In this paper an extension of the Adaptive Antoulas-Anderson (aaa) Model Order Reduction (MOR) method to time -domain data is defined, referred to as Time -Domain aaa (TDaaa). Inspired by other rational approximation ...
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In this paper an extension of the Adaptive Antoulas-Anderson (aaa) Model Order Reduction (MOR) method to time -domain data is defined, referred to as Time -Domain aaa (TDaaa). Inspired by other rational approximation time -domain MOR methods, like Time -Domain Vector Fitting (TDVF) and Time -Domain Loewner Framework (TDLF), TDaaa combines the adaptivity and flexibility of the aaa method in the frequency domain with an error minimization in the time domain. This combination makes the method an interesting alternative to fully time -domain or frequency -domain MOR methods. A combination of aaa and TDVF is also proposed, called aaa-TDVF, where the initial TDVF poles are selected by aaa. This new poles initialization improves both accuracy and convergence speed. Both TDaaa and TDVF are discussed in detail and their performance is compared on a benchmark LTI system.
In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form f(t) = & sum;(K)(j=1) gamma(j) cos (2 pi ...
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In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form f(t) = & sum;(K)(j=1) gamma(j) cos (2 pi a(j)t + b(j)), where the frequency parameters a(j) is an element of R (or a(j) is an element of iR) are pairwise different. Our method is based on the recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al.(SIAM J Sci Comput 40(3):A1494-A1522, 2018). For signal reconstruction we use a set of classical Fourier coefficients off with regard to a fixed interval (0,P) with P>0. Even though all terms off may be non-P-periodic, our reconstruction method requires at most 2 K + 2 Fourier coefficients c(n)(f) to recover all parameters off. We show that in the case of exact data, the proposed iterative algorithm terminates after at mostK+1 steps. The algorithm can also detect the number K of terms off, if K is a priori unknown and L >= 2 K + 2 Fourier coefficients are available. Therefore our method provides a new alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method.
Rational approximation schemes for reconstructing periodic signals from samples with poorly separated spectral content are described. These methods are automatic and adaptive, requiring no tuning or manual parameter s...
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Rational approximation schemes for reconstructing periodic signals from samples with poorly separated spectral content are described. These methods are automatic and adaptive, requiring no tuning or manual parameter selection. Collectively, they form a framework for fitting trigonometric rational models to data that is robust to various forms of corruption, including additive Gaussian noise, perturbed sampling grids, and missing data. Our approach combines a variant of Prony's method with a modified version of the adaptive Antoulas-Anderson algorithm. Using representations in both frequency and time space, a collection of algorithms is described for adaptively computing with trigonometric rationals. This includes procedures for differentiation, filtering, convolution, and more. A new MATLAB software system based on these algorithms is introduced. Its effectiveness is illustrated with synthetic and practical examples drawn from applications including biomedical monitoring, acoustic denoising, and feature detection.
Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation-precisely the case where rational functions outperform polynomials by a landslide...
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Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation-precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) an "aaa-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of vertical bar x vertical bar on [-1,1] in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter [Math. USSR Sb., 74 (1993), pp. 271-290] required 200-digit extended precision.
Burgers' equation is a well-studied model in applied mathematics with connections to the Navier- Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse...
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Burgers' equation is a well-studied model in applied mathematics with connections to the Navier- Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions to Burgers' equation in the complex plane, concentrating on the dynamics of the complex singularities and their relationship to the solution on the real line. For an initial condition with a simple pole in each of the upper-and lower-half planes, we apply formal asymptotics in the small-and large-time limits in order to characterise the initial and later motion of the singularities. The small-time limit highlights how infinitely many singularities are born at t = 0 and how they orientate themselves to lie increasingly close to anti-Stokes lines in the far field of the inner problem. This inner problem also reveals whether or not the closest singularity to the real axis moves toward the axis or away. For intermediate times, we use the exact solution, apply method of steepest descents, and implement the aaa approximation to track the complex singularities. Connections are made between the motion of the closest singularity to the real axis and the steepness of the solution on the real line. While Burgers' equation is integrable (and has an exact solution), we deliberately apply a mix of techniques in our analysis in an attempt to develop methodology that can be applied to other nonlinear partial differential equations that do not. & COPY;2023 Elsevier B.V. All rights reserved.
We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an oth...
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We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the aaa algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function's real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.
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