In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^be^{2πiωx}φ(x)d x$ with $ω\in \mathbb{R}$ in the Sobolev space $L_2^{(m)}[a,b]$ of complex-valued functions which are square integrable with $m$-th order derivative. Here, using the discrete analogue of the differential operator $\frac{d^{2m}}{d x^{2m}}$, the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is $O(h^m)$. As an application, we implement the filtered back-projection (FBP) algorithm, which is a well-known image reconstruction algorithm for computed tomography (CT). By approximating Fourier transforms and its inversion using the proposed optimal quadrature formula of the second and third orders, we observe that the accuracy of the reconstruction algorithm is improved. In numerical experiments, we compare the quality of the reconstructed image obtained by using the proposed optimal quadrature formulas with the conventional FBP, in which fast Fourier transform is used for the calculation of Fourier transform and its inversion. In the noise test, the proposed algorithm provides more reliable results against the noise than the conventional FBP.
The paper studies Sard's problem on construction of optimal quadrature formulas in the space $W_2^{(m,0)}$ by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space $W_2^{(m,0)}$ is obtained, moreover the existence and uniqueness of t…
We consider the numerical integration $${\rm INT}_d(f)=\int_{\mathbb{B}^{d}}f(x)w_μ(x)dx $$ for the weighted Sobolev classes $BW^{r}_{p,μ}$ and the weighted Besov classes $BB_τ^r(L_{p,μ})$ in the randomized case setting, where $w_μ, \,μ\ge0,$ is the classical Jacobi weight on the ball $\Bbb B^d$, $1\le p\le \infty$, $r>(d+2μ)/p$, and $0<τ\le\infty$. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are $n^{-r/d-1/2+(1/p-1/2)_+}$. Compared to the orders $n^{-r/d}$ of the optimal quadrature errors in the deterministic case setting, ra…
This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase…
An error estimate for the Gauss-Lobatto quadrature formula for integration over the interval $[-1, 1]$, relative to the Jacobi weight function $w^{α,β}(t)=(1-t)^α(1+t)^β$, $α,β>-1$, is obtained. This estimate holds true for functions belonging to some Sobolev-type subspaces of the weighted space $L_{w^{α,β}}^1([-1,1])$.
A randomised trapezoidal quadrature rule is proposed for continuous functions which enjoys less regularity than commonly required. Indeed, we consider functions in some fractional Sobolev space. Various error bounds for this randomised rule are established while an error bound for classical trapezoidal quadrature is obtained for comparison. The randomised trapezoidal quadrature rule is shown to improve the order of convergence by half.
We study Gauss quadrature for Freud weights and derive worst case error estimates for functions in a family of associated Sobolev spaces. For the Gaussian weight $e^{-πx^2}$ these spaces coincide with a class of modulation spaces which are well-known in (time-frequency) analysis and also appear under the name of Hermite spaces. Extensions are given to more general sets of nodes that are derived from Marcinkiewicz-Zygmund inequalities. This generalization can be interpreted as a stability result for Gauss quadrature.
The sub-optimality of Gauss--Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order $α$, where the optimality is in the sense of worst-case error. For Gauss--Hermite quadrature, we obtain matching lower and upper bounds, which turn out to be merely of the order $n^{-α/2}$ with $n$ function evaluations, although the optimal rate for the best possible linear quadrature is known to be $n^{-α}$. Our proof of the lower bound exploits the structure of the Gauss--Hermite nodes; the bound is independent of the qua…
As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. …
This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ of non-periodic, complex valued functions which are square integrable with first order derivative. Here the quadrature sum consists of linear combination of the given function values in a uniform grid. The difference between the integral and the quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. Analytic formula…
In this paper, we propose a new nonuniform mesh method to simulate acoustic scattering problems in two dimensional periodic structures with non-periodic incident fields numerically. As existing methods are difficult to extend to higher dimensions, we have designed the new method with such extensions in mind. With the help of the Floquet-Bloch transform, the solution to the original scattering problem is written as an integral of a family of quasi-periodic problems. These are defined in bounded domains for each value of the Floquet parameter which varies in a bounded interval. The key step in o…
We consider the numerical computation of finite-range singular integrals $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$ and $f(x)\in C^\infty(\mathbb{R}_t)$ is $T$-periodic with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$, $T=b-a$. Using a generalization of the Euler--Maclaurin expansion developed in [A. Sidi, {Euler--Maclaurin} expansions for integrals with arbitrary algebraic endpoint singularities. {\em Math. Comp.}, 81:2159--2173, 2012]…
In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals $I[f]=\intBar^b_a f(x)\,dx$, where $f(x)=g(x)/(x-t)^3,$ assuming that $g\in C^\infty[a,b]$ and $f(x)$ is $T$-periodic, $T=b-a$. With $h=T/n$, these numerical quadrature formulas read \begin{align*} \widehat{T}{}^{(0)}_n[f]&=h\sum^{n-1}_{j=1}f(t+jh) -\frac{π^2}{3}\,g'(t)\,h^{-1}+\frac{1}{6}\,g'''(t)\,h, \widehat{T}{}^{(1)}_n[f]&=h\sum^n_{j=1}f(t+jh-h/2) -π^2\,g'(t)\,h^{-1}, \widehat{T}{}^{(2)}_n[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4). \…
Monte Carlo Methods for Uniform Approximation on Periodic Sobolev Spaces with Mixed Smoothness
Hausdorff-Center for Mathematics（豪斯多夫数学中心）；Universität Osnabrück, Institut für Mathematik（奥斯纳布吕克大学数学研究所）；Department of Mathematics, University of Transport and Communications（交通与通信大学数学系）
We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $κ\in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min\{3/4,(1+κ)/2\}-ε$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $ε>0$). The cut-off of the convergence …
Let $L_{q,μ},\, 1\le q<\infty, \ μ\ge0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_μ$ on the ball $\Bbb B^d$. We consider the weighted least $\ell_q$ approximation problem for a given $L_{q,μ}$-Marcinkiewicz-Zygmund family on $\Bbb B^d$. We obtain the weighted least $\ell_q$ approximation errors for the weighted Sobolev space $W_{q,μ}^r$, $r>(d+2μ)/q$, which are order optimal. We also discuss the least squares quadrature induced by an $L_{2,μ}$-Marcinkiewicz-Zygmund family, and get the quadrature errors for $W_{2,μ}^r$, $r>(d+2μ)/2$, which are also order optimal. Mean…
The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. In addition, analytic formulas for optimal coefficients are obtained using the…
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski”（索非亚大学圣克利门特奥赫里斯基数学与信息学院）；Faculty of Pharmacy, Medical University of Sofia（索非亚医科大学药学院）
Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This paper presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centrepiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion (QBX), accelerated using a precomputation scheme. The…
Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces
Department of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学系）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院数学科学分部）；State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所计算机科学与技术国家重点实验室/并行计算实验室）
Lattice rules in non-periodic subspaces of Sobolev spaces
School of Engineering, University of Tokyo（东京大学工学部）；Graduate School of Science, Hiroshima University（广岛大学理学研究科）；Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University（京都大学情报学研究科应用数学物理系）
