arXiv subjects

Christopher Beattie

Publications and source records attributed to Christopher Beattie.

At least 19 recordsLinked to original sources

arXiv2026

Rational approximation and intrinsic Gaussian processes

Gaussian processes (GPs) defined through intrinsic random fields provide a flexible framework for modeling spatial phenomena, and have been advocated in a variety of applications over the past several decades. Nevertheless, their adoption has lagged behind traditional, covariance-based approaches, in part because the intrinsic formulation has lacked an accompanying toolkit of computational methods and dependence specifications that facilitate fitting and prediction. We develop here a systematic framework for modeling intrinsic GPs and introduce practical algorithms and dependence/variogram models for modeling, inference and computation that parallel those of traditional, stationary GPs. We explore a close connection between intrinsic GP models and rational approximation, which clarifies the underlying problem structure. Numerical examples illustrate how the new tools can be deployed in practice, highlighting the advantages of intrinsic-field modeling in terms of robustness, interpretability, and computational efficiency.

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arXiv2025

Partial Floquet Transformation and Model Order Reduction of Linear Time-Periodic Systems

Time-periodic dynamical systems occur commonly both in nature and as engineered systems. Large-scale linear time-periodic dynamical systems, for example, may arise through linearization of a nonlinear system about a given periodic solution (possibly as a consequence of a baseline periodic forcing) with subsequent spatial discretization. The potential need to simulate responses to a wide variety of input profiles (viewed as perturbations off a baseline periodic forcing) creates a potent incentive for effective model reduction strategies applicable to linear time-periodic (LTP) systems. Classical approaches that take into account the underlying time-periodic system structure often utilize the Floquet transform; however, computation of the Floquet transform is typically intractable for large order systems. In this paper, we develop the notion of a partial Floquet transformation connected to selected invariant subspaces of a time-varying differential operator associated with the LTP system. We modify and repurpose the Dominant Pole Algorithm of Rommes to identify effective invariant subspaces useful for model reduction. We discuss the construction of associated partial Floquet transformations and time-varying reduction bases with which to produce effective reduced-order LTP models and illustrate the process on a simple time-periodic system.

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arXiv2024

The Fundamental Subspaces of Ensemble Kalman Inversion

Ensemble Kalman Inversion (EKI) methods are a family of iterative methods for solving weighted least-squares problems, especially those arising in scientific and engineering inverse problems in which unknown parameters or states are estimated from observed data by minimizing the weighted square norm of the data misfit. Implementation of EKI requires only evaluation of the forward model mapping the unknown to the data, and does not require derivatives or adjoints of the forward model. The methods therefore offer an attractive alternative to gradient-based optimization approaches in inverse problem settings where evaluating derivatives or adjoints of the forward model is computationally intractable. This work presents a new analysis of the behavior of both deterministic and stochastic versions of basic EKI for linear observation operators, resulting in a natural interpretation of EKI's convergence properties in terms of ``fundamental subspaces'' analogous to Strang's fundamental subspaces of linear algebra. Our analysis directly examines the discrete EKI iterations instead of their continuous-time limits considered in previous analyses, and provides spectral decompositions that define six fundamental subspaces of EKI spanning both observation and state spaces. This approach verifies convergence rates previously derived for continuous-time limits, and yields new results describing both deterministic and stochastic EKI convergence behavior with respect to the standard minimum-norm weighted least squares solution in terms of the fundamental subspaces. Numerical experiments illustrate our theoretical results.

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arXiv2024

A non-intrusive data-based reformulation of a hybrid projection-based model reduction method

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, we do not necessarily require access to the original system matrices, but only to input/output data in the form of the system's transfer function, evaluated at particular values (frequencies). Numerical examples illustrate the efficiency of the new data-driven formulation.

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arXiv2022

Port-Hamiltonian Realizations of Nonminimal Linear Time Invariant Systems

Numerical methods for developing port-Hamiltonian representations of general linear time-invariant systems are studied. The approach extends previous port-Hamiltonian characterizations to include the general non-minimal case and the case where the feedthrough term fails to have an invertible symmetric part. The resulting construction is able to identify infeasibility when the system fails to be port-Hamiltonian, and allows for the incorporation of perturbations in order to arrive at a nearby port-Hamiltonian system. Results are illustrated via numerical examples.

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arXiv2021

Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference

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 distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions as expressed through the associated Gramians. We introduce Gramian definitions relevant to the inference setting and propose a balanced truncation approach based on these inference Gramians that yield a reduced dynamical system that can be used to cheaply approximate the posterior mean and covariance. Our definitions exploit natural connections between (i) the reachability Gramian and the prior covariance and (ii) the observability Gramian and the Fisher information. The resulting reduced model then inherits stability properties and error bounds from system theoretic considerations, and in some settings yields an optimal posterior covariance approximation. Numerical demonstrations on two benchmark problems in model reduction show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.

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arXiv2021

Data-driven balancing of linear dynamical systems

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 function {or the impulse response} corresponding to the original model. We discuss parallels that our approach bears with the Loewner framework, another popular data-driven model reduction method. We illustrate our approach numerically in both continuous-time and discrete-time cases.

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arXiv2019

Sampling-free parametric model reduction for structured systems

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 reduction methodology, which are then combined in various ways in order to interpolate the response from these representative models. The initial exploration of the parameter space can be a forbiddingly expensive task. A different approach is proposed here that does not require any parameter sampling or exploration of the parameter space. Instead, we represent the system response in terms of four subsystems that are nonparametric. One may apply any one of a number of standard (nonparametric) model reduction strategies to reduce the subsystems independently, and then conjoin these reduced models with the underlying parameterized representation to obtain an overall parameterized response. Our approach has elements in common with the parameter mapping approach of Baur et al. [8], but offers greater flexibility and potentially greater control over accuracy. In particular, a data-driven variation of our approach is described that exercises this flexibility through the use of limited frequency-sampling of the underlying nonparametric models. The parametric structure of our system representation allows for a priori guarantees of system stability the resulting parametric reduced models, uniformly across all parameter values. Incorporation of system theoretic error bounds allow us to determine appropriate approximation orders for the nonparametric systems sufficient to yield uniformly high accuracy with respect to parameter variation.

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arXiv2018

Empirical least-squares fitting of parametrized dynamical systems

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 practical; only response observations over a range of parameter settings might be known. Respecting these typical operational constraints, we propose a two phase approach that first encodes the response data into a high fidelity intermediate model of modest order, followed then by a compression stage that serves to eliminate redundancy in the intermediate model. For the first phase, we extend non-parametric least-squares fitting approaches so as to accommodate parameterized systems. This results in a (discrete) least-squares problem formulated with respect to both frequency and parameter that identifies "local" system response features. The second phase uses an $\mathbf{\mathcal{H}}_2$-optimal model reduction strategy accommodating the specialized parametric structure of the intermediate model obtained in the first phase. The final compressed model inherits the parametric dependence of the intermediate model and maintains the high fidelity of the intermediate model, while generally having dramatically smaller system order. We provide a variety of numerical examples demonstrating our approach.

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arXiv2018

Robust port-Hamiltonian representations of passive systems

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 linear matrix inequality (LMI). In this paper we analyze robustness measures for the different possible representations and relate it to quality functions defined in terms of the eigenvalues of the matrix associated with the LMI. In particular, we look at the analytic center of this LMI. From this, we then derive inequalities for the passivity radius of the given model representation.

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arXiv2017

Port-Hamiltonian descriptor systems

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 for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is demonstrated with several examples.

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arXiv2017

Damping optimization of parameter dependent mechanical systems by rational interpolation

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 optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates `interpolatory' reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the $\mathcal{H}_2$ system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.

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arXiv2016

Data-driven Structured Realization

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 from measurements of a transfer function. Our approach extends the Loewner realization framework to more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.

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arXiv2016

Interpolatory methods for $\mathcal{H}_\infty$ model reduction of multi-input/multi-output systems

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 (SISO) setting, which combined ideas originating in interpolatory $\mathcal{H}_2$-optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many large-scale settings. We are able to avoid computationally demanding $\mathcal{H}_\infty$ norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better $\mathcal{H}_\infty$ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation.

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arXiv2015

Vector Fitting for Matrix-valued Rational Approximation

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 framework to include the construction of effective rational approximations to matrix-valued functions, a problem which presents significant challenges that do not appear in the scalar case. Transfer functions associated with multi-input/multi-output (MIMO) dynamical systems typify the class of functions that we consider here. Others have also considered extensions of VF to matrix-valued functions and related numerical implementations are readily available. However to our knowledge, a detailed analysis of numerical issues that arise does not yet exist. We offer such an analysis including critical implementation details here. One important issue that arises for VF on matrix-valued functions that has remained largely unaddressed is the control of the McMillan degree of the resulting rational approximant; the McMillan degree can grow very high in the case of large input/output dimensions. We introduce two new mechanisms for controlling the McMillan degree of the final approximant, one based on alternating least-squares minimization and one based on ancillary system-theoretic reduction methods. Motivated in part by our earlier work on the scalar VF problem as well as by recent innovations for computing optimal $H_2$ approximation, we establish a connection with optimal $H_2$ approximation, and are able to improve significantly the fidelity of VF through numerical quadrature, with virtually no increase in cost or complexity. We provide several numerical examples to support the theoretical discussion and proposed algorithms.

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arXiv2014

Model Reduction by Rational Interpolation

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 the reduction of parametrized systems. This chapter offers a survey of interpolatory model reduction methods starting from basic principles and ranging up through recent developments that include weighted model reduction and structure-preserving methods based on generalized coprime representations. Our discussion is supported by an assortment of numerical examples.

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arXiv2014

Quadrature-Based Vector Fitting: Implications For H2 System Approximation

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 performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an H2 error measure. Our modified approach, designated here as QuadVF, helps recover the original transfer function with better global fidelity (as measured with respect to the H2 norm), than the localized least squares approximation implicit in VF. We extend the new framework also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. We consider the convergence behavior of both VF and QuadVF as well, and evaluate potential numerical ill-conditioning of the underlying least-squares problems. We investigate briefly VF in the case of noisy measurements and propose a new formulation for the resulting approximation problem. Several numerical examples are provided to support the theoretical discussion.

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arXiv2013

Nonlinear Parametric Inversion using Interpolatory Model Reduction

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 (PDEs) in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in DOT by drastically reducing the computational cost of solving the forward problems. The key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis; a similar observation holds for Jacobian evaluations as well. This motivates the use of system-theoretic model order reduction methods. We discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model. Within the DOT setting, these surrogate models can approximate both the cost functional and the associated Jacobian with very little loss of accuracy while significantly reducing the cost of the overall inversion process. Four numerical examples illustrate the efficiency of the proposed approach. Although we focus on DOT in this paper, we believe that our approach is applicable much more generally.

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