Projected Nonlinear Least Squares for Exponential Fitting

Hokanson, Jeffrey M.

doi:10.1137/16m1084067

Cited by 12 publications

(10 citation statements)

References 28 publications

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“…A projected nonlinear least squares method is studied in [58] to fit a model ψ to (noisy) measurements y for the exponential fitting problem:…”

Section: Subspace By Subsampling/sketchingmentioning

confidence: 99%

Subspace Methods for Nonlinear Optimization

Liu¹

2021

CSIAM-AM

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Subspace techniques such as Krylov subspace methods have been well known and extensively used in numerical linear algebra. They are also ubiquitous and becoming indispensable tools in nonlinear optimization due to their ability to handle large scale problems. There are generally two types of principals: i) the decision variable is updated in a lower dimensional subspace; ii) the objective function or constraints are approximated in a certain smaller subspace of their domain. The key ingredients are the constructions of suitable subspaces and subproblems according to the specific structures of the variables and functions such that either the exact or inexact solutions of subproblems are readily available and the corresponding computational cost is significantly reduced. A few relevant techniques include but not limited to direct combinations, block coordinate descent, active sets, limited-memory, Anderson acceleration, subspace correction, sampling and sketching. This paper gives a comprehensive survey on the subspace methods and their recipes in unconstrained and constrained optimization, nonlinear least squares problem, sparse and low rank optimization, linear and nonlinear eigenvalue computation, semidefinite programming, stochastic optimization and etc. In order to provide helpful guidelines, we emphasize on high level concepts for the development and implementation of practical algorithms from the subspace framework.

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“…A projected nonlinear least squares method is studied in [58] to fit a model ψ to (noisy) measurements y for the exponential fitting problem:…”

Section: Subspace By Subsampling/sketchingmentioning

confidence: 99%

Subspace Methods for Nonlinear Optimization

Liu¹

2021

CSIAM-AM

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“…, r}. Then we can reduce the computational cost of solving by (9) by projecting the measurements y onto the range space of Q [23]:…”

Section: Line Spectral Estimationmentioning

confidence: 99%

“…It is shown in [23] that the projected problem (10) has the same stationary points as the full problem (9) under certain conditions on the range space of Q. When applying an optimization method like Gauss-Newton, the advantage of the projected problem (10) over the full problem (9) is that each optimization step is much cheaper since the projected Jacobian has much smaller size.…”

Section: Line Spectral Estimationmentioning

confidence: 99%

“…When applying an optimization method like Gauss-Newton, the advantage of the projected problem (10) over the full problem (9) is that each optimization step is much cheaper since the projected Jacobian has much smaller size. Based on this observation, for the general case where the frequencies lie in multiple bands, [23] provides an iterative algorithm that in each iteration finds one underlying band and projects the signal onto this band, then applies Gauss-Newton to solve the projected problem. We also note that our Q can be further reduce the computational cost in [23] since Q can be efficiently applied to a vector, while the orthonormal basis utilized in [23] is a numerical approximation (obtained by performing PCA on a set of sinusoids) to the Slepian basis S K .…”

Section: Line Spectral Estimationmentioning

confidence: 99%

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ROAST: Rapid Orthogonal Approximate Slepian Transform

Zhu

Karnik

Wakin

et al. 2018

IEEE Trans. Signal Process.

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In this paper, we provide a Rapid Orthogonal Approximate Slepian Transform (ROAST) for the discrete vector that one obtains when collecting a finite set of uniform samples from a baseband analog signal. The ROAST offers an orthogonal projection which is an approximation to the orthogonal projection onto the leading discrete prolate spheroidal sequence (DPSS) vectors (also known as Slepian basis vectors). As such, the ROAST is guaranteed to accurately and compactly represent not only oversampled bandlimited signals but also the leading DPSS vectors themselves. Moreover, the subspace angle between the ROAST subspace and the corresponding DPSS subspace can be made arbitrarily small. The complexity of computing the representation of a signal using the ROAST is comparable to the FFT, which is much less than the complexity of using the DPSS basis vectors. We also give non-asymptotic results to guarantee that the proposed basis not only provides a very high degree of approximation accuracy in a mean squared error sense for bandlimited sample vectors, but also that it can provide high-quality approximations of all sampled sinusoids within the band of interest. Z. Zhu is with the Center for

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“…Here we introduce a new approach for solving this model reduction problem based on the projected nonlinear least squares framework introduced in [29]. The core of this approach consists of approximating the H 2 -norm using a sequence of orthogonal projections P (µ (n) ):…”

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confidence: 99%

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

Hokanson¹,

Magruder²

2018

Preprint

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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 is to find reduced order models that are locally optimal approximations in the H 2 norm, an approach taken by the Iterative Rational Krylov Algorithm (IRKA) and several others. Here we introduce a new approach for 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 H 2 optimization problem onto a finite-dimensional subspace yielding a weighted least 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, H 2 optimization problem. This enables us to build reduced order models with similar error in the H 2 norm as competing methods but using far fewer evaluations of the expensive, full order model. 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. This application of projected nonlinear least squares to the H 2 -optimal model reduction problem suggests extensions of this approach related model reduction problems.

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Projected Nonlinear Least Squares for Exponential Fitting

Cited by 12 publications

References 28 publications

Subspace Methods for Nonlinear Optimization

Subspace Methods for Nonlinear Optimization

ROAST: Rapid Orthogonal Approximate Slepian Transform

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

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