Multiwavelet Bases Research Articles

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained.

Galerkin multi-wavelet bases and mesh points method to solve integral equation with singular logarithmic kernel

In this article, we use an effective coupled technique to solve singular integral equations with logarithmic kernel which it occurs in different branches of sciences such as mathematical physics, mechanics, isotropic elastic, etc. At first, we approximate the singular kernel by choosing suitable mesh points and multi-wavelet bases then, Galerkin method is used for solving integral equations. In addition, for simplification and to achieve to a more accuracy, we convert integral equation with logarithmic kernel to integro-differential equation with weakly singular kernel which is solvable with the proposed method. The numerical results show the high accuracy of the solution.

Time-Varying Nonlinear Causality Detection Using Regularized Orthogonal Least Squares and Multi-Wavelets With Applications to EEG

A new transient Granger causality detection method is proposed based on a time-varying parametric modeling framework, and is applied to the real EEG signals to reveal the causal information flow during motor imagery (MI) tasks. The time-varying parametric modeling approach employs a nonlinear autoregressive with external input model, whose parameters are approximated by a set of multi-wavelet basis functions. A regularized orthogonal least squares (ROLS) algorithm is then used to produce a parsimonious or sparse regression model and estimate the associated model parameters. The time-varying Granger causality between nonstationary signals can be detected accurately by making use of both the good approximation properties of multi-wavelets and the good generalization performance of the ROLS in the presence of high-level noise. Two simulation examples are presented to demonstrate the effectiveness of the proposed method for both linear and nonlinear causal detection respectively. The proposed method is then applied to real EEG signals of MI tasks. It follows that transient causal information flow over the time course between various sensorimotor related channels can be successfully revealed during the whole reaction processes. Experimental results from these case studies confirm the applicability of the proposed scheme and show its utility for the understanding of the associated neural mechanism and the potential significance for developing MI tasks based brain-computer interface systems.

A Review of the Recent Advances Made in the Black-Scholes Models and Respective Solutions Methods

The Black-Scholes model has been a major advance in finance over a period of time; this paper examines this model in some detail, in terms of the latest developments in both analytical and numerical solutions. The paper initially briefly examines historical aspects but quickly moves to a critical analysis of the works in the applications, solutions and implications for the efficiency and use of these. More specifically, this paper critically reviews the existing literature on the proposed exact as well as the numerical solutions to the Black-Scholes model. For this purpose, the exact solution literature on the Black-Scholes model includes speed based solutions and techniques based on valuation issues, time varying instruments and stochastic volatility. Similarly, the key numerical solutions include finite difference methods, the semi-discretization technique, Crank-Nicolson and the R3C scheme, the cubic spline wavelets and multi-wavelet bases method, the two-step backward differentiation formula in the temporal discretization and a High-Order Difference approximation with Identity Expansion (HODIE) scheme and fractional Black-Scholes model (TFBSM) along with Fourier analysis. This analysis reveals that transaction costs, high volatility, illiquid markets and large investor preferences are the key issues of today’s financial derivatives markets, especially after the Global Financial Crisis (GFC). These issues require non-linear solutions to the Black-Scholes models; therefore, Crank-Nicolson and the R3C scheme should be focused upon more by incorporating more and more real-life assumptions of current day trading.

Use of Legendre multiwavelets to solve Carleman type singular integral equations

• Multiscale representation of Cauchy singular integral operator and weight ω ( x ) ( = x α ( 1 − x ) β , α , β > − 1 ) is given in LMW bases. • Estimation formula of local Holder exponent helps to recast the equation with bounded solution throughout the domain. • A posteriori L 2 error of approximate solution u J M S ( x ) is given in terms of wavelet coefficients of resolutions J 2, J 1. • Numerical examples from the theory of elasticity having real as well as complex exponents of the solution are presented. • The stress intensity factor at the tip of the crack can be efficiently estimated with the aid of the approximate solution. This paper is concerned with obtaining approximate numerical solutions of Carleman type singular integral equations by using Legendre multiwavelet basis. The properties of Legendre multiwavelets are first given and the low- and high-pass filters for two-scale relations involving Legendre multiwavelets have been derived. The scheme proposed here consists of two basic steps. In the first step the multiscale representation of the given integral operator is obtained and explicit expressions for the elements of the matrix associated with the multiscale representation are derived. In the second step the given integral equation is converted into a system of linear algebraic equations. The convergence of the method is proved in L 2 space. Two illustrative examples from elasticity are given to show the efficiency of the method developed here.

Identification of time-varying neural dynamics from spike train data using multiwavelet basis functions

BackgroundTracking the changes of neural dynamics based on neuronal spiking activities is a critical step to understand the neurobiological basis of learning from behaving animals. These dynamical neurobiological processes associated with learning are also time-varying, which makes the modeling problem challenging. New methodWe developed a novel multiwavelet-based time-varying generalized Laguerre–Volterra (TVGLV) modeling framework to study the time-varying neural dynamical systems using natural spike train data. By projecting the time-varying parameters in the TVGLV model onto a finite sequence of multiwavelet basis functions, the time-varying identification problem is converted into a time invariant linear-in-the-parameters one. An effective forward orthogonal regression (FOR) algorithm aided by mutual information (MI) criterion is then applied for the selection of significant model regressors or terms and the refinement of model structure. A generalized linear model fit approach is finally employed for parameter estimation from spike train data. ResultsThe proposed multiwavelet-based TVGLV approach is used to identify both synthetic input-output spike trains and spontaneous retinal spike train recordings. The proposed method gives excellent the performance of tracking either sharply or slowly changing parameters with high sensitivity and accuracy regardless of the a priori knowledge of spike trains, which these results indicate that the proposed method is shown to deal well with spike train data. Comparison with existing methodsThe proposed multiwavelet-based TVGLV approach was compared with several state-of-art parametric estimation methods like the steepest descent point process filter (SDPPF) or Chebyshev polynomial expansion method. The conventional SDPPF algorithm, or SDPPF with B-splines wavelet expansion method was shown to have the poor performance of tracking the time-varying system changes with the synthetic spike train data due to the slow convergence of the adaptive filter methods. Although the Chebyshev polynomial basis function method gave the good parametric estimation results, it requires prior parameter estimation. It was shown that the proposed multiwavelet-based TVGLV method can track the time-varying parameter changes rapidly and accurately. ConclusionsThe multiwavelet-based TVGLV modeling framework developed in this paper can not only provide a computational modeling scheme for investigating such nonstationary properties, track more general forms of changes in time-varying neural dynamics, and but also may potentially be applied to investigate the spatial-temporal information underlying biomedical spiking signals.

Time-Varying System Identification Using an Ultra-Orthogonal Forward Regression and Multiwavelet Basis Functions With Applications to EEG.

A new parametric approach is proposed for nonlinear and nonstationary system identification based on a time-varying nonlinear autoregressive with exogenous input (TV-NARX) model. The TV coefficients of the TV-NARX model are expanded using multiwavelet basis functions, and the model is thus transformed into a time-invariant regression problem. An ultra-orthogonal forward regression (UOFR) algorithm aided by mutual information (MI) is designed to identify a parsimonious model structure and estimate the associated model parameters. The UOFR-MI algorithm, which uses not only the observed data themselves but also weak derivatives of the signals, is more powerful in model structure detection. The proposed approach combining the advantages of both the basis function expansion method and the UOFR-MI algorithm is proved to be capable of tracking the change of TV parameters effectively in both numerical simulations and the real EEG data.

Stochastic Galerkin framework with locally reduced bases for nonlinear two-phase transport in heterogeneous formations

The generalized polynomial chaos method with multiwavelet basis functions is applied to the Buckley–Leverett equation. We consider a spatially homogeneous domain modeled as a random field. The problem is projected onto stochastic basis functions which yields an extended system of partial differential equations. Analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations.The accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions. Adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic Galerkin setting since the change of basis affects the system matrix itself. To achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell, we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices.Results are presented for problems in one and two spatial dimensions, with varying number of stochastic dimensions. We show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic Galerkin method with local basis reduction. The system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through Monte Carlo sampling.

Multiscale approximation of the solution of weakly singular second kind Fredholm integral equation in Legendre multiwavelet basis

Numerical solution of Fredholm integral equation of second kind with weakly singular kernel is obtained in this paper by employing Legendre multi-wavelet basis. The low- and high-pass filters for two-scale relations involving Legendre multiwavelets having four or five vanishing moments of their wavelets have been derived and are used in the evaluation of integrals for the multiscale representation of the integral operator. Explicit expressions for the elements of the matrix associated with the multiscale representation are given. An estimate for the Hölder exponent of the solution of the integral equation at any point in its domain is obtained. A number of examples is provided to illustrate the efficiency of the method developed here.

Fault Diagnosis of Demountable Disk-Drum Aero-Engine Rotor Using Customized Multiwavelet Method.

The demountable disk-drum aero-engine rotor is an important piece of equipment that greatly impacts the safe operation of aircraft. However, assembly looseness or crack fault has led to several unscheduled breakdowns and serious accidents. Thus, condition monitoring and fault diagnosis technique are required for identifying abnormal conditions. Customized ensemble multiwavelet method for aero-engine rotor condition identification, using measured vibration data, is developed in this paper. First, customized multiwavelet basis function with strong adaptivity is constructed via symmetric multiwavelet lifting scheme. Then vibration signal is processed by customized ensemble multiwavelet transform. Next, normalized information entropy of multiwavelet decomposition coefficients is computed to directly reflect and evaluate the condition. The proposed approach is first applied to fault detection of an experimental aero-engine rotor. Finally, the proposed approach is used in an engineering application, where it successfully identified the crack fault of a demountable disk-drum aero-engine rotor. The results show that the proposed method possesses excellent performance in fault detection of aero-engine rotor. Moreover, the robustness of the multiwavelet method against noise is also tested and verified by simulation and field experiments.

Customized maximal-overlap multiwavelet denoising with data-driven group threshold for condition monitoring of rolling mill drivetrain

Fault identification timely of rolling mill drivetrain is significant for guaranteeing product quality and realizing long-term safe operation. So, condition monitoring system of rolling mill drivetrain is designed and developed. However, because compound fault and weak fault feature information is usually sub-merged in heavy background noise, this task still faces challenge. This paper provides a possibility for fault identification of rolling mills drivetrain by proposing customized maximal-overlap multiwavelet denoising method. The effectiveness of wavelet denoising method mainly relies on the appropriate selections of wavelet base, transform strategy and threshold rule. First, in order to realize exact matching and accurate detection of fault feature, customized multiwavelet basis function is constructed via symmetric lifting scheme and then vibration signal is processed by maximal-overlap multiwavelet transform. Next, based on spatial dependency of multiwavelet transform coefficients, spatial neighboring coefficient data-driven group threshold shrinkage strategy is developed for denoising process by choosing the optimal group length and threshold via the minimum of Stein’s Unbiased Risk Estimate. The effectiveness of proposed method is first demonstrated through compound fault identification of reduction gearbox on rolling mill. Then it is applied for weak fault identification of dedusting fan bearing on rolling mill and the results support its feasibility.

Construction of orthogonal multiwavelet bases

We propose a method for constructing orthogonal multiwavelet bases of the space L 2(ℝ) for any known multiscaling functions that generate a multiresolution analysis of dimension greater than 1.

Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree-Fock and density functional theory via linear response.

A fully numerical method for the time-dependent Hartree-Fock and density functional theory (TD-HF/DFT) with the Tamm-Dancoff (TD) approximation is presented in a multiresolution analysis (MRA) approach. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. The integral equation is efficiently and adaptively solved using a numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H2, Be, N2, H2O, and C2H4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. We introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.

Planetary gearbox condition monitoring of ship-based satellite communication antennas using ensemble multiwavelet analysis method

Satellite communication antennas are key devices of a measurement ship to support voice, data, fax and video integration services. Condition monitoring of mechanical equipment from the vibration measurement data is significant for guaranteeing safe operation and avoiding the unscheduled breakdown. So, condition monitoring system for ship-based satellite communication antennas is designed and developed. Planetary gearboxes play an important role in the transmission train of satellite communication antenna. However, condition monitoring of planetary gearbox still faces challenges due to complexity and weak condition feature. This paper provides a possibility for planetary gearbox condition monitoring by proposing ensemble a multiwavelet analysis method. Benefit from the property on multi-resolution analysis and the multiple wavelet basis functions, multiwavelet has the advantage over characterizing the non-stationary signal. In order to realize the accurate detection of the condition feature and multi-resolution analysis in the whole frequency band, adaptive multiwavelet basis function is constructed via increasing multiplicity and then vibration signal is processed by the ensemble multiwavelet transform. Finally, normalized ensemble multiwavelet transform information entropy is computed to describe the condition of planetary gearbox. The effectiveness of proposed method is first validated through condition monitoring of experimental planetary gearbox. Then this method is used for planetary gearbox condition monitoring of ship-based satellite communication antennas and the results support its feasibility.

Linear scaling Coulomb interaction in the multiwavelet basis, a parallel implementation

We present a parallel and linear scaling implementation of the calculation of the electrostatic potential arising from an arbitrary charge distribution. Our approach is making use of the multi-resolution basis of multiwavelets. The potential is obtained as the direct solution of the Poisson equation in its Green's function integral form. In the multiwavelet basis, the formally non local integral operator decays rapidly to negligible values away from the main diagonal, yielding an effectively banded structure where the bandwidth is only dictated by the requested accuracy. This sparse operator structure has been exploited to achieve linear scaling and parallel algorithms. Parallelization has been achieved both through the shared memory (OpenMP) and the message passing interface (MPI) paradigm. Our implementation has been tested by computing the electrostatic potential of the electronic density of long-chain alkanes and diamond fragments showing (sub)linear scaling with the system size and efficent parallelization.

A Multiwavelet Treatment of the Quantum Subsystem in Quantum Wavepacket Ab Initio Molecular Dynamics through an Hierarchical Partitioning of Momentum Space.

We present an hierarchical scheme where the propagator in quantum dynamics is represented using a multiwavelet basis. The approach allows for a recursive refinement methodology, where the representation in momentum space can be adaptively improved through additional, decoupled layers of basis functions. The method is developed within the constructs of quantum-wavepacket ab initio molecular dynamics (QWAIMD), which is a quantum-classical method and involves the synergy between a time-dependent quantum wavepacket description and ab initio molecular dynamics. Specifically, the current development is embedded within an "on-the-fly" multireference electronic structural generalization of QWAIMD. The multiwavelet treatment is used to study the dynamics and spectroscopy in a small hydrogen bonded cluster. The results are in agreement with previous calculations and with experiment. The studies also allow an interpretation of the shared proton dynamics as one that can be modeled through the dynamics of dressed states.

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.

Numerical Solution Of Singular Fredholm Integrodifferential Equations Of The Second Kind Via Petrovgalerkin Method By Using Legendre Multiwavelet

It is of interest to know whether we can solve a singular Fredholm integro-differential equation of the second kind with an infinite or semi-infinite range of integrate via Petrov–Galerkin method by using Legendre multiwavelet. For this purpose, we directly deal with infinite range of integrate. We introduce some change of variables for mapping infinite interval into a finite interval. After that, we use Petrov– Galerkin method with Legendre multiwavelet basis that yields linear system. Numerical results of our example will demonstrate accuracy and efficiency of the proposed method.

Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes

In this paper, we introduce a new global troubled-cell indicator for the discontinuous Galerkin (DG) method in one- and two-dimensions. This is done by taking advantage of the global expression of the DG method and re-expanding it in terms of a multiwavelet basis, which is a sum of the global average and finer details on different levels. Examining the higher level difference coefficients acts as a troubled-cell indicator, thus avoiding unnecessary increased computational cost of a new expansion. In two-dimensions the multiwavelet decomposition uses combinations of scaling functions and multiwavelets in the x- and y-directions for improved troubled-cell indication. By using such a troubled-cell indicator, we are able to reduce the computational cost by avoiding limiting in smooth regions. We present numerical examples in one- and two-dimensions and compare our troubled-cell indicator to the subcell resolution technique of Harten (1989) [11] and the shock detector of Krivodonova et al. (2004) [18], which were previously investigated by Qiu and Shu (2005) [23].

BIORTHOGONAL MULTIWAVELETS RELATED BY DIFFERENTIATION

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1,1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition.