Mixed estimator of spline truncated, Fourier series, and kernel in biresponse semiparametric regression model

Regression analysis is a method of analysis to determine the relationship between the response and the predictor variables. There are three approaches in regression analysis, namely the parametric, nonparametric, and semiparametric approaches. Biresponse Semiparametric regression model is a regression model that uses a combination approach between parametric and nonparametric components, where two response variables are correlated with each other. For data cases with several predictor variables, different estimation technique approaches can be used for each variable. In this study, the parametric component is assumed to be linear. At the same time, the nonparametric part is approached using a mixture of three estimation techniques, namely, spline truncated, Fourier series, and the kernel. The unknown data pattern is assumed to follow the criteria of each of these estimation techniques. The spline is used when the data pattern tends to change at certain time intervals, the Fourier series is used when the data pattern tends to repeat itself, and the kernel is used when the data does not have a specific way. This study aims to obtain parameter estimates for the mixed semiparametric regression model of spline truncated, Fourier series, and the kernel on the biresponse data using the Weighted Least Square (WLS) method. The formed model depends on the selection of knot points, oscillation parameters, and optimal bandwidth. The best model is based on the smallest Generalized Cross Validation (GCV).

Coupled Time and Passage Spectral Method for an Efficient Resolution of Turbomachinery Far Upstream Wakes

The paper proposes a novel numerical method called coupled time and passage spectral method for an efficient resolution of far upstream wakes in an unsteady analysis of flow field within generic multiple blade rows. The proposed method is a very simple and natural extension of the time spectral form harmonic balance method. By including inter-blade phase angle and passage index in a truncated Fourier series, the proposed method is capable of circumventing the limitations of the harmonic balance method in dealing with zero frequency harmonics and time harmonics with the same frequency but different inter blade phase angles. Different from the harmonic balance method, which seeks solutions at different time instants of the same passage, the coupled spectral method seeks solutions at different passages and time instants. Same as the harmonic balance method, all these solutions are connected via a spectral operator. Coupled time and passage sampling is performed using the modified Gram Schmidt process to choose the time and passage pairs which give the most orthogonal rows of an inverse Fourier transform matrix. The coupled time and passage spectral method requires minimum change to an existing harmonic balance solver. A numerical case study has been provided in the paper to demonstrate the expected capability against the harmonic balance method.

Efficient Bayesian phase estimation using mixed priors

We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.

Coupled Time and Passage Spectral Method for an Efficient Resolution of Turbomachinery Far Upstream Wakes

The paper proposes a novel numerical method called coupled time and passage spectral method for an efficient resolution of far upstream wakes in an unsteady analysis of flow field within generic multiple blade rows. The proposed method is a very simple and natural extension of the time spectral form harmonic balance method. By including inter blade phase angle and passage index in a truncated Fourier series, the proposed method is capable of circumventing the limitations of the harmonic balance method in dealing with zero frequency harmonics and time harmonics with same frequency but different inter blade phase angles. Different from the harmonic balance method, which seeks solution at different time instants of the same passage, the coupled spectral method seeks solutions at different passages and time instants. Same as the harmonic balance method, all these solutions are connected via a spectral operator. Coupled time and passage sampling is performed using the modified Gram Schmidt process to choose the time and passage pairs which give the most orthogonal rows of an inverse Fourier transform matrix. The coupled time and passage spectral method requires minimum change to an existing harmonic balance solver. A numerical case study has been provided in the paper to demonstrate the expected capability against the harmonic balance method.

Dynamics of Fourier Modes in Torus Generative Adversarial Networks

Generative Adversarial Networks (GANs) are powerful machine learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable, and typically, it is necessary to implement several accessory heuristics to the networks to reach acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of generative adversarial networks. For this purpose, we propose to decompose the objective function of the adversary min–max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous alternating gradient descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of GAN. This approach is confirmed empirically by studying the training flow in a 2-parametric GAN, aiming to generate an unknown exponential distribution. As a by-product, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.

A Fast Method for Numerical Realization of Fourier Tools

This chapter presents new application of author’s recent algorithms for fast summations of truncated Fourier series. A complete description of this method is given, and an algorithm for numerical implementation with a given accuracy for the Fourier transform is proposed.

A Floquet-Based Analysis of Parametric Excitation Through the Damping Coefficient

The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu’s equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.

Fourier neural networks: A comparative study

We review neural network architectures which were motivated by Fourier series and integrals and which are referred to as Fourier neural networks. These networks are empirically evaluated in synthetic and real-world tasks. Neither of them outperforms the standard neural network with sigmoid activation function in the real-world tasks. All neural networks, both Fourier and the standard one, empirically demonstrate lower approximation error than the truncated Fourier series when it comes to approximation of a known function of multiple variables.

Convergence of a series associated with the convexification method for coefficient inverse problems

This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L^{2} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H^{1}-norm for a sequence of L^{2}-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

Response Characteristics of Systems With Parametric Excitation Through Damping and Stiffness

Floquet theory is combined with harmonic balance to study parametrically excited systems with combination of both time varying damping and stiffness. An approximated solution having an exponential part with unknown exponents and a periodic term consisting of a truncated Fourier series is considered. When applied to a system with parametric damping and stiffness the analysis shows that combination of parametric damping and stiffness alters stability characteristics, particularly in the primary and superharmonic instabilities comparing to the system with only parametric damping or stiffness. We also look at the initial conditions response and its frequency content. The second excitation harmonic in the system with parametric damping is seen to disrupt the coexistence phenomenon which is observed in the parametric damping case.