arXiv subjects

Jeffrey M. Hokanson

Publications and source records attributed to Jeffrey M. Hokanson.

10 recordsLinked to original sources

arXiv2023

Simultaneous Identification and Denoising of Dynamical Systems

In recent years there has been a push to discover the governing equations dynamical systems directly from measurements of the state, often motivated by systems that are too complex to directly model. Although there has been substantial work put into such a discovery, doing so in the case of large noise has proved challenging. Here we develop an algorithm for Simultaneous Identification and Denoising of a Dynamical System (SIDDS). We infer the noise in the state measurements by requiring that the denoised data satisfies the dynamical system with an equality constraint. This is unlike existing work where the mismatch in the dynamical system is treated as a penalty in the objective. We assume the dynamics is represented in a pre-defined basis and develop a sequential quadratic programming approach to solve the SIDDS problem featuring a direct solution of KKT system with a specialized preconditioner. In addition, we show how we can include sparsity promoting regularization using an iteratively reweighted least squares approach. The resulting algorithm leads to estimates of the dynamical system that approximately achieve the Cramér-Rao lower bound and, with sparsity promotion, can correctly identify the sparsity structure for higher levels of noise than existing techniques. Moreover, because SIDDS decouples the data from the evolution of the dynamical system, we show how to modify the problem to accurately identify systems from low sample rate measurements. The inverse problem approach and solution framework used by SIDDS has the potential to be expanded to related problems identifying governing equations from noisy data.

math.OC↗

arXiv2022

An integrated heterogeneous computing framework for ensemble simulations of laser-induced ignition

An integrated computational framework is introduced to study complex engineering systems through physics-based ensemble simulations on heterogeneous supercomputers. The framework is primarily designed for the quantitative assessment of laser-induced ignition in rocket engines. We develop and combine an implicit programming system, a compressible reacting flow solver, and a data generation/management strategy on a robust and portable platform. We systematically present this framework using test problems on a hybrid CPU/GPU machine. Efficiency, scalability, and accuracy of the solver are comprehensively assessed with canonical unit problems. Ensemble data management and autoencoding are demonstrated using a canonical diffusion flame case. Sensitivity analysis of the ignition of a turbulent, gaseous fuel jet is performed using a simplified, three-dimensional model combustor. Our approach unifies computer science, physics and engineering, and data science to realize a cross-disciplinary workflow. The framework is exascale-oriented and can be considered a benchmark for future computational science studies of real-world systems.

cs.CE↗

arXiv2020

A Lipschitz Matrix for Parameter Reduction in Computational Science

We introduce the Lipschitz matrix: a generalization of the scalar Lipschitz constant for functions with many inputs. Among the Lipschitz matrices compatible a particular function, we choose the smallest such matrix in the Frobenius norm to encode the structure of this function. The Lipschitz matrix then provides a function-dependent metric on the input space. Altering this metric to reflect a particular function improves the performance of many tasks in computational science. Compared to the Lipschitz constant, the Lipschitz matrix reduces the worst-case cost of approximation, integration, and optimization; if the Lipschitz matrix is low-rank, this cost no longer depends on the dimension of the input, but instead on the rank of the Lipschitz matrix defeating the curse of dimensionality. Both the Lipschitz constant and matrix define uncertainty away from point queries of the function and by using the Lipschitz matrix we can reduce uncertainty. If we build a minimax space-filling design of experiments in the Lipschitz matrix metric, we can further reduce this uncertainty. When the Lipschitz matrix is approximately low-rank, we can perform parameter reduction by constructing a ridge approximation whose active subspace is the span of the dominant eigenvectors of the Lipschitz matrix. In summary, the Lipschitz matrix provides a new tool for analyzing and performing parameter reduction in complex models arising in computational science.

math.NA↗

arXiv2020

Multivariate Rational Approximation Using a Stabilized Sanathanan-Koerner Iteration

The Sanathanan-Koerner iteration developed in 1963 is classical approach for rational approximation. This approach multiplies both sides of the approximation by the denominator polynomial yielding a linear problem and then introduces a weight at each iteration to correct for this linearization. Unfortunately this weight introduces a numerical instability. We correct this instability by constructing Vandermonde matrices for both the numerator and denominator polynomials using the Arnoldi iteration with an initial vector that enforces this weighting. This Stabilized Sanathanan-Koerner iteration corrects the instability and yields accurate rational approximations of arbitrary degree. Using a multivariate extension of Vandermonde with Arnoldi, we can apply the Stabilized Sanathanan-Koerner iteration to multivariate rational approximation problems. The resulting multivariate approximations are often significantly better than existing techniques and display a more uniform accuracy throughout the domain.

math.NA↗

arXiv2020

A Data-Driven McMillan Degree Lower Bound

Given measurements of a linear time-invariant system, the McMillan degree is the dimension of the smallest such system that reproduces these observed dynamics. Using impulse response measurements where the system has been started in some (unknown) state and then allowed to evolve freely, a classical result by Ho and Kalman reveals the McMillan degree as the rank of a Hankel matrix built from these measurements. However, if measurements are contaminated by noise, this Hankel matrix will almost surely be full rank. Hence practitioners often estimate the rank of this matrix---and thus the McMillan degree---by manually setting a threshold between the large singular values that correspond to the non-zero singular values of the noise-free Hankel matrix and the small singular values that are pertubations of the zero singular values. Here we introduce a probabilistic upper bound on the perturbation of the singular values of this Hankel matrix when measurements are corrupted by additive Gaussian noise, and hence provide guidance on setting the threshold to obtain a lower bound on the McMillan degree. This result is powered by a new, probabilistic bound on the 2-norm of a random Hankel matrix with normally distributed entries. Unlike existing results for random Hankel matrices, this bound features no unknown constants and, moreover, is within a small factor of the empirically observed bound when entries are independent and identically distributed. This bound on the McMillan degree provides an inexpensive alternative to more general model order selection techniques such as the Akaike Information Criteria (AIC).

math.NA↗

arXiv2020

$\mathcal{H}_2$-Optimal Model Reduction Using Projected Nonlinear Least Squares

In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full order model. One approach to model reduction finds reduced order models that are locally optimal approximations in the $\mathcal{H}_2$ norm, an approach taken by the Iterative Rational Krylov Algorithm (IRKA) among others. Here we introduce a new approach for $\mathcal{H}_2$-optimal model reduction using the projected nonlinear least squares framework previously introduced in [J. M. Hokanson, SIAM J. Sci. Comput. 39 (2017), pp. A3107--A3128]. At each iteration, we project the $\mathcal{H}_2$ optimization problem onto a finite-dimensional subspace yielding a weighted least squares rational approximation problem. Subsequent iterations append this subspace such that the least squares rational approximant asymptotically satisfies the first order necessary conditions of the original, $\mathcal{H}_2$ optimization problem. This enables us to build reduced order models with similar error in the $\mathcal{H}_2$ norm but using far fewer evaluations of the expensive, full order model compared to competing methods. Moreover, our new algorithm only requires access to the transfer function of the full order model, unlike IRKA which requires a state-space representation or TF-IRKA which requires both the transfer function and its derivative. Applying the projected nonlinear least squares framework to the $\mathcal{H}_2$-optimal model reduction problem open new avenues for related model reduction problems.

math.NA↗

arXiv2019

A Numerical Investigation of the Minimum Width of a Neural Network

Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These theorems assume requirements, such as infinite data, that must be discretized in practice. Through numerical experiments, we seek to test the lower bounds established by Hanin in 2017.

cs.LG↗

arXiv2018

Least Squares Rational Approximation

Rational approximation appears in many contexts throughout science and engineering, playing a central role in linear systems theory, special function approximation, and many others. There are many existing methods for solving the rational approximation problem, from fixed point methods like the Sanathanan-Koerner iteration and Vector Fitting, to partial interpolation methods like Adaptive Anderson Antoulas (AAA). While these methods can often find rational approximations with a small residual norm, they are unable to find optimizers with respect to a weighted l2 norm with a square dense weighting matrix. Here we develop a nonlinear least squares approach constructing rational approximations with respect to this norm. We explore this approach using two parameterizations of rational functions: a ratio of two polynomials and a partial fraction expansion. In both cases, we show how we can use Variable Projection (VARPRO) to reduce the dimension of the optimization problem. As many applications seek a real rational approximation that can be described as a ratio of two real polynomials, we show how this constraint can be enforced in both parameterizations. Although this nonlinear least squares approach often converge to suboptimal local minimizers, we find this can be largely mitigated by initializing the algorithm using the poles of the AAA algorithm applied to the same data. This combination of initialization and nonlinear least squares enables us to construct rational approximants using dense and potentially ill-conditioned weight matrices such as those that appear as a step in new H2 model reduction algorithm recently developed by the authors.

math.NA↗

arXiv2018

Data-driven polynomial ridge approximation using variable projection

Inexpensive surrogates are useful for reducing the cost of science and engineering studies involving large-scale, complex computational models with many input parameters. A ridge approximation is one class of surrogate that models a quantity of interest as a nonlinear function of a few linear combinations of the input parameters. When used in parameter studies (e.g., optimization or uncertainty quantification), ridge approximations allow the low dimensional structure to be exploited, reducing the effective dimension. We introduce a new, fast algorithm for constructing a ridge approximation where the nonlinear function is a polynomial. This polynomial ridge approximation is chosen to minimize least squared mismatch between the surrogate and the quantity of interest on a given set of inputs. Naively, this would require optimizing both the polynomial coefficients and the linear combination of weights; the latter of which define a low-dimensional subspace of the input space. However, given a fixed subspace the optimal polynomial can be found by solving a linear least-squares problem, and hence by using variable projection the polynomial can be implicitly found leaving an optimization problem over the subspace alone. We provide an algorithm that finds this polynomial ridge approximation by minimizing over the Grassmann manifold of low-dimensional subspaces using a Gauss-Newton method. We provide details of this optimization algorithm and demonstrate its performance on several numerical examples. Our Gauss-Newton method has superior theoretical guarantees and faster convergence than the alternating approach for polynomial ridge approximation earlier proposed by Constantine, Eftekhari, Hokanson, and Ward [https://doi.org/10.1016/j.cma.2017.07.038] that alternates between (i) optimizing the polynomial coefficients given the subspace and (ii) optimizing the subspace given the coefficients.

math.NA↗

arXiv2017

Projected nonlinear least squares for exponential fitting

The modern ability to collect vast quantities of data poses a challenge for parameter estimation problems. When posed as a nonlinear least squares problem fitting a model to data, the cost of each iteration grows linearly with the amount of data and it can easily become prohibitively expensive to perform many iterations. Here we develop an approach that projects the data onto a low-dimensional subspace of the high-dimensional data that preserves the information in the original data. We provide results from both optimization and statistical perspectives showing that the information is preserved when the subspace angles between this projection and the Jacobian of the model at the current iterate remain small. However, for this approach to reduce computational complexity, both the projected model and Jacobian must be computed inexpensively. This is a constraint on the pairs of models and subspaces for which this approach provides a computational speedup. Here we consider the exponential fitting problem projected onto the range of Vandermonde matrix, for which the projected model and Jacobian can be computed in closed form using a generalized geometric sum formula. We further provide an inexpensive heuristic that picks this Vandermonde matrix so that the subspace angles with the Jacobian remain small and use this heuristic to update the subspace during optimization. Although the asymptotic cost still depends on the data dimension, the overall cost of this sequence of projected nonlinear least squares problems is less expensive than the original nonlinear least squares problem. Applied to the exponential fitting problem, this provides an algorithm that is not only faster in the limit of large data than the conventional nonlinear least squares approach, but is also faster than subspace based approaches such as HSVD.

math.NA↗