Haar Expansion Research Articles

Uncertainty Quantification of Self-Breakdown Switches in a Circuit Model Using Wiener–Haar Expansion

Since many electromagnetic compatibility tests are resourcelimited, it is important to quantify the uncertainties of instruments beforehand. One of the common instruments is the trigger pulse generator with self-breakdown switches. Due to the abrupt changing caused by self-breakdown switches, stochastic methods based on the generalized polynomial chaos suffer from Gibbs-type phenomena and become erroneous around abrupt transition point. This article proposes an effective technique based on the Haar wavelets for quantifying the time delay of self-breakdown switches in stochastic simulations. Wiener–Haar wavelet expansion method is suitable for dealing with abrupt-changing problems and is applied to the nodal-analysis-based circuit model. The stochastic circuit model is expanded to an augmented model with deterministic coefficients by exploiting wavelets’ orthogonality. Furthermore, the distribution of time delay is represented as the mean value of a Heaviside function, which is implemented by an auxiliary circuit in the model. The statistical mean can be directly obtained from the 0th coefficient of Wiener–Haar expansion. A case study of a trigger generator for electromagnetic pulse generators is provided to validate the effectiveness of the proposed method. It is shown that, compared with Monte Carlo methods, Wiener–Haar representation provides robust results with less computational time and a faster convergence rate.

Wavelet Filtering in Shock Stochastic Systems with High Availability

For filtering problems in StSHA under nonGaussian ShD methodological and algorithmically WL support is developed. 3 types of filters are considered: KBF (WLKBF), LPF (WLLPF) and SOLF (WLSOLF). These filters have the following advantages: on-line regime, high accuracy and possibility of algorithmically description of complex ShD. Wavelet filter modifications are based on Galerkin method and Haar wavelet expansions. WLF unlike KBF, LPF and SOLF do not need to integrate system of ordinary differential Eqs. These filters must solve system of linear algebraic Eqs with constant coefficients. KBF (WLKBF) and SOLF (WLSOLF) are recommended for StSHA with additive ShD whereas LPF (WLLPF) are recommended for StSHA with parametric and additive ShD. Basic applications are: on-line identification and calibration of nonstationary processes in StSHA of ShD. Methods are illustrated by example of 3 dimensional differential linear information control system at complex ShD. Basic algorithms and error analysis for KBF (WLKBF) and LPF (WLLPF) are presented and 15 Figure; illustrate filters peculiarities for small and fin damping. These filters allow to estimate the accumulation effects for systematic and random errors. Results may be generalized for filtration, extrapolation with interpolation problems in StSHA and multiple ShD.

Data-Driven Uncertainty Quantification and Characterization for Household Energy Demand Across Multiple Time-Scales

Uncertainty has become a key challenge in energy profiles at the domestic level, which is influenced by a variety of factors, such as behaviors, technologies, weather conditions, energy prices, and so on. These factors influence the household demand at various time horizons, hence result in diverse uncertainty natures across time-scales. Therefore, it is crucial to understand the temporal natures of the demand uncertainty at different time-scales, particularly intra-day and inter-days. This paper first attempts to quantify and characterize uncertainties of household electricity demand across multiple time scales. An advanced data-driven temporal-dependency Haar expansions uncertainty quantification approach is thus proposed by this paper. The proposed approach consists of two stages. At first stage, uncertainty will be decomposed into multiple components of different time scales and then quantified by Monte Carlo methods. The second stage will further characterize natures of each uncertainty component in terms of threefold: 1) temporal uncertainty distribution at a certain time scale; 2) coupling degree between two time scales; and 3) uncertainty propagation natures to the system level of uncertainty component at a certain time scale. The proposed analysis is demonstrated on Irish smart meter database. To prove the efficiency and validity of the proposed approach, the error bound and computational cost is analyzed.

Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation

The aim of this paper is to derive a numerical scheme for solving stochastic differential equations (SDEs) via Wong-Zakai approximation. One of the most important methods for solving SDEs is Milstein method, but this method is not so popular because the cost of simulating the double stochastic integrals is high. For overcoming this complexity, we present an implicit Milstein scheme based on Wong-Zakai approximation by approximating the Brownian motion with its truncated Haar expansion. The main advantages of this method lie in the fact that it preserves the convergence order and also stability region of the Milstein method while its simulation is much easier than Milstein scheme. We show the convergence rate of the method by some numerical examples.

Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions

In the present paper, two numerical methods are analyzed for the solution of two-dimensional Poisson equation with two different types of nonlocal boundary conditions. The first numerical method is a collocation method based on Haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions (RBFs). A two-point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work. For the collocation method based on Haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a Haar expansion which is integrated subsequently to get wavelet approximation of the solution. For the meshless method based on RBFs, the algorithm is implemented using two different splitting schemes (with and without shape parameter splitting) for numerical solution of the model. The comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the Haar wavelet. Accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests.

Linear Multifractional Stable Motion: Representation via Haar basis

The goal of this paper is to provide a wavelet series representation for Linear Multifractional Stable Motion (LMSM). Instead of using Daubechies wavelets, which are not given in closed form, we use a Haar wavelet, thus yielding a more explicit expression than that in Ayache and Hamonier (in press).The main ingredient of this construction is a Haar expansion of the integrands which define the high and low frequency components of the SαS random field X generating LMSM. Then, by using Abel summation, we show that these series are a.s. convergent in the space of continuous functions. Finally, we determine their a.s. rates of convergence in the latter space.In the end, the representations of the high and low frequency components of X, provide a new method for simulating the high and low frequency components of LMSM. Moreover, this new way is faster than the way based on Fast Fourier Transform (Le Guevel and Levy-vehel, 2012; Stoev and Taqqu, 2004).

Wiener–Askey and Wiener–Haar Expansions for the Analysis and Prediction of Limit Cycle Oscillations in Uncertain Nonlinear Dynamic Friction Systems

This paper is devoted to the robust modeling and prediction of limit cycle oscillations in nonlinear dynamic friction systems with a random friction coefficient. In recent studies, the Wiener–Askey and Wiener–Haar expansions have been proposed to deal with these problems with great efficiency. In these studies, the random dispersion of the friction coefficient is always considered within intervals near the Hopf bifurcation point. However, it is well known that friction induced vibrations—with respect to the distance of the friction dispersion interval to the Hopf bifurcation point—have different properties in terms of tansient, frequency and amplitudes. So, the main objective of this study is to analyze the capabilities of the Wiener–Askey (general polynomial chaos, multielement generalized polynomial chaos) and Wiener–Haar expansions to be efficient in the modeling and prediction of limit cycle oscillations independently of the location of the instability zone with respect to the Hopf bifurcation point.

Uncertainty quantification in hybrid dynamical systems

Uncertainty quantification (UQ) techniques are frequently used to ascertain output variability in systems with parametric uncertainty. Traditional algorithms for UQ are either system-agnostic and slow (such as Monte Carlo) or fast with stringent assumptions on smoothness (such as polynomial chaos and Quasi-Monte Carlo). In this work, we develop a fast UQ approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems. To capture discontinuities, we use a wavelet-based Wiener–Haar expansion. We develop a boundary layer approach to propagate uncertainty through separable reset conditions. We also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems. Here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics. The solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems. The above methods are demonstrated on example problems.

Wiener–Haar Expansion for the Modeling and Prediction of the Dynamic Behavior of Self-Excited Nonlinear Uncertain Systems

This paper deals with the modeling and the prediction of the dynamic behavior of uncertain nonlinear systems. An efficient method is proposed to treat these problems. It is based on the Wiener–Haar chaos concept resulting from the polynomial chaos theory and it generalizes the use of the multiresolution analysis well known in the signal processing theory. The method provides a powerful tool to describe stochastic processes as series of orthonormal piecewise functions whose weighting coefficients are identified using the Mallat pyramidal algorithm. This paper shows that the Wiener–Haar model allows an efficient description and prediction of the dynamic behavior of nonlinear systems with probabilistic uncertainty in parameters. Its contribution, compared to the representation using the generalized polynomial chaos model, is illustrated by evaluating the two models via their application to the problems of the modeling and the prediction of the dynamic behavior of a self-excited uncertain nonlinear system.

A decomposition theorem for singular integral operators on spaces of homogeneous type

Let ( X , d , μ ) be a space of homogeneous type. Under the assumption μ ( { x } ) = 0 for all x ∈ X , we prove a decomposition theorem for singular integral operators on ( X , d , μ ) . Isotropic Haar expansion gives a representation of the integral operator as a series of simple shifts and rearrangements plus two paraproducts. This yields a UMD-valued T ( 1 ) theorem on spaces of homogeneous type.

On Haar expansion of Riemann–Liouville process in a critical case

We show that Haar-based series representation of the critical Riemann–Liouville process Rα with α =3/2 is rearrangement non-optimal in the sense of convergence rate in C[0, 1]. Bibliography: 10 titles.

Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness

AbstractWe study the discrepancy function of two‐dimensional Hammersley type point sets in the unit square. It is well‐known that the symmetrized Hammersley point set achieves the asymptotically best possible rate for the L2‐norm of the discrepancy function. In this paper we consider the norm of the discrepancy function of Ham¬mersley type point sets in Besov spaces of dominating mixed smoothness and show that it achieves the optimal rate under appropriate assumptions on the set and the smoothness parameter of the Besov space. Our proof relies on a characterization of the Besov spaces of dominating mixed smoothness via coefficients in the Haar expansion and the computation of the Haar expansion of the discrepancy function of Hammersley type point sets (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Gaussian approximation of the empirical process under random entropy conditions

We obtain rates of strong approximation of the empirical process indexed by functions by a Brownian bridge under only random entropy conditions. The results of Berthet and Mason [P. Berthet, D.M. Mason, Revisiting two strong approximation results of Dudley and Philipp, in: High Dimensional Probability, in: IMS Lecture Notes-Monograph Series, vol. 51, 2006, pp. 155–172] under bracketing entropy are extended by combining their method to properties of the empirical entropy. Our results show that one can improve the universal rate v n = o ( log log n ) from Dudley and Philipp [R.M. Dudley, W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrsch. Verw. Gebiete 62 (1983) 509–552] into v n → 0 at a logarithmic rate, under a weak random entropy assumption which is close to necessary. As an application the results of Koltchinskii [V.I. Kolchinskii, Komlós–Major–Tusnády approximation for the general empirical process and Haar expansions of classes of functions, J. Theoret. Probab. 7 (1994) 73–118] are revisited when the conditions coming in addition to random entropy are relaxed.

Spectral and multiresolution Wiener expansions of oscillatory stochastic processes

Wiener chaos expansions are being evaluated for the representation of stochastic variability in the response of nonlinear aeroelastic systems, which often exhibit limit cycles. Preliminary studies with a simple nonlinear aeroelastic computational model have shown that the standard non-intrusive Wiener–Hermite expansion fails to maintain time accuracy as the simulation evolves. Wiener–Hermite expansions faithfully reproduce the short-term characteristics of the process but consistently lose energy after several mean periods of oscillation. This energy loss remains even for very high-order expansions. To uncover the cause of this energy loss and to explore potential remedies, the more elementary problem of a sinusoid with random frequency is used herein to simulate the periodic response of an uncertain system. As time progresses, coefficients of the higher order terms in both the Wiener–Hermite and Wiener–Legendre expansions successively gain and lose dominance over the lower-order coefficients in a manner that causes any fixed-order expansion in terms of global basis functions to fail over a simulation time of sufficient duration. This characteristic behavior is attributed to the continually increasing frequency of the process in the random dimension. The recently developed Wiener–Haar expansion is found to almost entirely eliminate the loss of energy at large times, both for the sinusoidal process and for the response of a two degree-of-freedom nonlinear system, which is examined as a prelude to the stochastic simulation of aeroelastic limit cycles. It is also found that Mallat's pyramid algorithm is more efficient and accurate for evaluating Wiener–Haar expansion coefficients than Monte Carlo simulation or numerical quadrature.

Time adaptive time-domain techniques for the design of microwave circuits

A novel time adaptive time-domain technique based on the Haar expansion basis is proposed and validated for a specific circuit problem. The modeling of active and passive lumped and distributed elements, as well as of excitation and boundary conditions, is performed effectively. This scheme, based on a combination of absolute and relative thresholding, provides a real-time time adaptive grid with improved time resolution in comparison to conventional time-domain schemes (FDTD) while maintaining a similar accuracy.

Calculus on cantor triadic set (I)—derivative

For real-valued functions defined on Cantor triadic set, a derivative with corresponding formula of Newton-Leibniz’s type is given. In particular, for the self-similar functions and alternately jumping functions defined in this paper, their derivative and exceptional sets are studied accurately by using ergodic theory on Σ2 and Duffin-Schaeffer’s theorem concerning metric diophantine approximation. In addition, Haar basis of L2 (Σ2) is constructed and Haar expansion of standard self-similar function is given.

Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions

Rates of convergence of the Komlos-Major-Tusnady type in the invariance principle for empirical processes indexed by functions are obtained. The conditions are given in terms of the empirical entropy and the accuracy of the Haar type approximation for the corresponding classes of functions. The recent results of Massart(25) as well as some new results for empirical characteristic functions are obtained as a corollary.

Haar expansions of a class of fractal interpolation functions and their logical derivatives

In this paper, we study a special class of fractal interpolation functions, and give their Haar-wavelet expansions. On the basis of the expansions, we investigate the Holder smoothness of such funstions and their logical derivatives of order α.

The Convergence of Series for Various Choices of Sign in Banach Spaces

1. Let (xn, Xn) denote a basis for a Banach space (X, ∥ • ∥) of measurable functions in (0, 1).It is shown in [2] and [9] that the equivalence of the normsand ∥ • ∥ is equivalent to the unconditionality of the basis (xn, Xn). In [8] a weaker relationship between these norms is exploited to establish the existence of an element of L1(E) for each E ⊂ (0, 1), |£| > 0, whose Haar series expansion is conditionally convergent in the norm of L\(E).In this note, a Lemma of Orlicz [7] is generalized to provide a relationship between , and the changes in sign that are tolerated in without disruption of norm convergence.