Hadamard Finite Part Research Articles

Efficiently parallelized modeling of tightly focused, large bandwidth laser pulses

The Stratton–Chu integral representation of electromagnetic fields is used to study the spatio-temporal properties of large bandwidth laser pulses focused by high numerical aperture mirrors. We review the formal aspects of the derivation of diffraction integrals from the Stratton–Chu representation and discuss the use of the Hadamard finite part in the derivation of the physical optics approximation. By analyzing the formulation we show that, for the specific case of a parabolic mirror, the integrands involved in the description of the reflected field near the focal spot do not possess the strong oscillations characteristic of diffraction integrals. Consequently, the integrals can be evaluated with simple and efficient quadrature methods rather than with specialized, more costly approaches. We report on the development of an efficiently parallelized algorithm that evaluates the Stratton–Chu diffraction integrals for incident fields of arbitrary temporal and spatial dependence. This method has the advantage that its input is the unfocused field coming from the laser chain, which is experimentally known with high accuracy. We use our method to show that the reflection of a linearly polarized Gaussian beam of femtosecond duration off a high numerical aperture parabolic mirror induces ellipticity in the dominant field components and generates strong longitudinal components. We also estimate that future high-power laser facilities may reach intensities of .

Erratum: “The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals” [J. Math. Phys. 57, 033502 (2016)

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Simple Method for Evaluating Singular Integrals

In this paper, we develop a method for evaluating one dimensional singular integrals (weakly, strongly, and hyper-singular) that converge in the sense of Cauchy principal value and Hadamard finite part integrals. A proof of convergence of this method is also provided.

High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(tau ^{3-alpha } + h^2), 0<alpha <1 are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where tau and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.

Computation of the Madelung constant for hypercubic crystal structures in any dimension

A new method of computing the Madelung constants for hypercubic crystal structures in any dimension $$n\ge 2$$ is given. It is shown for $$n\ge 3$$ that the Madelung constant may be obtained in a simple, efficient and unambiguous way as the Hadamard finite part of the integral representation of the potential within the crystal which is divergent at any point charge location. Such a regularization method fails in the bidimensional case due to the logarithmic nature of singularities for the potential. In that case, a specific approach is proposed taking in account the scale invariance of the Poisson equation and the existence of a finite horizon for each point charge in the plane. Since a closed-form exact solution for the 2D electrostatic potential may be derived, one shows that the Madelung constant may be defined via an appropriate limit calculation as the mean value of potential energies of charges composing the unit cell.

The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals.

Regularization and derivatives of multipole potentials

The only harmonic homogeneous functions defined in Rn∖{0} are the harmonic polynomials and the so-called multipole potentials, namely functions of the type P(x)=p(x)/|x|2k+n−2 for some harmonic polynomial p of degree k. The first aim of this article is to study the distributional regularization of multipole potentials. We show that even though the Hadamard regularization Pf(p(x)/|x|2k+n−2) exists for any homogeneous polynomial of degree k, the principal value p.v.(p(x)/|x|2k+n−2) exists if and only if p is harmonic; this means that if p is harmonic then for any test function ϕ the divergent integral ∫Rnp(x)ϕ(x)/|x|2k+n−2dx can be computed by employing polar coordinates and performing the angular integral first. We also find the first and second order distributional derivatives of these regularizations and, more generally, of the regularizations of functions of the form Pl(x)=p(x)/|x|k+l. We find many interesting formulas that hold precisely when p is a harmonic polynomial of degree k. In particular, we prove thatΔ‾p.v.(p(x)r2k+n−2)=(−1)k+1πn/22k−2Γ(n2+k−1)p(∇)δ(x), generalizing the well known relation Δ‾(r2−n)=(2−n)Cδ(x), where C is the area of a sphere of radius 1. Actually formulas like this one hold for a homogeneous polynomial p of degree k if and only if p is harmonic.

Error estimates of a high order numerical method for solving linear fractional differential equations

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O(Δt2−α),0<α<1, where α is the order of the fractional derivative and Δt is the step size. We then use a similar idea to prove the error estimates of the high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(Δt3−α),0<α<1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals

The divergent integral ∫abf(x)(x−x0)−n−1dx, for −∞ &amp;lt; a &amp;lt; x0 &amp;lt; b &amp;lt; ∞ and n = 0, 1, 2, …, is assigned, under certain conditions, the value equal to the simple average of the contour integrals ∫C±f(z)(z − x0)−n−1dz, where C+ (C−) is a path that starts from a and ends at b and which passes above (below) the pole at x0. It is shown that this value, which we refer to as the analytic principal value, is equal to the Cauchy principal value for n = 0 and to the Hadamard finite-part of the divergent integral for positive integer n. This implies that, where the conditions apply, the Cauchy principal value and the Hadamard finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox theorem with integrals along some arbitrary paths. The utility of the analytic principal value in the numerical, analytical, and asymptotic evaluations of the principal value and the finite-part integral is discussed and demonstrated.

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Abstract We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order O ( Δ x 3 - α ) ${O(\Delta x^{3- \alpha })}$ , 1 &lt; α &lt; 2 ${1&amp;lt;\alpha &amp;lt;2}$ . A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O ( Δ t + Δ x min ( 3 - α , β ) ) ${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$ , 1 &lt; α &lt; 2 ${1&amp;lt; \alpha &amp;lt;2}$ , β &gt; 0 ${\beta &amp;gt;0}$ , where Δ t , Δ x ${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

Extrapolations for hypersingular integrals

By using the mildpoint trapezoidal rule a generalized Euler-Maclaurin expansion for computing Hadamard finite part values of the hypersingular integrals with a singular point on the inteval interior is shown, where there are some negative exponential terms of the step size in the expansion. It means that the rule is divergent. However, by means of succession Richandson extrapolations a high accuracy of Hadamard finite part values of the hypersingular integrals is obtained.

3D electrostatic charge distribution on finitely thick busbars in micro-acoustic devices: combined regularization in the near- and far-field

The original work for 3-D charge distributions in micro-acoustic devices has been manifestly extended to account for finitely thick busbars. The work has been initiated to create a platform for simulating the electric charge localization and field enhancement at the electrode/busbar gaps depending on the thickness of the metalization in submicrometer geometries. A recipe for the construction of relevant Green's functions has been provided. A universal function (UF) for setting up system matrices in the method-of-moments' implementations has been constructed. Universal functions (moments of Green's functions) are by construction highly smooth and easy to compute. This work also presents a comprehensive completion of earlier work. For the first time, the calculation of the UF for a 3-D problem has been presented in great detail, highlighting the underlying regularization techniques. It is shown that the singular Fourier-type integrals involved can be regularized simultaneously in the near- and far-field. The pinnacle of the work is the detailed demonstration of the property that Hadamard's finite part regularization naturally arises in the construction of UFs. Three lemmata facilitate the understanding of the underlying concepts.

Numerical quadrature for computing of singular integrals

AbstractThe paper is concerned with the superconvergence of numerical evaluation of Hadamard finite-part integral. Following the works [6-9], we studied the second-order and the third-order quadrature formulae of Newton-Cotes type and introduced new rules. The rule for the second-order gives the same convergence rate as the rule [6] but in more general cases, the rule for the third-order gives better results than the rule in [9] In this work, first we mention the main results on the superconvergence of the Newton-Cotes rules, we mention trapezoidal and Simpson’s rules and then we introduce a rule based on the cubic approximation. In the second part we describe important error estimates and in the last section we demonstrate theoretical results by numerical examples.

Gauss rules associated with nearly singular weights

We consider the problem of evaluating ∫−11f(x)G(x)(1−x2)−1/2dx, when f is smooth and G is nearly singular and non-negative. For this we construct a Gauss quadrature formula w.r.t. the weight G(x)(1−x2)−1/2. Once the factor G has been chosen, the procedure is relatively simple and mainly involves the application of FFT to compute a finite number of coefficients of the Chebyshev series expansion of G which in turn are used to calculate modified moments.It is shown that this approach is very effective when the complexity of f is high, or when f is parametric and the integral must be calculated for many values of the parameters. For this, there is presented a selection of numerical examples which allows comparison with other methods. In particular, there is considered the evaluation of Hadamard finite part integrals when the regular part of the integrand is nearly singular.

Foundational aspects of singular integrals

We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-L1 element in the class, and the Hadamard finite part integral (p.f.) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs.The Lebesgue integral as well as p.f., starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if f=g on (0,a), then their integrals from 0 to x coincide for any x∈(0,a).We find that integrals from zero with the essential properties of p.f., plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on (0,1] which are L1((ε,1]) for all ε>0. However, this existence proof does not provide a satisfactory construction. Without some regularity at 0, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-L1 element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator.Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze p.f. on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.

A new method for numerical evaluation of nearly singular integrals over high-order geometry elements in 3D BEM

This work presents a new method for numerical computation of the two-dimensional nearly singular integrals by using the eight-node second-order quadrilateral surface elements in 3D BEM. A new indirect regularized boundary element formulation excluding the CPV (Cauchy Principal Value) and HFP (Hadamard-Finite-Part) integrals is proposed. Based on this, a new approximation formula of the distance from the fixed calculation point to a generic point of the aforementioned surface geometry elements is developed firstly, and then the exponential transformation, which has been widely employed in 2D BEM, is extended to 3D BEM to remove the near singularities of integrands for considered integrals. Several numerical examples are given to verify the high efficiency and the stability of the proposed scheme.

Analysis of errors in some recent numerical quadrature formulas for periodic singular and hypersingular integrals via regularization

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the singular integrals I(1)[u]=∫ab(cotπ(x−t)T)u(x)dx and I(2)[u]=∫ab(csc2π(x−t)T)u(x)dx, with b−a=T and u(x) a T-periodic continuous function on R. These integrals are not defined in the regular sense, but are defined in the sense of Cauchy Principal Value and Hadamard Finite Part, respectively. With h=(b−a)/n, n=1,2,…, the numerical quadrature formulas Qn(1)[u] for I(1)[u] and Qn(2)[u] for I(2)[u] areQn(1)[u]=h∑j=1nf(t+jh−h/2),f(x)=(cotπ(x−t)T)u(x), andQn(2)[u]=h∑j=1nf(t+jh−h/2)−T2u(t)h−1,f(x)=(csc2π(x−t)T)u(x). We provided a complete analysis of the errors in these formulas under the assumption that u∈C∞(R) and is T-periodic. We actually showed that,I(1)[u]−Qn(1)[u]=O(n−μ)andI(2)[u]−Qn(2)[u]=O(n−μ)as n→∞,∀μ>0. In this note, we analyze the errors in these formulas under the weaker assumption that u∈Cs(R) for some finite integer s. By first regularizing these integrals, we prove that, if u(s+1) is piecewise continuous, thenI(1)[u]−Qn(1)[u]=o(n−s−1/2)as n→∞, if s⩾1,andI(2)[u]−Qn(2)[u]=o(n−s+1/2)as n→∞, if s⩾2. We also extend these results by imposing different smoothness conditions on u(s+1). Finally, we append suitable numerical examples.

Quadrature rules for evaluation of hyper singular integrals

Some Newton-Cote’s type of quadrature rules have been formulated for the numerical evaluation of hyper singular integrals, which are interpreted as of Hadamard finite part type. Rules have been tested numerically by some standard test integrals and their respective error bounds have been determined. Mathematics Subject Classification: 65D30; 65D32

Hierarchical quadrature for multidimensional singular integrals – Part II

In a previous part I, we introduced a method for the evaluation of singular integrals arising in the discretization of integral equations based on the repeated subdivision of domains. The integrals defined on these subdomains are classified such that a class of integrals can be expressed as a sum of regular integrals and representatives of other classes. A system of equations describes the relations between the classes. Furthermore the approximate value of the singular integrals only depends on the accuracy of the calculation of regular integrals. Part I left a gap on certain parameter configurations on which the mentioned system of equations is irregular. This paper shall close this gap. To this end, we introduce an alternative splitting strategy based on a modified Hadamard partie finie integral.

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals $$I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx$$ I ( 1 ) [ g ] = ? a b g ( x ) x - t d x and $$I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx$$ I ( 2 ) [ g ] = ? a b g ( x ) ( x - t ) 2 d x . These integrals are not defined in the regular sense; $$I^{(1)}[g]$$ I ( 1 ) [ g ] is defined in the sense of Cauchy Principal Value while $$I^{(2)}[g]$$ I ( 2 ) [ g ] is defined in the sense of Hadamard Finite Part. With $$h=(b-a)/n, \,n=1,2,\ldots $$ h = ( b - a ) / n , n = 1 , 2 , ? , and $$t=a+kh$$ t = a + k h for some $$k\in \{1,\ldots ,n-1\}, \,t$$ k ? { 1 , ? , n - 1 } , t being fixed, the numerical quadrature formulas $${Q}^{(1)}_n[g]$$ Q n ( 1 ) [ g ] for $$I^{(1)}[g]$$ I ( 1 ) [ g ] and $$Q^{(2)}_n[g]$$ Q n ( 2 ) [ g ] for $$I^{(2)}[g]$$ I ( 2 ) [ g ] are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ Q n ( 1 ) [ g ] = h ? j = 1 n f ( a + j h - h / 2 ) , f ( x ) = g ( x ) x - t , and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ Q n ( 2 ) [ g ] = h ? j = 1 n f ( a + j h - h / 2 ) - ? 2 g ( t ) h - 1 , f ( x ) = g ( x ) ( x - t ) 2 . We provided a complete analysis of the errors in these formulas under the assumption that $$g\in C^\infty [a,b]$$ g ? C ? [ a , b ] . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ I ( k ) [ g ] - Q n ( k ) [ g ] ~ ? i = 1 ? c i ( k ) h 2 i as n ? ? , the constants $$c^{(k)}_i$$ c i ( k ) being independent of $$h$$ h . In this work, we apply the Richardson extrapolation to $${Q}^{(k)}_n[g]$$ Q n ( k ) [ g ] to obtain approximations of very high accuracy to $$I^{(k)}[g]$$ I ( k ) [ g ] . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function $$g(x)$$ g ( x ) , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.