The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind

Plato, Robert

doi:10.1007/s10444-011-9217-0

Cited by 13 publications

(10 citation statements)

References 20 publications

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“…Remark 3.6 (a) We mention some results on other quadrature schemes for the approximate solution of Abel-type integral equations of the first kind. The product trapezoidal method is considered, e.g., in Weiss [28], Eggermont [8], and in [21]. Fractional multistep methods are treated in Lubich [18,19] and in [20].…”

Section: The Main Resultsmentioning

confidence: 99%

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The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data

Plato

2018

Trends in Mathematics

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In the present paper we consider the regularizing properties of the product midpoint rule for the stable solution of Abel-type Volterra integral equations of the first kind with perturbed right-hand sides. The Hölder continuity of the solution and its derivative is carefully taken into account, and correction weights are considered to get rid of initial conditions. The proof of the inverse stability of the quadrature weights relies on Banach algebra techniques. Finally, numerical results are presented.

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Section: The Main Resultsmentioning

confidence: 99%

“…The following lemma is closely related to results in Erdős, Feller and Pollard [10]. A detailed proof can be found in [21]. Lemma 6.9 Let c 1 , c 2 , .…”

Section: The Proof Of the Lower Bound (614)mentioning

confidence: 91%

The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data

Plato

2018

Trends in Mathematics

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“…✷ Remark 3. 8 The stability analysis presented in the third part of the proof of Theorem 3.7 uses techniques similar to those used in Eggermont [5]; see also Lubich [18] as well as [22] and [23].…”

Section: The Main Resultsmentioning

confidence: 99%

The Regularizing Properties of Multistep Methods for First Kind Volterra Integral Equations with Smooth Kernels

Plato

2016

Computational Methods in Applied Mathematics

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We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a version which sometimes requires less amount of computational work than the standard version of this principle. Numerical results are included.

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“…The main issue in the reconstruction procedure consists in solving the integral equation (2.8). For that purpose we use the product integration method with the mid-point rule [23,29,40], that is presented in the following subsection.…”

Section: Basic Reconstruction Strategymentioning

confidence: 99%

Inversion of the Attenuated V-Line Transform With Vertices on the Circle

Haltmeier

Moon

Schiefeneder

2017

IEEE Trans. Comput. Imaging

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The Compton camera is a promising alternative to the Anger camera for imaging gamma radiation, with the potential to significantly increase the sensitivity of SPECT. Two-dimensional Compton camera image reconstruction can be implemented by inversion of the V-line transform, which integrates the emission distribution over V-lines (unions of two half-lines), that have vertices on a surrounding detector array. Inversion of the V-line transform without attenuation has recently been addressed by several authors. However, it is well known from standard SPECT that ignoring attenuation can significantly degrade the quality of the reconstructed image. In this paper we address this issue and study the attenuated V-line transform accounting for attenuation of photons in SPECT with Compton cameras. We derive an analytic inversion approach based on circular harmonics expansion, and show uniqueness of reconstruction for the attenuated V-line transform. We further develop a discrete image reconstruction algorithm based on our analytic studies, and present numerical results that demonstrate the effectiveness of our algorithm.

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The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind

Cited by 13 publications

References 20 publications

The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data

The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data

The Regularizing Properties of Multistep Methods for First Kind Volterra Integral Equations with Smooth Kernels

Inversion of the Attenuated V-Line Transform With Vertices on the Circle

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