Numerical Methods of Multifractal Analysis in Information Communication Systems and Networks

In this chapter, the main principles of the theory of fractals and multifractals are stated. A singularity spectrum is introduced for the random telecommunication traffic, concepts of fractal dimensions and scaling functions, and methods used in their determination by means of Wavelet Transform Modulus Maxima (WTMM) are proposed. Algorithm development methods for estimating multifractal spectrum are presented. A method based on multifractal data analysis at network layer level by means of WTMM is proposed for the detection of traffic anomalies in computer and telecommunication networks. The chapter also introduces WTMM as the informative indicator to exploit the distinction of fractal dimen- sions on various parts of a given dataset. A novel approach based on the use of multifractal spectrum parameters is proposed for estimating queuing performance for the generalized multifractal traffic on the input of a buffering device. It is shown that the multifractal character of traffic has significant impact on queuing performance characteristics

Numerical Methods for Quasicrystals

Quasicrystals are one kind of space-filling structures. The traditional crystalline approximant method utilizes periodic structures to approximate quasicrystals. The errors of this approach come from two parts: the numerical discretization, and the approximate error of Simultaneous Diophantine Approximation which also determines the size of the domain necessary for accurate solution. As the approximate error decreases, the computational complexity grows rapidly, and moreover, the approximate error always exits unless the computational region is the full space. In this work we focus on the development of numerical method to compute quasicrystals with high accuracy. With the help of higher-dimensional reciprocal space, a new projection method is developed to compute quasicrystals. The approach enables us to calculate quasicrystals rather than crystalline approximants. Compared with the crystalline approximant method, the projection method overcomes the restrictions of the Simultaneous Diophantine Approximation, and can also use periodic boundary conditions conveniently. Meanwhile, the proposed method efficiently reduces the computational complexity through implementing in a unit cell and using pseudospectral method. For illustrative purpose we work with the Lifshitz-Petrich model, though our present algorithm will apply to more general systems including quasicrystals. We find that the projection method can maintain the rotational symmetry accurately. More significantly, the algorithm can calculate the free energy density to high precision.Comment: 27 pages, 8 figures, 6 table

Numerical Methods for Multilattices

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices (Tadmor et al, 1999). Another existing numerical approach to modeling multilattices is homogenization. In the present paper we review the existing numerical methods for multilattices and propose another concurrent macro-to-micro method in the numerical homogenization framework. We give a unified mathematical formulation of the new and the existing methods and show their equivalence. We then consider extensions of the proposed method to time-dependent problems and to random materials.Comment: 31 page

Models for TMDs and numerical methods

We study the connection between the quark orbital angular momentum and the pretzelosity transverse-momentum dependent parton distribution function. We discuss the origin of this relation in quark models, identifying as key ingredient for its validity the assumption of spherical symmetry for the nucleon in its rest frame. Finally we show that the individual quark contributions to the orbital angular momentum obtained from this relation can not be interpreted as the intrinsic contributions, but include the contribution from the transverse centre of momentum which cancels out only in the total orbital angular momentum.Comment: 43 pages, 8 figures; proceedings of International School of Physics "Enrico Fermi", Course CLXXX - "Three-dimensional Partonic Structure of the Nucleon", 28 June - 8 July 2011, Varenna (Italy

Numerical methods for multiscale inverse problems

We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the solutions. Here, $\epsilon$ indicates the smallest characteristic wavelength in the problem ($0<\epsilon\ll1$). In addition to the general difficulty of finding an inverse, the oscillatory nature of the forward problem creates an additional challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed and one common approach is to replace the original problem with an effective parameter estimation problem. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, $b = 0$, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, $b > 0$, and exploration seismology, $b < 0$