Showing papers on "Fourier series published in 2020"

A numerical method for computing the overall response of nonlinear composites with complex microstructure

Abstract: The local and overall responses of nonlinear composites are classically investigated by the Finite Element Method. We propose an alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images. It is based on the exact expression of the Green function of a linear elastic and homogeneous comparison material. First, the case of elastic nonhomogeneous constituents is considered and an iterative procedure is proposed to solve the Lippman-Schwinger equation which naturally arises in the problem. Then, the method is extended to non-linear constituents by a step-by-step integration in time. The accuracy of the method is assessed by varying the spatial resolution of the microstructures. The flexibility of the method allows it to serve for a large variety of microstructures. (C) 1998 Elsevier Science S.A.

Convexification and experimental data for a 3D inverse scattering problem with the moving point source

Abstract: Inverse scattering problems of the reconstructions of physical properties of a medium from boundary measurements are substantially challenging ones. This work aims to verify the performance on experimental data of a newly developed convexification method for a 3D coefficient inverse problem for the case of objects buried in a sandbox a fixed frequency and the point source moving along an interval of a straight line. Using a special Fourier basis, the method of this work strongly relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic equations. This problem, in turn, is solved via the minimization of a Tikhonov-like functional weighted by a Carleman Weight Function. The global convergence of the numerical procedure is established analytically. The numerical verification is performed using experimental data, which are raw backscatter data of the electric field. These data were collected using a microwave scattering facility at The University of North Carolina at Charlotte.

A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data

Abstract: We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded domain. Although this problem is highly nonlinear, we do not require an initial guess of the true solution. The key in our method is the derivation of a boundary value problem for a system of coupled quasilinear elliptic equations whose solution is the vector function of the spatially dependent Fourier coefficients of the solution to the governing parabolic equation. We solve this problem by an iterative method. The global convergence of the system is rigorously established using a Carleman estimate. Numerical examples are presented.

Nonlinear Optimal Control of Thermal Processes in a Nonlinear Inverse Problem

Abstract: The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation are studied. The parabolic equation is considered under initial and boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. Moreover, the recovery function nonlinearly enters into the differential equation. Is applied the method of variable separation based on the search for a solution to the mixed inverse problem in the form of a Fourier series. It is assumed that the recovery function and nonlinear term of the given differential equation are also expressed as a Fourier series. For fixed values of the control function, the unique solvability of the inverse problem is proved by the method of compressive mappings. The quality functional has a nonlinear form. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complicated functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and nonlinear integral equations. Nonlinear functional and integral equations are solved by the method of successive approximations. Formulas are obtained for the approximate calculation of the state function of the controlled process, the recovery function, and the optimal control function. Is proved the absolutely and uniformly convergence of the obtained Fourier series.

The phase diagram of approximation rates for deep neural networks

Abstract: We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to functional classes of arbitrary positive smoothness, and identify the boundary between the feasible and infeasible rates. Moreover, we show that all networks with a piecewise polynomial activation function have the same phase diagram. Next, we demonstrate that standard fully-connected architectures with a fixed width independent of smoothness can adapt to smoothness and achieve almost optimal rates. Finally, we consider deep networks with periodic activations ("deep Fourier expansion") and prove that they have very fast, nearly exponential approximation rates, thanks to the emerging capability of the network to implement efficient lookup operations.

Rademacher expansions and the spectrum of 2d CFT

Abstract: A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin j ≠ 0. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator

Abstract: In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

On a Boundary-value Problem for Boussinesq type Nonlinear Integro-Differential Equation with Reflecting Argument

Abstract: In the three-dimensional domain a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and reflecting argument is considered. The solution of this partial integro-differential equation is studied in the class of generality functions. The method of separation of variables and the method of a degenerate kernels are used. Using these methods, the nonlocal boundary value problem is integrated as a countable system of ordinary differential equations. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving countable system of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed and we obtained the countable system of nonlinear integral equations for each of five cases. To establish the unique solvability of this countable system of nonlinear integral equations we use the method of successive approximations and the method of compressing mappings. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution of the boundary value problem with respect to given functions in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed as Fourier series. For other cases, the absence of nontrivial solutions of the problem is proved. The corresponding theorems are formulated.

Nonlinear Systems With Uncertain Periodically Disturbed Control Gain Functions: Adaptive Fuzzy Control With Invariance Properties

Abstract: This paper proposes a novel adaptive fuzzy dynamic surface control (DSC) method for an extended class of periodically disturbed strict-feedback nonlinear systems. The peculiarity of this extended class is that the control gain functions are not bounded a priori but simply taken to be continuous and with a known sign. In contrast with existing strategies, controllability must be guaranteed by constructing appropriate compact sets ensuring that all trajectories in the closed-loop system never leave these sets. We manage to do this by means of invariant set theory in combination with the Lyapunov theory. In other words, boundedness is achieved a posteriori as a result of stability analysis. The approximator composed of fuzzy logic systems and Fourier series expansion is constructed to deal with the unknown periodic disturbance terms.

Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs

Abstract: The book is devoted to the strong approximation of iterated stochastic integrals (ISIs) in the context of numerical integration of Ito SDEs and non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise. The presented monograph open a new direction in researching of ISIs. For the first time we successfully use the generalized multiple Fourier series (Fourier-Legendre series as well as trigonometric Fourier series) converging in the sense of norm in the space $L_2([t, T]^k)$ for the expansion and strong approximation of Ito ISIs of multiplicity $k,$ $k\in{\bf N}$ (Chapter 1). This result is adapted for Stratonovich ISIs of multiplicities 1 to 5 (Chapter 2) as well as for some other types of ISIs (Chapter 1). Two theorems on expansions of Stratonovich ISIs of multiplicity $k$ ($k\in{\bf N}$) based on generalized iterated Fourier series (converging pointwise) are formulated and proved (Chapter 2). The integration order replacement technique for the class of Ito ISIs has been introduced (Chapter 3). This result is generalized for the class of ISIs with respect to martingales. We derived the exact and approximate expressions for the mean-square error of approximation of Ito ISIs of multiplicity $k$, $k\in{\bf N}$ (Chapter 1). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and approximations of specific Ito and Stratonovich ISIs of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4) using the system of Legendre polynomials and the system of trigonometric functions. The methods formulated in this book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of Ito ISIs of multiplicity $k,$ $k\in{\bf N}$ with respect to the $Q$-Wiener process.

Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters

Abstract: The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.

On the leading-edge suction and stagnation-point location in unsteady flows past thin aerofoils

Abstract: Unsteady thin-aerofoil theory is a low-order method for calculating the forces and moment developed on a camber line undergoing arbitrary motion, based on potential-flow theory. The vorticity distribution is approximated by a Fourier series, with a special ‘ and independent of the leading-edge radius. Closed-form expressions for these in simplified scenarios such as quasi-steady flow and small-amplitude harmonic oscillations are derived.

Mixture Models for the Analysis, Edition, and Synthesis of Continuous Time Series

Abstract: This chapter presents an overview of techniques used for the analysis, edition, and synthesis of continuous time series, with a particular emphasis on motion data. The use of mixture models allows the decomposition of time signals as a superposition of basis functions. It provides a compact representation that aims at keeping the essential characteristics of the signals. Various types of basis functions have been proposed, with developments originating from different fields of research, including computer graphics, human motion science, robotics, control, and neuroscience. Examples of applications with radial, Bernstein, and Fourier basis functions are presented, with associated source codes to get familiar with these techniques.

Adaptive Transmit Waveform Design using Multi-Tone Sinusoidal Frequency Modulation

Abstract: This paper presents an adaptive waveform design method using Multi-Tone Sinusoidal Frequency Modulation (MTSFM). The MTSFM waveform's modulation function is represented as a finite Fourier series expansion. The Fourier coefficients are utilized as a discrete set of design parameters that may be modified to adapt the waveform's properties. The MTSFM's design parameters are adjusted to shape the spectrum, Auto-Correlation Function (ACF), and Ambiguity Function (AF) shapes of the waveform. The MTSFM waveform model naturally possesses the constant envelope and spectrally compact waveforms that make it well suited for transmission on practical radar/sonar transmitters which utilize high power amplifiers. The MTSFM has an exact mathematical definition for its time-series using Generalized Bessel Functions which allow for deriving closed-form analytical expressions for its spectrum, AF, and ACF. These expressions allow for establishing well-defined optimization problems that finely tune the MTSFM's properties. This adaptive waveform design model is demonstrated by optimizing MTSFM waveforms that initially possess a "thumbtack-like" AF shape. The resulting optimized designs possess substantially improved sidelobe levels over specified regions in the range-Doppler plane without increasing the Time-Bandwidth Product (TBP) that the initialized waveforms possessed. Simulations additionally demonstrate that the optimized thumbtack-like MTSFM waveforms are competitive with thumbtack-like phase-coded waveforms derived from design algorithms available in the published literature.

Nanoparticle lattices with bases: Fourier modal method and dipole approximation

Abstract: The utilization of periodic structures such as photonic crystals and metasurfaces is common for light manipulation at nanoscales. One of the most widely used computational approaches to consider them and design effective optical devices is the Fourier modal method (FMM) based on Fourier decomposition of electromagnetic fields. Nevertheless, calculating periodic structures with small inclusions is often a difficult task since they induce lots of high-${k}_{\ensuremath{\parallel}}$ harmonics that should be taken into account. In this paper, we consider small-particle lattices with bases (complex unit cells) and construct their scattering matrices via discrete dipole approximation. Afterwards, these matrices are implemented in FMM for consideration of complicated layered structures. We show the performance of the proposed hybrid approach by its application to a lattice, which routes left and right circularly polarized incident light to guided modes propagating in opposite directions. We also demonstrate its precision by spectra comparison with finite-element method calculations. The high speed and precision of this approach enable the calculation of angle-dependent spectra with very high resolution in a reasonable time, which allows resolving narrow lines unobservable by other methods.

Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

Abstract: In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed.

On the Rankin--Selberg problem

Abstract: In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

Abstract: We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L 2 . We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.

Observer-based adaptive fractional-order control of flexible-joint robots using the Fourier series expansion: theory and experiment

Abstract: In this paper, fractional-order observer/controller design for flexible-joint robots is developed. In order to eliminate the need for obtaining the regressor matrix, the Fourier series expansion is applied for uncertainty estimation. Voltage saturation nonlinearities are compensated in the control law; hence, knowledge of the actuator/manipulator dynamics model is not required in the proposed method. Uniformly ultimately boundedness of observer estimation error and joint position tracking error are guaranteed through Lyapunov stability concept. The case study is a single-link flexible-joint robot actuated by permanent magnet DC motors. Experimental results are presented to emphasize the successful practical implementation of the proposed algorithm. Based on the experimental results, the proposed controller considerably outperforms some previous related works using various criteria.

Drift estimation for discretely sampled SPDEs

Abstract: The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consider the usual setup of continuous time observations of the Fourier coefficients of the solutions, and show that the MLE is consistent, asymptotically normal and optimal in the mean-square sense. In the second part of the paper we investigate the natural time discretization of the MLE, by assuming that the first N Fourier modes are measured at M time grid points, uniformly spaced over the time interval [0, T]. We provide a rigorous asymptotic analysis of the proposed estimators when $$N\rightarrow \infty $$ and/or $$T,M\rightarrow \infty $$ . We establish sufficient conditions on the growth rates of N, M and T, that guarantee consistency and asymptotic normality of these estimators.

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

Abstract: We formulate an explicit refinement of Bocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.

Nonlocal Inverse Problem for a Pseudohyperbolic- Pseudoelliptic Type Integro-Differential Equations

Abstract: The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered. Using the method of the Fourier series, a system of countable systems of ordinary integro-differential equations is obtained. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system regular and irregular values of the spectral parameters were calculated. The unique solvability of the inverse boundary value problem for regular values of spectral parameters is proved. For irregular values of spectral parameters is established a criterion of existence of an infinite set of solutions of the inverse boundary value problem. The results are formulated as a theorem.

A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems

Abstract: In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.