Optimal Quadrature Formulas with Positive Coefficients in L2(m)(0,1)L_2^{(m)}(0,1) Space

In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing parameter $\eta_0$ in (1.5) the optimal quadrature formulas with positive coefficients are obtained and compared with well known optimal formulas.Comment: 32 pages, submitted to the Journal of computational and applied mathematic

On the Randomization of Frolov's Algorithm for Multivariate Integration

We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\in\mathbb{N}$ over the $d$-dimensional unit cube. In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \, (\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant multiple of $n^{-r-1/2} \, (\log n)^{(d-1)/2}$ in expectation and by $n^{-r} \, (\log n)^{(d-1)/2}$ almost surely. The main term $n^{-r-1/2}$ is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on $H^{r,\text{mix}}([0,1]^d)$ is even bounded above by $n^{-r-1/2}$. This thesis is a review of the mentioned upper bounds and their proofs.Comment: Master Thesi

On an optimal quadrature formula for approximation of Fourier integrals in the space L2(1)L_2^{(1)}

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 formulas for optimal coefficients can also be obtained using discrete analogue of the differential operator $d^2/d x^2$. In addition, the convergence order of the optimal quadrature formula is studied. It is proved that the obtained formula is exact for all linear polynomials. Thus, it is shown that the convergence order of the optimal quadrature formula for functions of the space $C^2[a,b]$ is $O(h^2)$. Moreover, several numerical results are presented and the obtained optimal quadrature formula is applied to reconstruct the X-ray Computed Tomography image by approximating Fourier transforms.Comment: 27 pages, 6 figure

On filtered polynomial approximation on the sphere

This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a real-valued continuous function on $[0,\infty)$ such that $h(t)=1$ for $t\in[0,1]$ and $h(t)=0$ for $t\ge2$. The resulting "filtered polynomial approximation" (a spherical polynomial of degree $2L-1$) is then made fully discrete by approximating the inner product integrals by an $N$-point cubature rule of suitably high polynomial degree of precision, giving an approximation called "filtered hyperinterpolation". In this paper we require that the filter $h$ and all its derivatives up to $\lfloor\tfrac{d-1}{2}\rfloor$ are absolutely continuous, while its right and left derivatives of order $\lfloor \tfrac{d+1}{2}\rfloor$ exist everywhere and are of bounded variation. Under this assumption we show that for a function $f$ in the Sobolev space $W^s_p(\mathbb{S}^d),\ 1\le p\le \infty$, both approximations are of the optimal order $L^{-s}$, in the first case for $s>0$ and in the second fully discrete case for $s>d/p$

Upper and lower estimates for numerical integration errors on spheres of arbitrary dimension

In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error have been obtained By Brauchart, Hesse, and Sloan earlier in papers. We investigate the behaviour for $s\to\frac d2$ by introducing spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ with an extra logarithmic weight. For these spaces we obtain similar upper and lower bounds for the worst case integration error

Complexity of Oscillatory Integrals on the Real Line

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. We find tight upper and lower bounds for the worst case error of optimal algorithms that use $n$ function values. More specifically, we study integrals of the form I_k^\rho (f) = \int_{ {\mathbb{R}}} f(x) \,e^{-i\,kx} \rho(x) \, {\rm d} x\ \ \ \mbox{for}\ \ f\in H^s({\mathbb{R}})\ \ \mbox{or}\ \ f\in C^s({\mathbb{R}}) with $k\in {\mathbb{R}}$ and a smooth density function $\rho$ such as $\rho(x) = \frac{1}{\sqrt{2 \pi}} \exp( -x^2/2)$. The optimal error bounds are $\Theta((n+\max(1,|k|))^{-s})$ with the factors in the $\Theta$ notation dependent only on $s$ and $\rho$.Comment: 21 page

Optimal quadrature formulas with derivatives in Sobolev space

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number $N$ and for any $m\geq 4$ using S.L. Sobolev method which is based on discrete analogue of the differential operator $d^{2m}/dx^{2m}$. In particular, for $m=4,\ 5$ optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from $m=6$ new optimal quadrature formulas are obtained

Lattice rules in non-periodic subspaces of Sobolev spaces

We investigate quasi-Monte Carlo (QMC) integration over the $s$-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ and their subspaces of high order smoothness $\alpha>1$, where $\boldsymbol{\gamma}$ denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter $\alpha>1/2$ consisting of non-periodic smooth functions, denoted by $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{cos}})$, and also in the sum of half-period cosine spaces and Korobov spaces with common parameter $\alpha$, denoted by $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$. Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer $\alpha$, we provide their corresponding norm-equivalent subspaces of $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$. This implies that $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$ is strictly smaller than $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ as sets for $\alpha \geq 2$, which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in $\mathcal{H}(K_{2,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ and also the worst-case error of symmetrized lattice rules in an intermediate space between $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$ and $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$. We show that the almost optimal rate of convergence can be achieved for both cases, while a weak dependence of the worst-case error bound on the dimension can be obtained for the former case

Construction of optimal quadrature formulas exact for exponentional-trigonometric functions by Sobolev's method

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 the solution of this system are discussed. Next, the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{d x^{2m}}-1$ is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space $W_2^{(m,0)}$, which based on the discrete analogue $D_m(h\beta)$, is described. Finally, for $m=1$ and $m=3$ the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.Comment: 22 page

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$ with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets achieve the optimal rate of convergence $N^{-r} (\log N)^{(d-1)(1-1/q)}$. The logarithmic term does not depend on $r$ and hence improves the known bound provided by J. Dick for the special case of Sobolev spaces $H^r_{\text{mix}}(\mathbb{T}^d)$. Secondly, the rate of convergence is independent of the integrability $p$ of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of higher order digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case $r=2$.Comment: 7 figures in Num. Mathem., online 201