A fully discrete Galerkin method for Abel-type integral equations

Vögeli, Urs; Nedaiasl, Khadijeh; Sauter, Stéfan A.

doi:10.1007/s10444-018-9598-4

Cited by 13 publications

(13 citation statements)

References 13 publications

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“…Galerkin methods for Abel-type integral equations are considered, e.g., in Eggermont [8] and in Vögeli, Nedaiasl and Sauter [27]. Some general references are already given in the beginning of this paper.…”

Section: The Main Resultsmentioning

confidence: 99%

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

Plato

2018

Trends in Mathematics

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“…Here the first term on the left hand can be estimated from below by means of coercivity of the Abel integral operator [16] t…”

Section: 20) With (21analysis Of the Forward Problemmentioning

confidence: 99%

Determining the nonlinearity in an acoustic wave equation

Kaltenbacher,

Rundell

2021

Preprint

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We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form pp t . However, this should be considered as a low order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f (p) p t and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consists of time trace observations of the acoustic pressure at a single point or on a one dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of wellposedness of the resulting PDE, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f . Numerical simulations will also be shown to illustrate the efficiency of the methods.

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“…Lemma 3. Let N and ∆t > 0 be arbitrary, and let ρ n (N ; ∆t) be the spectral radius of W n (N ; ∆t) defined in (42). Then for all n = 0,…”

Section: Tighter Boundsmentioning

confidence: 99%

“…For previous work on Abel-type equations with singular kernels, we refer the reader to [6,21,22,42]. These papers, however, are mostly concerned with implicit marching schemes.…”

Section: The Robin Problem In One Dimensionmentioning

confidence: 99%

Explicit unconditionally stable methods for the heat equation via potential theory

Barnett

Epstein

Greengard

et al. 2019

Pure Appl. Analysis

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We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step ∆t is of the order O(∆x 2 ), where ∆x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L 2 -norm of the solution to the integral equation is bounded by c d T d/2 times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if ∆t < C/κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in L ∞ -norm.

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A fully discrete Galerkin method for Abel-type integral equations

Cited by 13 publications

References 13 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

Determining the nonlinearity in an acoustic wave equation

Explicit unconditionally stable methods for the heat equation via potential theory

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