Hadamard Finite Part Research Articles

Asymptotic Expansions for Two-Dimensional Hypersingular Integrals

We define and examine two-dimensional hypersingular integrals on [0,1)2 and on [0,∞)2 and relate their Hadamard finite-part (HFP) values to Mellin transforms. These integrands have algebraic singularities of a possibly unintegrable nature on the axes and at the origin. Extending our work on one-dimensional integrals reported in 1998, we obtain variants of the classical Euler-Maclaurin expansion for various two-dimensional integrals. In many cases, the constant term in the expansion (which is not necessarily the leading term) provides the value of the HFP integral. These expansions may be used as the basis for the numerical evaluation of a class of HFP integrals by extrapolation.

Hadamard regularization of the third post-Newtonian gravitational wave generation of two point masses

Continuing previous work on the 3PN-accurate gravitational-wave generation from point-particle binaries, we obtain the binary's 3PN mass-type quadrupole and dipole moments for general (not necessarily circular) orbits in harmonic coordinates. The final expressions are given in terms of their core parts, resulting from the application of the pure-Hadamard-Schwartz self-field regularization scheme, and augmented by an ambiguous part. In the case of the 3PN quadrupole we find three ambiguity parameters, $\ensuremath{\xi}$, $\ensuremath{\kappa}$ and $\ensuremath{\zeta}$, but only one for the 3PN dipole, in the form of the particular combination $\ensuremath{\xi}+\ensuremath{\kappa}$. Requiring that the dipole moment agree with the center-of-mass position deduced from the 3PN equations of motion in harmonic coordinates yields the relation $\ensuremath{\xi}+\ensuremath{\kappa}=\ensuremath{-}9871/9240$. Our results will form the basis of the complete calculation of the 3PN radiation field of compact binaries by means of dimensional regularization.

Standard and geometric approaches to quantum Liouville theory on the pseudosphere

We compare the standard and geometric approaches to quantum Liouville theory on the pseudosphere by performing perturbative calculations of the one and two point functions up to the third order in the coupling constant. The choice of the Hadamard regularization within the geometric approach leads to a discrepancy with the standard approach. On the other hand, we find complete agreement between the results of the standard approach and the bootstrap conjectures for the one point function and the auxiliary two point function.

Equations of motion of compact binaries at the third post-Newtonian order

The equations of motion of two point masses in harmonic coordinates are derived through the third post-Newtonian (3PN) approximation. The problem of self-field regularization (necessary for removing the divergent self-field of point particles) is dealt with in two separate steps. In a first step the extended Hadamard regularization is applied, resulting in equations of motion which are complete at the 3PN order, except for the occurence of one and only one unknown parameter. In a second step the dimensional regularization (in d dimensions) is used as a powerful argument for fixing the value of this parameter, thereby completing the 3-dimensional Hadamard-regularization result. The complete equations of motion and associated energy at the 3PN order are given in the case of circular orbits.

Improvement of the asymptotic behaviour of the Euler–Maclaurin formula for Cauchy principal value and Hadamard finite‐part integrals

AbstractIn the recent works (Commun. Numer. Meth. Engng 2001; 17: 881; to appear), the superiority of the non‐linear transformations containing a real parameter b ≠ 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so‐called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finite‐part integrals by using the Euler–Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b.Through the asymptotic error analysis of the Euler–Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method. Copyright © 2004 John Wiley & Sons, Ltd.

Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective

In this note, we provide a new perspective on Euler–Maclaurin expansions of (offset) trapezoidal rule approximations of the finite-range integrals I[f]=∫baf(x),dx, where f ∈ C∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms * where Ps(y) and Qs(y) are some polynomials in y. Here the γs and δs are complex in general and different from −1,−2,... . The results we obtain in this work generalize, and include as special cases, those pertaining to the known special cases in which f(x)=(x−a)γ[ log (x−a)]pga(x)=(b−x)δ[log (b−x)]qgb(x), where p and q are nonnegative integers and ga ∈ C∞[a,b) and gb ∈ C∞(a,b]. In addition, they have the pleasant feature that they are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. With h=(b−a)/n, where n is a positive integer, and with * one of these results reads, as h→0,* where ζ(z) is the Riemann Zeta function.

Dimensional regularization of the third post-Newtonian dynamics of point particles in harmonic coordinates

Dimensional regularization is used to derive the equations of motion of two point masses in harmonic coordinates. At the third post-Newtonian (3PN) approximation, it is found that the dimensionally regularized equations of motion contain a pole part [proportional to $(d\ensuremath{-}{3)}^{\ensuremath{-}1}]$ which diverges as the space dimension d tends to 3. It is proven that the pole part can be renormalized away by introducing suitable shifts of the two world-lines representing the point masses, and that the same shifts renormalize away the pole part of the ``bulk'' metric tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}{(x}^{\ensuremath{\lambda}}).$ The ensuing, finite renormalized equations of motion are then found to belong to the general parametric equations of motion derived by an extended Hadamard regularization method, and to uniquely determine the 3PN ambiguity parameter $\ensuremath{\lambda}$ to be $\ensuremath{\lambda}=\ensuremath{-}1987/3080.$ This value is fully consistent with the recent determination of the equivalent 3PN ``static ambiguity'' parameter, ${\ensuremath{\omega}}_{s}=0,$ by a dimensional-regularization derivation of the Hamiltonian in Arnowitt-Deser-Misner coordinates. Our work provides a new, powerful check of the consistency of the dimensional regularization method within the context of the classical gravitational interaction of point particles.

Direct Evaluation of Hypersingular Galerkin Surface Integrals

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. The singular integrals are defined as limits to the boundary, and by integrating two of the four dimensions analytically, the coincident integral is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals. A single analytic integration is employed for the edge and vertex singular integrals. This is sufficient to display the divergent term in the edge-adjacent integral and to show that the vertex integral is finite. By explicitly identifying the divergent quantities, we can compute the hypersingular integral without recourse to Stokes's theorem or the Hadamard finite part. The algorithms are developed in the context of a linear element approximation for the Laplace equation but are expected to be generally applicable. As an example, the algorithms are applied to solve a thermal problem in an exponentially graded material.

Uniform approximations to finite Hilbert transform and its derivative

Interpolatory integration rules of numerical stability are presented for approximating Cauchy principal value (p.v.) integrals ⨍ −1 1f(t)/(t−c) dt and Hadamard finite part (f.p.) integrals ∫= −1 1f(t)/(t−c) 2 dt , −1< c<1, respectively, for a given smooth function f( t). Present quadrature rules consist of interpolating f( t) at abscissae in the interval of integration [−1,1] except for the pole c, where neither the function value f( c) nor its derivative f′( c) is required, followed by subtracting out the singularities. We demonstrate that the use of both endpoints ±1 as abscissae in interpolating f( t) is essential for uniformly approximating the integrals, namely for bounding the approximation errors independently of the values of c. In fact, for the f.p. integrals the use of double abscissae at both endpoints ±1 as well as simple abscissae in (−1,1) enables the uniform approximations, while the use of simple abscissae at both endpoints ±1 and those in (−1,1) is sufficient for the p.v. integrals. These facts suggest that finite Hilbert transforms (p.v. integrals) and their derivatives (f.p. integrals) with varied values of c could be approximated efficiently with the same number of abscissae, respectively. Some numerical examples are given.

On the issue of imposing boundary conditions on quantum fields

An interesting example of the deep interrelation between Physics and Mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields. This procedure has recently been re-examined with care. Comments on that and previous analysis are here provided, together with considerations on the results of the purely mathematical zeta-function method, in an attempt at clarifying the issue. Hadamard regularization is invoked in order to fill the gap between the infinities appearing in the QFT renormalized results and the finite values obtained in the literature with other procedures.

Computational Aspects Of Numerical Integration Based On Optimal Nodal Splines

The numerical construction of nodal spline integration rules is considered for the evaluation both of weakly singular integrals and of certain integrals defined in the Hadamard finite part sense. A bound for the quadrature sum condition number is also given.

Chaotic expansion and smoothness of some functionals of the fractional Brownian motion

This paper deals with some additive functionals of the fractional Brownian motion that arise as limits in law of some occupation times of this process. In concrete, this functionals are obtained via the Cauchy principal value and the Hadamard finite part. We derive some regularity properties of theses functionals in Sobolev-Watanabe sense.

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.

Gravitational waves from inspiraling compact binaries: Energy flux to third post-Newtonian order

The multipolar-post-Minkowskian approach to gravitational radiation is applied to the problem of the generation of waves by the compact binary inspiral. We investigate specifically the third post-Newtonian (3PN) approximation in the total energy flux. The new results are the computation of the mass quadrupole moment of the binary to the 3PN order, and the current quadrupole and mass octupole to the 2PN order. Wave tails and tails of tails in the far zone are included up to the 3.5PN order. The recently derived 3PN equations of binary motion are used to compute the time derivatives of the moments. We find perfect agreement to the 3.5PN order with perturbation calculations of black holes in the test-mass limit for one body. Technical inputs in our computation include a model of point particles for describing the compact objects, and the Hadamard self-field regularization. Because of a physical incompleteness of the Hadamard regularization at the 3PN order, the energy flux depends on one unknown physical parameter, which is a combination of a parameter \ensuremath{\lambda} in the equations of motion, and a new parameter \ensuremath{\theta} coming from the quadrupole moment.

Transformations for evaluating singular boundary element integrals

Accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method. Usually, the regular integrals arising from a boundary element method implementation are evaluated using standard Gaussian quadrature. However, the singular integrals which arise are often evaluated in another way, sometimes using a different integration method with different nodes and weights. This paper presents a straightforward transformation to improve the accuracy of evaluating singular integrals. The transformation is, in a sense, a generalisation of the popular method of Telles with the underlying idea being to utilise the same Gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation. The new transformation is also shown to be equivalent to other existing transformations in certain situations. Comparison of the new method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained. The technique can also be extended to evaluate certain Hadamard finite-part integrals. Based on the observation of several integrals considered, guidelines are suggested for the best transformation order to use (i.e. the degree to which nodes should be clustered near the singular point).

Nodal Spline Integration Rules for Certain 2-D Cauchy Principal Value Integrals

In this paper we consider integration rules based on products of one-dimensional nodal spline quadratures for the numerical evaluation of 2-D singular integrals defined in the Hadamard finite part sense. We derive convergence results and error bounds. Some numerical tests are also presented.

Lorentzian regularization and the problem of point-like particles in general relativity

The two purposes of the article are (1) to present a regularization of the self-field of point-like particles, based on Hadamard’s concept of “partie finie,” that permits in principle to maintain the Lorentz covariance of a relativistic field theory, and (2) to use this regularization for defining a model of stress-energy tensor that describes point-particles in post-Newtonian expansions (e.g., 3PN) of general relativity. We consider specifically the case of a system of two point-particles. We first perform a Lorentz transformation of the system’s variables which carries one of the particles to its rest frame, next implement the Hadamard regularization within that frame, and finally come back to the original variables with the help of the inverse Lorentz transformation. The Lorentzian regularization is defined in this way up to any order in the relativistic parameter 1/c2. Following a previous work of ours, we then construct the delta-pseudo-functions associated with this regularization. Using an action principle, we derive the stress-energy tensor, made of delta-pseudo-functions, of point-like particles. The equations of motion take the same form as the geodesic equations of test particles on a fixed background, but the role of the background is now played by the regularized metric.

Galerkin trigonometric wavelet methods for the natural boundary integral equations

A simply supported boundary value problem of biharmonic equations in the unit disk is reduced into an equivalent second kind natural boundary integral equation (NBIE) with hypersingular kernel (in the sense of Hadamard finite part). Trigonometric Hermite interpolatory wavelets introduced by Quak [Math. Comput. 65 (1996) 683–722] as trial functions are used to its Galerkin discretization with 2 J+1 degrees of freedom on the boundary. It is proved that the stiffness matrix is a block diagonal matrix and its diagonal elements are some symmetric and block circulant submatrices. The simple computational formulae of the entries in stiffness matrix are obtained. These show that we only need to compute 2(2 J + J+1) elements of a 2 J+2 ×2 J+2 stiffness matrix. The error estimates for the approximation solutions are established. Finally, numerical examples are given.

Evaluations of certain hypersingular integrals on interval

AbstractIn this paper, we investigate a hypersingular integral on an interval. The definition of Hadamard's finite‐part integrals and some of its properties are given. Some numerical methods on approximate computation of the finite‐part integrals are constructed. The new method is very simple, easy to implement, reliable, and above all, not affected by the location of singular point. Some numerical experiments are carried out using the current formulae, and numerical results show that the current methods are feasible and effective. Copyright © 2001 John Wiley &amp; Sons, Ltd.

General relativistic dynamics of compact binaries at the third post-Newtonian order

The general relativistic corrections in the equations of motion and associated energy of a binary system of pointlike masses are derived at the third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian expansion of the metric in harmonic coordinates at the 3PN approximation. The metric is parametrized by appropriate nonlinear potentials, which are evaluated in the case of two point particles using a Lorentzian version of a Hadamard regularization which has been defined in previous works. Distributional forms and distributional derivatives constructed from this regularization are employed systematically. The equations of motion of the particles are geodesiclike with respect to the regularized metric. Crucial contributions to the acceleration are associated with the nondistributivity of the Hadamard regularization and the violation of the Leibniz rule by the distributional derivative. The final equations of motion at the 3PN order are invariant under global Lorentz transformations, and admit a conserved energy (neglecting the radiation reaction force at the 2.5PN order). However, they are not fully determined, as they depend on one arbitrary constant, which probably reflects a physical incompleteness of the point-mass regularization. The results of this paper should be useful when comparing theory to the observations of gravitational waves from binary systems in future detectors VIRGO and LIGO.