A novel sparse feature extraction method based on sparse signal via dual-channel self-adaptive TQWT

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix–vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension.

On a Sparse Representation of an n -Dimensional Laplacian in Wavelet Coordinates

Galerkin method with new quadratic spline wavelets for integral and integro-differential equations

The Black-Scholes equation is used in financial mathematics for computation of market values of options at a given time. We use the θ–scheme for time discretization and an adaptive scheme based on wavelets for discretization on the given time level. Advantages of the proposed method are small number of degrees of freedom, high-order accuracy with respect to variables representing prices and relatively small number of iterations needed to resolve the problem with a desired accuracy. We use several cubic spline wavelet and multi-wavelet bases and discuss their advantages and disadvantages. We also compare an isotropic and anisotropic approach. Numerical experiments are presented for the two-dimensional Black-Scholes equation.

Numerical solution of the Black-Scholes equation using cubic spline wavelets

We revisit the construction of wavelets on the interval with various degrees of polynomial exactness, and explain how existing schemes for orthogonal- and Spline wavelets can be extended to compactly supported delay-normalized wavelets. The contribution differs substantially from previous ones in how results are stated and deduced: linear algebra notation is exploited more heavily, and the use of sums and complicated index notation is reduced. This extended use of linear algebra eases translation to software, and a general open source implementation, which uses the deductions in this paper as a reference, has been developed. Key features of this implementation is its flexibility w.r.t. the length of the input, as well as its generality regarding the wavelet transform.

/pdf/on-the-unification-of-schemes-and-software-for-wavelets-on-12ovzh84x1.pdf

On the Unification of Schemes and Software for Wavelets on the Interval

In 1989, R. Coifman suggested the design of orthonormal wavelet systems with vanishing moments for both scaling and wavelet functions. They were first constructed by I. Daubechies [15, 16], and she named them coiflets. In this paper, we propose a system of necessary conditions which is redundant free and simpler than the known system due to the elimination of some quadratic conditions, thus the construction of coiflets is simplified and enables us to find the exact values of the scaling coefficients of coiflets up to length 8 and two further with length 12. Furthermore for scaling coefficients of coiflets up to length 14 we obtain two quadratic equations, which can be transformed into a polynomial of degree 4 for which there is an algebraic formula to solve them.

On the exact values of coefficients of coiflets

The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions The w

https://www.worldscientific.com/doi/pdf/10.1142/S0219691318500613

Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model

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