Retrieval of Single-Doppler Radar Wind Field by Nonlinear Approximation

The methods employed in recent years to retrieve vector wind information from single-Doppler radar observation are reviewed briefly. These methods are based on a linearity hypothesis for the wind field, so the retrieved wind field is sometimes negatively affected by the non-linearity of wind. This paper proposes a new method based on a non-linear approximation technique. This method, which relies on the piecewise smooth property of the wind field and makes full use of the radar velocity data, is applied to two cases of the Huaihe River Basin Energy and Water Cycle Experiment (HUBEX) in 1998. Checked against the wind field observed by dual-Doppler radar, the retrieved wind field by the method presented in this paper yields a relatively accurate horizontal vector wind field with high resolution, as well as a reasonable estimate of the magnitude of vertical velocity.

Nonlinear approximation of image based on monoscale orthonormal ridgelets frame

A new tight frame called as monoscale orthonormal ridgelet frame （MORF） is proposed. The localization principle and the orthonormal ridgelet constructed by Donoho are applied to construct the MORF, which are used to evaluate the order of nonlinear approximation for image with edge. Because the new tight frame not only has directionality but also bears orthonormality. It overcomes redundancy of Candes＇s monoscale ridgelets and provides many excellent properties in practical application. Theoretical analysis and experiments demonstrate that the new frame has remarkable potential for image compression, image reconstruction, and image denoising with the simple refinement for MORF.

NONLINEAR APPROXIMATION OF A VARIATION OF L^1 UNDER THE LOCAL HAAR CONDITION

In this paper, we discuss the nonlinear approximation of a variation of L1 under the local Haar condition. Theorems on characterizations (including alternations), unicity and strong unicity of the nonlinear approximation are obtained, the associated linear problem equivalent to the nonlinear approximation are given.

NONLINEAR APPROXIMATION WITH GENERAL WAVE PACKETS

We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete characterization of the approximation spaces is derived.

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems , which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besom spaces B(?)(0,1) with 0<σ<∞ and (1+σ)-1<γ<∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.

Adaptive Algorithms of Nonlinear Approximation with Finite Terms

This paper deals with realizable adaptive algorithms of the nonlinear approximation with finite terms based on wavelets. We present a concrete algorithm by which we may find the required index set Am for the greedy algorithm Gm^P（., Ψ）. This makes the greedy algorithm realize the near best approximation in practice. Moreover, we study the efficiency of the finite-term approximation of another Mgorithm introduced by Birge and Massart.

SOME NONLINEAR APPROXIMATIONS FOR MATRIX-VALUED FUNCTIONS

Some nonlinear approximants, i.e., exponential-sum interpolation with equal distance or at origin, (0,1)-type, (0,2)-type and (1,2)-type fraction-sum approximations, for matrix-valued functions are introduced. All these approximation problems lead to a same form system of nonlinear equations. Solving methods for the nonlinear system are discussed. Conclusions on uniqueness and convergence of the approximants for certain class of functions are given.

Convergence Properties of Piecewise Power Approximations

We address the problem of convergence of approximations obtained from two versions of the piecewise power-law representations arisen in Systems Biology. The most important cases of mean-square and uniform convergence are studied in detail. Advantages and drawbacks of the representations as well as properties of both kinds of convergence are discussed. Numerical approximation algorithms related to piecewise power-law representations are described in Appendix.

Entropy generation analysis of tangent hyperbolic fluid in quadratic Boussinesq approximation using spectral quasi-linearization method

In many industrial applications,heat transfer and tangent hyperbolic fluid flow processes have been garnering increasing attention,owing to their immense importance in technology,engineering,and science.These processes are relevant for polymer solutions,porous industrial materials,ceramic processing,oil recovery,and fluid beds.The present tangent hyperbolic fluid flow and heat transfer model accurately predicts the shear-thinning phenomenon and describes the blood flow characteristics.Therefore,the entropy production analysis of a non-Newtonian tangent hyperbolic material flow through a vertical microchannel with a quadratic density temperature fluctuation(quadratic/nonlinear Boussinesq approximation)is performed in the present study.The impacts of the hydrodynamic flow and Newton’s thermal conditions on the flow,heat transfer,and entropy generation are analyzed.The governing nonlinear equations are solved with the spectral quasi-linearization method(SQLM).The obtained results are compared with those calculated with a finite element method and the bvp4c routine.In addition,the effects of key parameters on the velocity of the hyperbolic tangent material,the entropy generation,the temperature,and the Nusselt number are discussed.The entropy generation increases with the buoyancy force,the pressure gradient factor,the non-linear convection,and the Eckert number.The non-Newtonian fluid factor improves the magnitude of the velocity field.The power-law index of the hyperbolic fluid and the Weissenberg number are found to be favorable for increasing the temperature field.The buoyancy force caused by the nonlinear change in the fluid density versus temperature improves the thermal energy of the system.

ON NONLINEAR GALERKIN APPROXIMATION

Nonlinear Galerkin methods are numerical schemes adapted well to the long time integration of evolution partial differential equations. The aim of this paper is to discuss such schemes for reaction diffusion equations. The convergence results are proved.

Self-similar solutions to Lin-Reissner-Tsien equation

The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation. In the present paper, the equation is reduced to an ordinary differential equation via a similarity transformation. The resulting equation is then solved analytically and even exactly in some cases. Numerical simulations are provided for the cases in which there is no exact solution. Travelling wave solutions are also obtained.

APPROXIMATE INERTIAL MANIFOLDS UNDERNONSELFADJOINT

This paper sets up the approximate inertias manifold(AIM) in the nouselfadjoint nonlinear evolutionary equation and Ands AIMs which are explitly dafined in the weally damped forced KdV equation (WDF KdV).

Approximating Nonlinear Relations Between Susceptibility and Magnetic Contents in Rocks Using Neural Networks

Correlations between magnetic susceptibility and contents of magnetic minerals in rocks are important in interpreting magnetic anomalies in geophysical exploration and understanding magnetic behaviors of rocks in rock magnetism studies. Previous studies were focused on describing such correlations using a sole expression or a set of expressions through statistical analysis. In this paper, we use neural network techniques to approximate the nonlinear relations between susceptibility and magnetite and/or hematite contents in rocks. This is the first time that neural networks are used for such study in rock magnetism and magnetic petrophysics. Three multilayer perceptrons are trained for producing the best possible estimation on susceptibility based on magnetic contents. These trained models are capable of producing accurate mappings between susceptibility and magnetite and/or hematite contents in rocks. This approach opens a new way of quantitative simulation using neural networks in rock magnetism and petrophysical research and applications.

Ways to sparse representation:An overview

Many algorithms have been proposed to find sparse representations over redundant dictionaries or transforms. This paper gives an overview of these algorithms by classifying them into three categories： greedy pursuit algorithms, lp norm regularization based algorithms, and iterative shrinkage algorithms. We summarize their pros and cons as well as their connections. Based on recent evidence, we conclude that the algorithms of the three categories share the same root： lp norm regularized inverse problem. Finally, several topics that deserve further investigation are also discussed.

Ways to Sparse Representation: A Comparative Study

Many algorithms have been proposed to achieve sparse representation over redundant dictionaries or transforms. A comprehensive understanding of these algorithms is needed when choosing and designing algorithms for particular applications. This research studies a representative algorithm for each category, matching pursuit （MP）, basis pursuit （BP）, and noise shaping （NS）, in terms of their sparsifying capability and computational complexity. Experiments show that NS has the best performance in terms of sparsifying ca- pability with the least computational complexity. BP has good sparsifying capability, but is computationally expensive. MP has relatively poor sparsifying capability and the computations are heavily dependent on the problem scale and signal complexity. Their performance differences are also evaluated for three typical ap- plications of time-frequency analyses, signal denoising, and image coding. NS has good performance for time-frequency analyses and image coding with far fewer computations. However, NS does not perform well for signal denoising. This study provides guidelines for choosing an algorithm for a given problem and for designing or improving algorithms for sparse representation.

FINITE ELEMENT EXTERIOR CALCULUS FOR EVOLUTION PROBLEMS

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 （2010）, 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus （FEEC） for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12：3 （2012）, 263 293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian man- ifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time- parameterized Hilbert spaces （or Bochner spaces）. Building on classical approaches by Thom^e for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomée and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their re- sults to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.