Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions

Černá, Dana; Finěk, Václav

doi:10.1142/s0219691315500149

Cited by 10 publications

(11 citation statements)

References 18 publications

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“…Some results can be found in [28]. In the case of cubic spline wavelets, the smallest number of iterations was required for the wavelet basis from [36]. The discretization matrix for most spline wavelet bases is not sparse, but only quasi-sparse, and thus, the above-mentioned routine APPLY has to be used.…”

Section: Examplementioning

confidence: 99%

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

Černá

Finěk

2017

Axioms

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Abstract:We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black-Scholes equation with a quadratic volatility.

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Section: Examplementioning

confidence: 99%

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

Černá

Finěk

2017

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“…For more details about constructions of well-conditioned wavelet bases in the sense of the above definition, we refer to [15,16,17]. A detail comparison of efficiency of numerical solution of the Black-Scholes equation in two dimensions for different cubic spline wavelet basis as well as a comparison of isotropic and anisotropic approach was performed in [6].…”

Section: Waveletsmentioning

confidence: 99%

Fourth order scheme for wavelet based solution of Black-Scholes equation

Finěk¹

2017

AIP Conference Proceedings

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Abstract. The present paper is devoted to the numerical solution of the Black-Scholes equation for pricing European options. We apply the Crank-Nicolson scheme with Richardson extrapolation for time discretization and Hermite cubic spline wavelets with four vanishing moments for space discretization. This scheme is the fourth order accurate both in time and in space. Computational results indicate that the Crank-Nicolson scheme with Richardson extrapolation significantly decreases the amount of computational work. We also numerically show that optimal convergence rate for the used scheme is obtained without using startup procedure despite the data irregularities in the model.

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“…Luego se consideraron solo aquellas traslaciones y dilaciones contenidas en el intervalo llamadas wavelets interiores, y se analizaron diferentes clases de wavelets de borde. En primer lugar, se utilizaron las definidas por Černá y col. [24] y luego, se diseñaron wavelets de borde con mejores resultados de convergencia imponiendo condiciones de ortogonalidad sobre sus derivadas. Esto constituye uno de los aportes más importantes de esta tesis.…”

Section: Desarrollo De La Investigaciónunclassified

“…En la revisión bibliográfica correspondiente se encuentran diversas estrategias para adaptar un AMR de L 2 (R) al intervalo ([14]- [24], [28], [36], [43]). A continuación se describen algunas de las más importantes.…”

Section: Amr En L 2 [0 1]unclassified

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Utilización de bases wavelet para la resolución numérica de ecuaciones diferenciales

Calderon¹

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En este trabajo de tesis se estudiaron estrategias matemáticas orientadas en particular, a la resolución numérica de operadores diferenciales del tipo Lu = f mediante el método Wavelet-Galerkin (W-G). El marco proporcionado por el Análisis Multirresolución (AMR) permitió analizar y formalizar el diseño de una técnica numérica con cuya implementación se lograron óptimos resultados. La estructura AMR de las B-splines cúbicas sugiere la definición de una base sobre intervalo con un requerimiento extra de ortogonalidad sobre sus derivadas entre distintas escalas de aproximación. Se desarrolló en esta tesis un método para la confección de esta base, teniendo en cuenta su estructura sobre intervalo y estudiando en particular las wavelets de borde y sus específicas condiciones de suavidad y soporte. Al aplicar W-G esta particular estrategia conduce a matrices de rigidez ralas o esparcidas, o bien diagonales por bloques, donde además cada bloque suele ser una matriz banda. Así estas propiedades garantizan matrices bien condicionadas, con número de condición pequeño, acotado e independiente del nivel de aproximación. Esto permite definir algoritmos de alta eficiencia computacional logrando resultados con elevada convergencia a la solución y bajo costo computacional.

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Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions

Cited by 10 publications

References 18 publications

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

Fourth order scheme for wavelet based solution of Black-Scholes equation

Utilización de bases wavelet para la resolución numérica de ecuaciones diferenciales

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