Haar Basis Research Articles

Memory Effects in Diffusion Like Equation Via Haar Wavelets

The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.

Efficient wavelets-based valuation of synthetic CDO tranches

We present new formulae for the valuation of synthetic collateralized debt obligation (CDO) tranches under a one-factor model. These formulae are based on the wavelet theory and the method used is called WA[a,b]. We approximate the cumulative distribution function (CDF) of the underlying pool by a finite combination of jth order B-spline basis, where the B-spline basis of order zero is typically called a Haar basis. We provide an error analysis and we show that for this type of distributions, the rate of convergence in the approximation is similar regardless of the order of the B-spline basis employed. The resulting formula for the Haar basis case is much easier to implement and performs better than the formula for the B-spline basis of order one in terms of computational time. The numerical experiments confirm the impressive speed and accuracy of the WA[a,b] method equipped with a Haar basis, independently of the inhomogeneity features of the underlying pool. The method appears to be particularly fast for multiple tranche valuation.

Haar functions in weighted Besov and Triebel–Lizorkin spaces

The paper deals with Haar wavelet bases in function spaces of Besov and Triebel–Lizorkin type with local Muckenhoupt weights. We show that Haar wavelets can be used to characterize such function spaces as far as absolute value of smoothness parameter is small enough and weights fulfill some conditions. The result is based on mapping properties of linear operators involving characteristic functions of dyadic cubes in related spaces and on local means characterization of weighted Besov and Triebel–Lizorkin spaces.

SU‐F‐BRB‐12: A Novel Haar Wavelet Based Approach to Deliver Non‐Coplanar Intensity Modulated Radiotherapy Using Sparse Orthogonal Collimators

Purpose:Existing efforts to replace complex multileaf collimator (MLC) by simple jaws for intensity modulated radiation therapy (IMRT) resulted in unacceptable compromise in plan quality and delivery efficiency. We introduce a novel fluence map segmentation method based on compressed sensing for plan delivery using a simplified sparse orthogonal collimator (SOC) on the 4π non‐coplanar radiotherapy platform.Methods:4π plans with varying prescription doses were first created by automatically selecting and optimizing 20 non‐coplanar beams for 2 GBM, 2 head & neck, and 2 lung patients. To create deliverable 4π plans using SOC, which are two pairs of orthogonal collimators with 1 to 4 leaves in each collimator bank, a Haar Fluence Optimization (HFO) method was used to regulate the number of Haar wavelet coefficients while maximizing the dose fidelity to the ideal prescription. The plans were directly stratified utilizing the optimized Haar wavelet rectangular basis. A matching number of deliverable segments were stratified for the MLC‐based plans.Results:Compared to the MLC‐based 4π plans, the SOC‐based 4π plans increased the average PTV dose homogeneity from 0.811 to 0.913. PTV D98 and D99 were improved by 3.53% and 5.60% of the corresponding prescription doses. The average mean and maximal OAR doses slightly increased by 0.57% and 2.57% of the prescription doses. The average number of segments ranged between 5 and 30 per beam. The collimator travel time to create the segments decreased with increasing leaf numbers in the SOC. The two and four leaf designs were 1.71 and 1.93 times more efficient, on average, than the single leaf design.Conclusion:The innovative dose domain optimization based on compressed sensing enables uncompromised 4π non‐coplanar IMRT dose delivery using simple rectangular segments that are deliverable using a sparse orthogonal collimator, which only requires 8 to 16 leaves yet is unlimited in modulation resolution.This work is supported in part by Varian Medical Systems, Inc. and NIH R43 CA18339.

Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis

In this article we approximate the solutions of the nonlinear Fredholm integral equations of the second kind, by the method based on using the properties of RH wavelets and matrix operator. Also, the Banach fixed point theorem guarantees the convergence of the method. Also we get an upper bound for the error. Furthermore, the order of convergence is analyzed. The algorithm to compute the solutions and some numerical examples are also illustrated. The numerical results obtained by our method have been compared with other methods.

Walsh and wavelet methods for differential equations on the Cantor group

Ordinary and partial differential equations for unknown functions defined on the Cantor dyadic group are studied. We consider two types of equations: related to the Gibbs derivatives and to the fractional pseudo differential operators. Since the Cantor group is an ultrametric space, pseudo differential operators have spacial properties and are of interest for some applications to models of complex systems, e.g., ultrametric diffusion models in biophysics. We find solutions to the equations in classes of distributions and analyse under what assumptions these solutions are regular functions with some “good” properties. Haar wavelets are used to solve pseudo differential equation. It is very important that the Haar MRA coincides with the Shannon MRA on the Cantor group. To analyse solutions, specific computational method based on the multiresolution structure of the Haar basis was developed.

Numerical approximation of 2D Fredholm integral eigenvalue problems by orthogonal wavelets

We investigate the numerical approximation of two-dimensional, second kind Fredholm integral eigenvalue problems by the Galerkin method with the Cohen–Daubechies–Vial (CDV) wavelet family. This choice provides us orthogonal bases for bounded domains, avoiding the need of periodization or domain truncation. The CDV family is indexed by the number of vanishing moments, which drives the regularity of the basis. We generate the Galerkin basis from tensorized scaling functions and employ weighted Gaussian quadratures derived from refinement equations. Numerical experiments address the relative computational cost of this approach with respect to the Haar basis and the relationship between convergence rate and number of vanishing moments.

Lp- and Sp,qrB-discrepancy of (order 2) digital nets

Dick proved that all order $2$ digital nets satisfy optimal upper bounds of the $L_2$-discrepancy. We give an alternative proof for this fact using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds of the $S_{p,q}^r B$-discrepancy for a certain parameter range and enlarge that range for order $2$ digitals nets. $L_p$-, $S_{p,q}^r F$- and $S_p^r H$-discrepancy is considered as well.

Multilevel Wavelet-based Numerical Method of Local Structural Analysis for Three-dimensional Problem

High-accuracy solution of three-dimensional problems of structural analysis is normally required in some pre-known domains (regions of structure with the risk of significant stresses that could potentially lead to the destruction, regions subjected to specific operational requirements). This paper is devoted to wavelet-based numerical method of local analysis of three-dimensional structures. Initial discrete operational and variational formulation of the considering problem and corresponding formulations with the use of discrete Haar basis are presented. Due to special algorithms of averaging within multigrid approach, correct reduction of the problem is presented.

Ортогональная система Франклина и ортогональная система финитных функций в численных методах решения краевых задач

Возможности классических рядов Фурье, связанных с тригонометрическими функциями, существенно ограничены в двумерных и трехмерных краевых задачах. Граничные условия таких краевых задач для областей с криволинейными границами часто не удается выполнить при использовании классических рядов Фурье. Решение этой проблемы дает использование ортогональных финитных базисных функций. Однако ортогональные базисные функции Хаара не являются непрерывными. Ортогональные вейвлеты Добеши имеют компактные носители, но не записываются в аналитической форме и имеют низкую гладкость. Непрерывные финитные функции Фабера-Шаудера не являются ортогональными. Ортогональные непрерывные функции Франклина не являются финитными. Здесь установлена связь ортогональной системы функций Франклина с последовательностью сеточных наборов кусочно-линейных ортогональных финитных базисных функций (ОФФ). Сформирован ряд Фурье-ОФФ на основе таких непрерывных ОФФ, который позволяет выполнять граничные условия типа Дирихле на криволинейных границах в рамках интегральных постановок краевых задач. Аналогичная проблема, связанная с удовлетворением граничных условий типа Неймана, также устраняется при дополнительном использовании смешанных интегральных постановок краевых задач. Ряд Фурье-ОФФ повышает эффективность смешанных численных методов решения краевых задач.

A remark on the multipliers of the Haar basis of L1[0,1

A proof of a necessary and sufficient condition for a sequence to be a multiplier of the normalized Haar basis of $L^1 [0, 1]$ is given. This proof depends only on the most elementary properties of this system and is an alternative proof to that recently

Comparative analysis of the solution of linear continuous inverse problems using different basis expansions

Geophysical inverse problems try to infer the value of a physical property of the earth from data measured at the boundary of the domain. Model parameterization is a key concept to make the inverse problem less ill conditioned. In this contribution we compare the performance of four different basis functions, Fourier, Pixel, Haar and Daubechies Wavelets, for a 1D linear continuous inverse problem. Adopting the right parameterization also reduces the number of dimensions in which the inverse problem is going to be solved, allowing easy posterior analysis. To compare the different basis expansions we have studied a simple 1D linear inverse problem in gravimetric inversion for a density anomaly with Gaussian shape. For this simple toy-problem we show that the Fourier basis gives a better reconstruction using Filon's quadrature to avoid numerical instabilities caused by highly oscillating functions, both, in the noise-free and noisy cases. Besides, the Fourier base is the one that provides the lowest reconstruction error for the number of basis terms and the highest condition number. The Haar and pixel (piecewise-continuous functions) bases provide similar results, although the pixel base needs more terms to achieve lower reconstruction errors. Also, the pixel basis is the one that has the lowest condition number that is obviously related to the way the energy in the system matrix is distributed. Besides, the Daubechies Db2 basis expansion is the one that has the highest system rank, it is more difficult to apply to project the integral kernel, and provides the worst results, compared to the other basis expansions. Finally we propose how to generalize this methodology to linear inverse problems in several dimensions and to non-linear problems. A separate paper will be devoted to this subject.

On orthogonal p-adic wavelet bases

A variety of different orthogonal wavelet bases has been found for L2(R) for the last three decades. Such bases are not only interesting by themselves, a number of theoretical and applied problems were solved using them. It appeared that similar non-trivial wavelet constructions also exist for functions defined on some other algebraic structures, such as local fields of positive characteristic including the Vilenkin/Cantor groups. We note that most of the bases involved consist of Bruhat–Schwartz functions, i.e. compactly supported and band-limited ones. There were several attempts to develop wavelet theory in a more general setting, e.g., for zero-dimensional groups. Analyzing these papers, one can see that actually nothing except for the Haar basis and its trivial modifications was constructed. In the present paper we give an explanation for these failures. Namely, we describe the situation for the field Qp of p-adic numbers and prove that any orthogonal wavelet basis for L2(Qp) that consists of band-limited (periodic) functions, is a modification of the Haar basis.

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).

Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients

In this article, the authors proposed a numerical scheme based on Crank–Nicolson finite difference scheme and Haar wavelets to find numerical solutions of different types of second order hyperbolic telegraph equations (i.e. telegraph equation with constant coefficients, with variable coefficients, and singular telegraph equation). This work is an extension of the scheme by Jiwari (2012) for hyperbolic equations. The use of Haar basis function is made with multiresolution analysis to get the fast and accurate results on collocation points. The convergence of the proposed scheme is proved by doing its error analysis. Four test examples are considered to demonstrate the accuracy and efficiency of the scheme. The scheme is easy and very suitable for computer implementation and provides numerical solutions close to the exact solutions and available in the literature.

Modified Wavelet-Based Multilevel Discrete-Continual Finite Element Method for Local Structural Analysis - Part 2: Reduced Formulations of the Problems in Haar Basis

The distinctive paper is devoted to wavelet-based discrete-continual finite element method (WDCFEM) of structural analysis. Discrete-continual formulations of multipoint boundary problems of two-dimensional and three-dimensional structural analysis are transformed to corresponding localized formulations by using the discrete Haar wavelet basis and finally, with the use of averaging and reduction algorithms, the localized and reduced governing equations are obtained. Special algorithms of localization with respect to each degree of freedom are presented.

Constructive Characteristic of ho¨ Lder Classes and M-Term Approximations in the Multiple Haar Basis

In terms of the best polynomial approximations in the multiple Haar basis, we obtain a constructive characteristic of the Holder classes H of functions defined on the unit cube $$ \mathbb{I} $$ d of the space ℝ d under the restriction $$ 0<\alpha <\frac{1}{p}\le 1 $$ . We also solve the problem of order estimates of the best m-term approximations in the Haar basis of classes H in the Lebesgue spaces L q ( $$ \mathbb{I} $$ d ).

A class of Laplacian multiwavelets bases for high-dimensional data

We introduce a framework for representing functions defined on high-dimensional data. In this framework, we propose to use the eigenvectors of the graph Laplacian to construct a multiresolution analysis on the data. We assume the dataset to have an associated hierarchical tree partition, together with a function that measures the similarity between pairs of points in the dataset. The construction results in a one parameter family of orthonormal bases, which includes both the Haar basis as well as the eigenvectors of the graph Laplacian, as its two extremes. We describe a fast discrete transform for the expansion in any of the bases in this family, and estimate the decay rate of the expansion coefficients. We also bound the error of non-linear approximation of functions in our bases. The properties of our construction are demonstrated using various numerical examples.

Strategies for High-Performance Resource-Efficient Compression of Neural Spike Recordings

Brain-machine interfaces (BMIs) based on extracellular recordings with microelectrodes provide means of observing the activities of neurons that orchestrate fundamental brain function, and are therefore powerful tools for exploring the function of the brain. Due to physical restrictions and risks for post-surgical complications, wired BMIs are not suitable for long-term studies in freely behaving animals. Wireless BMIs ideally solve these problems, but they call for low-complexity techniques for data compression that ensure maximum utilization of the wireless link and energy resources, as well as minimum heat dissipation in the surrounding tissues. In this paper, we analyze the performances of various system architectures that involve spike detection, spike alignment and spike compression. Performance is analyzed in terms of spike reconstruction and spike sorting performance after wireless transmission of the compressed spike waveforms. Compression is performed with transform coding, using five different compression bases, one of which we pay special attention to. That basis is a fixed basis derived, by singular value decomposition, from a large assembly of experimentally obtained spike waveforms, and therefore represents a generic basis specially suitable for compressing spike waveforms. Our results show that a compression factor of 99.8%, compared to transmitting the raw acquired data, can be achieved using the fixed generic compression basis without compromising performance in spike reconstruction and spike sorting. Besides illustrating the relative performances of various system architectures and compression bases, our findings show that compression of spikes with a fixed generic compression basis derived from spike data provides better performance than compression with downsampling or the Haar basis, given that no optimization procedures are implemented for compression coefficients, and the performance is similar to that obtained when the optimal SVD based basis is used.

Scale-Invariant V-Transform Signal De-Noising

Approximation,reconstruction and denoising of signals with traditional wavelet transform sometimes exhibit visual artifacts,for example,Gibbs phenomena in the vicinity of discontinuities,which are caused by the continuous property of the wavelet basis.We present a novel method for signal de-noising using Scale-Invariant V-Transform(SIVT).The V-system is the generalization of the well-known Haar wavelet basis function and it is multi-wavelets in invariant set.An aporia of signal denoising is that the local signal reconstruction at the singular points.Based on the analysis for the signal singular points,combining local signal resampling and orthogonal transform,a novel technical framework—SIVT has been proposed for signal de-noising.The test results reveal the SIVT reconstructions exhibit higher visual quality and numerical measurement of SNR than wavelet-based reconstructions.Existing theory of SIVT suggests that these new approaches can perform significantly better than wavelet methods in certain signal reconstruction problems.