space systems. In this paper, we develop the framework for model reduction of large-scale multi-input/multi-output port-Hamiltonian systems via tangential rational interpolation. The resulting reduced-order model not on...

We introduce an interpolatory approach to ℋ∞ model reduction for large-scale dynamical systems. Guided by the optimality conditions of [26] for best uniform rational approximants on the unit disk, our proposed method uses the freedom in choosing the d-term in the reduced order model to enforce 2r + 1 interpolation conditions in the right-half plane...

We develop and describe an iteratively corrected rational Krylov algorithm for the solution of the optimal H2 model reduction problem. The formulation is based on finding a reduced order model that satisfies interpolation based first-order necessary conditions for H2 optimality and results in a method that is numerically effective and suited for la...

We investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem. We give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods. We show that when a Petrov-Galerkin framework is e...

Large scale dynamical systems are a common framework for the modeling and control of many complex phenomena of scientific interest and industrial value, with examples of diverse origin that include signal propagation and interference in electric circuits, storm surge prediction before an advancing hurricane, vibration suppression in large structure...

We present a trust-region approach for optimal H₂ model reduction of multiple-input/multiple-output (MIMO) linear dynamical systems. The proposed approach generates a sequence of reduced order models producing monotone improving H₂ error norms and is globally convergent to a reduced order model guaranteed to satisfy first-orde...

Port network modeling of physical systems leads directly to an important class of passive state space systems: port-Hamiltonian systems. We consider here methods for model reduction of large scale port-Hamiltonian systems that preserve port-Hamiltonian structure and are capable of yielding reduced order models that satisfy first-order optimality co...

We present a framework for interpolatory model reduction that treats systems having a generalized coprime factorization C(s)(s)−1B(s)+D. This includes rational Krylov-based interpolation methods as a special case. The broader framework allows retention of special structure in reduced models such as symmetry, second- and higher order structure, stat...

We develop a general framework for interpolation-based model reduction that includes rational Krylov-based methods as a special case. This new broader framework allows retention of special structure in the reduced order models such as symmetry, second order structure, internal delays, and infinite dimensional subsystems.

We describe the development of a reliable parallel algorithm and software tools that utilize flow-adapted KLE/POD representations and that are able to take advantage of distributed data formats on cluster/grid computer architectures. The associated module functions efficiently within the context of current best practices of fluid flow simulation. A...

We present an approach to model reduction for linear dynamical systems that is numerically stable, computationally tractable even for very large order systems, produces a sequence of monotone decreasing H ₂ error norms, and (under modest hypotheses) is globally convergent to a reduced order model that is guaranteed to satisfy first-order...

The optimal H 2 model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches....

The proper orthogonal decomposition (POD) is a popular approach for building reduced-order models for nonlinear distributed parameter systems. The approach is based on developing a reduced basis by post-processing one, and often multiple, high fidelity simulations of a nonlinear partial differential equation. The computational overhead required to...

We investigate the use of inexact solves in a Krylov-based model reduction setting and present the resulting perturbation effects on the underlying model reduction problem. We show that for a good selection of interpolation points, Krylov-based model reduction is robust with respect to the perturbations due to inexact solves. On the other hand, whe...

In this note, we examine Krylov-based model reduction of second order systems where proportional damping is used to model energy dissipation. We give a detailed analysis of the distribution of system poles, and then, through a connection with potential theory, we are able to exploit the structure of these poles to obtain an optimal single shift str...

In this paper, we introduce a computationally efficient controller reduction approach using a rational Krylov method. We show that a reduced-order controller obtained via a Krylov projection is guaranteed to match the desired full-order closed loop system response at shifts used in the Krylov reduction of the controller. Two different shift selecti...

The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degre...

CK DK be an n th κ order controller with transfer function K(s )= CK(sI − AK)−1BK + DK .W e seek a reduced-order controller Kr(s) of order r with r � nκ to replace K(s). Assume that both G(s )a ndK(s )a re single-input single-output (SIS0), and that K(s) is a stable stabilizing controller. (The general case follows simply and will be presented in t...

The convergence of Krylov subspace eigenvalue algorithms can be robustly measured by the angle the approximating Krylov space makes with a desired invariant subspace. This paper describes a new bound on this angle that handles the complexities introduced by non-Hermitian matrices, yet has a simpler derivation than similar previous bounds. The new b...

We review Lehmann’s inclusion bounds and provide extensions to general (non-normal) matrices. Each inclusion region has a diameter related to the singular values of a restriction of the matrix to a subspace and dependent on either an eigenvector condition number or the departure of the matrix from normality. The inclusion regions are optimal for no...

Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in p...

The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the eects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree...

The chief tool for design of viscoelastic-based damping treatments over the past 20 years has been the modal strain energy (MSE) approach. This approach to damping design traditionally has involved a practitioner to vary placement and stiffness of add-on elements using experience and trial and error so as to maximize the add-on element's share of s...

A simple extension of a method by Calogero and Marchioro for constructing lower bound problems for ground states of systems of indistinguishable particles is applied to atomic systems. Their method is extended to yield an improved lower bound problem, which raises the ground-state estimate and yields nontrivial lower bounds to excited states that w...

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace -- and this occurs more rapidly than the underlying rate of convergence...

Eigenvalue estimates that are optimal in some sense have self-evident appeal and leave estimators with a sense of virtue and economy. So, it is natural that ongoing searches for effective strategies for difficult tasks such as estimating matrix eigenvalues that are situated well into the interior of the spectrum revisit from time to time methods th...

The paper presents a control scheme based on the sliding-mode-control approach. The analytical formulation focuses on the development of (1) a convenient, systematic and general scheme to achieve the so-called regular form of the equations of motion required to uncouple the control actions from the sliding motion description, (2) a systematic treat...

This article reviews a variety of results related to optimal bounds for matrix eigenvalues — some re- sults presented here are well-known; others are less known; and a few are new. The focus rests especially on Ritz and harmonic Ritz values, and right- and left-definite variants of Lehmann’s optimal bounds. Two new computationally advantageous refo...

New approaches for computing tight lower bounds to the eigenvalues of a class of semibounded self-adjoint operators are presented that require comparatively little a priori spectral information and permit the effective use of (among others) finite-element trial functions. A variant of the method of intermediate problems making use of operator decom...

The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. It is not obvious from their original form how they can be differentiated and how an algorithm for solving nonlinear equations can be applied to them. A contragredient transformat...

Problems of model correlation and system identification are central in the design, analysis, and control of large space structures. Of the numerous methods that have been proposed, many are based on finding minimal adjustments to a model matrix sufficient to introduce some desirable quality into that matrix. In this work, several of these methods a...

Improved convergence rate estimates are derived for a variant of Aronszajn-type intermediate problems that is both computationally feasible and convergent for problems with nontrivial essential spectra. In a previous paper the authors obtained rate of convergence estimates for this method in terms of containment gaps between subspaces. In the prese...

This work examines techniques under the general approach of optimal-update identification which produce optimally adjusted, or updated, property matrices (i.e., mass, stiffness and damping matrices) to more closely match the structure modal response. For practical applications, the techniques must perform when the modal response is inconsistent wit...