Peaks and jumps reconstruction with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.gif" display="inline" overflow="scroll"><mml:mi>B</mml:mi></mml:math>-splines scaling functions

Ortiz-Gracia, Luis; Masdemont, Josep J.

doi:10.1016/j.cam.2014.05.015

Cited by 13 publications

(9 citation statements)

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“…The linear B-spline functions [ 58 ] can be combined with other applications, e.g., Haar function when BLac-wavelet is accomplished [ 59 ], two operators (smoothing and error) were applied for computing the coarse mesh and to determine the difference between the approximation and the original meshes [ 60 ]. B-spline with Haar scaling functions of 0, 1st, or 2nd order can be quite invaluable when round off errors (also called rounding errors) are minimized [ 61 ]. An increase in filtration time might be one of the undue complications.…”

Section: Introductionmentioning

confidence: 99%

Suppression of the High-Frequency Errors in Surface Topography Measurements Based on Comparison of Various Spline Filtering Methods

Podulka

2021

Materials

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The metrology of so-called “engineering surfaces” is burdened with a substantial risk of both measurement and data analysis errors. One of the most encouraging issues is the definition of frequency-defined measurement errors. This paper proposes a new method for the suppression and reduction of high-frequency measurement errors from the surface topography data. This technique is based on comparisons of alternative types of noise detection procedures with the examination of profile (2D) or surface (3D) details for both measured and modelled surface topography data. In this paper, the results of applying various spline filters used for suppressions of measurement noise were compared with regard to several kinds of surface textures. For the purpose of the article, the influence of proposed approaches on the values of surface topography parameters (from ISO 25178 for areal and ISO 4287 for profile standards) was also performed. The effect of the distribution of some features of surface texture on the results of suppressions of high-frequency measurement noise was also closely studied. Therefore, the surface topography analysis with Power Spectral Density, Autocorrelation Function, and novel approaches based on the spline modifications or studies of the shape of an Autocorrelation Function was presented.

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Section: Introductionmentioning

confidence: 99%

Suppression of the High-Frequency Errors in Surface Topography Measurements Based on Comparison of Various Spline Filtering Methods

Podulka

2021

Materials

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“…If we integrate by parts, We aim at recovering F c from its Fourier transformF c . For this purpose, we use the metod initially developed in [12] for Laplace transform inversion and further extended in [15] for Fourier transform inversion, where numerical errors are studied in detail as well. It is based on Haar wavelets (for a deep insight in wavelets we refer the reader to [4]) and called WA.…”

Section: Inversion Methodsmentioning

confidence: 99%

“…The evaluation of the characteristic function (9) in Section 2.2 in a certain point w, which is a particular case of (15) for d = 1, involves the computation of a double integral that we solve efficiently in Section 3.2.1 by means of numerical quadrature. However, looking at the expressions of the characteristic functions (8) and (15), corresponding to the multi-factor Gaussian and tcopula models, we realize that a direct attempt of solving the d-and (d + 1)-dimensional integrals, respectively, at fixed points w is not affordable with numerical integration. For these challenging tasks, we rely on the QTA put forward in [10] for computing the Laplace transform of the portfolio loss within the multi-factor Gaussian copula model.…”

Section: Efficient Computation Of Characteristic Functionsmentioning

confidence: 99%

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Quantifying credit portfolio losses under multi-factor models

Colldeforns-Papiol

Ortiz-Gracia

Oosterlee

2018

International Journal of Computer Mathematics

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In this work, we investigate the challenging problem of estimating credit risk measures of portfolios with exposure concentration under the multi-factor Gaussian and multi-factor t-copula models. It is well-known that Monte Carlo (MC) methods are highly demanding from the computational point of view in the aforementioned situations. We present efficient and robust numerical techniques based on the Haar wavelets theory for recovering the cumulative distribution function (CDF) of the loss variable from its characteristic function. To the best of our knowledge, this is the first time that multi-factor t-copula models are considered outside the MC framework. The analysis of the approximation error and the results obtained in the numerical experiments section show a reliable and useful machinery for credit risk capital measurement purposes in line with Pillar II of the Basel Accords.

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“…This method was originally developed within a credit risk environment to recover a CDF on a bounded domain from its Laplace transform by means of a Haar basis (see [Mas11]). The method was extended in [Ort12b] to invert Fourier transforms over the entire real line with B-splines up to order one. Later on, it was applied to an option pricing problem in [Ort13] and it was called WA [a,b] method.…”

Section: Introductionmentioning

confidence: 99%

Efficient Wavelets-Based Valuation of Synthetic CDO Tranches

Ortiz-Gracia

2014

SSRN Journal

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We present new formulae for the valuation of synthetic collateralized debt obligation (CDO) tranches under the one-factor Gaussian copula model. These formulae are based on the wavelet theory and the method used is called WA [a,b] . We approximate the cumulative distribution function (CDF) of the underlying pool by a finite combination of jth order B-spline basis, where the B-spline basis of order zero is typically called a Haar basis. We provide an error analysis and we show that for this type of distributions, the accuracy in the approximation is the same regardless of the order of the B-spline basis employed. The resulting formula for the Haar basis case is much easier to implement and performs better than the formula for the B-spline basis of order one in terms of computational time. The numerical experiments confirm the impressive speed and accuracy of the WA [a,b] method equipped with a Haar basis, independently of the inhomogeneity features of the underlying pool. The method appears to be particularly fast for multiple tranche valuation.

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Peaks and jumps reconstruction withB-splines scaling functions

Cited by 13 publications

References 14 publications

Suppression of the High-Frequency Errors in Surface Topography Measurements Based on Comparison of Various Spline Filtering Methods

Suppression of the High-Frequency Errors in Surface Topography Measurements Based on Comparison of Various Spline Filtering Methods

Quantifying credit portfolio losses under multi-factor models

Efficient Wavelets-Based Valuation of Synthetic CDO Tranches

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