1,741 research outputs found

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

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    In the Sobolev space L2(m)(0,1)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 Ξ·0\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

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    We are concerned with the numerical integration of functions from the Sobolev space Hr,mix([0,1]d)H^{r,\text{mix}}([0,1]^d) of dominating mixed smoothness r∈Nr\in\mathbb{N} over the dd-dimensional unit cube. In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order nβˆ’r (log⁑n)(dβˆ’1)/2n^{-r} \, (\log n)^{(d-1)/2} with respect to the number nn of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using dd random dilations. We showed that its error is bounded above by a constant multiple of nβˆ’rβˆ’1/2 (log⁑n)(dβˆ’1)/2n^{-r-1/2} \, (\log n)^{(d-1)/2} in expectation and by nβˆ’r (log⁑n)(dβˆ’1)/2n^{-r} \, (\log n)^{(d-1)/2} almost surely. The main term nβˆ’rβˆ’1/2n^{-r-1/2} is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space Hs([0,1]d)H^s([0,1]^d) of smoothness s>d/2s>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 Hr,mix([0,1]d)H^{r,\text{mix}}([0,1]^d) is even bounded above by nβˆ’rβˆ’1/2n^{-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)}

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    This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space L2(1)[a,b]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 d2/dx2d^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 C2[a,b]C^2[a,b] is O(h2)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

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    This paper considers filtered polynomial approximations on the unit sphere SdβŠ‚Rd+1\mathbb{S}^d\subset \mathbb{R}^{d+1}, obtained by truncating smoothly the Fourier series of an integrable function ff with the help of a "filter" hh, which is a real-valued continuous function on [0,∞)[0,\infty) such that h(t)=1h(t)=1 for t∈[0,1]t\in[0,1] and h(t)=0h(t)=0 for tβ‰₯2t\ge2. The resulting "filtered polynomial approximation" (a spherical polynomial of degree 2Lβˆ’12L-1) is then made fully discrete by approximating the inner product integrals by an NN-point cubature rule of suitably high polynomial degree of precision, giving an approximation called "filtered hyperinterpolation". In this paper we require that the filter hh and all its derivatives up to ⌊dβˆ’12βŒ‹\lfloor\tfrac{d-1}{2}\rfloor are absolutely continuous, while its right and left derivatives of order ⌊d+12βŒ‹\lfloor \tfrac{d+1}{2}\rfloor exist everywhere and are of bounded variation. Under this assumption we show that for a function ff in the Sobolev space Wps(Sd),Β 1≀pβ‰€βˆžW^s_p(\mathbb{S}^d),\ 1\le p\le \infty, both approximations are of the optimal order Lβˆ’s L^{-s}, in the first case for s>0s>0 and in the second fully discrete case for s>d/ps>d/p

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

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    In this paper we study the worst-case error of numerical integration on the unit sphere SdβŠ‚Rd+1\mathbb{S}^{d}\subset\mathbb{R}^{d+1}, dβ‰₯2d\geq2, for certain spaces of continuous functions on Sd\mathbb{S}^{d}. For the classical Sobolev spaces Hs(Sd)\mathbb{H}^s(\mathbb{S}^d) (s>d2s>\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 sβ†’d2s\to\frac d2 by introducing spaces Hd2,Ξ³(Sd)\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

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    We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space Hs(R)H^s({\mathbb{R}}) and from the space Cs(R)C^s({\mathbb{R}}) with an arbitrary integer sβ‰₯1s\ge1. We find tight upper and lower bounds for the worst case error of optimal algorithms that use nn 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∈Rk\in {\mathbb{R}} and a smooth density function ρ\rho such as ρ(x)=12Ο€exp⁑(βˆ’x2/2) \rho(x) = \frac{1}{\sqrt{2 \pi}} \exp( -x^2/2) . The optimal error bounds are Θ((n+max⁑(1,∣k∣))βˆ’s)\Theta((n+\max(1,|k|))^{-s}) with the factors in the Θ\Theta notation dependent only on ss and ρ\rho.Comment: 21 page

    Optimal quadrature formulas with derivatives in Sobolev space

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    In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1)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 NN and for any mβ‰₯4m\geq 4 using S.L. Sobolev method which is based on discrete analogue of the differential operator d2m/dx2md^{2m}/dx^{2m}. In particular, for m=4,Β 5m=4,\ 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6m=6 new optimal quadrature formulas are obtained

    Lattice rules in non-periodic subspaces of Sobolev spaces

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    We investigate quasi-Monte Carlo (QMC) integration over the ss-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces H(KΞ±,Ξ³,ssob)\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}}) and their subspaces of high order smoothness Ξ±>1\alpha>1, where Ξ³\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 Ξ±>1/2\alpha>1/2 consisting of non-periodic smooth functions, denoted by H(KΞ±,Ξ³,scos)\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 Ξ±\alpha, denoted by H(KΞ±,Ξ³,skor+cos)\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 Ξ±\alpha, we provide their corresponding norm-equivalent subspaces of H(KΞ±,Ξ³,ssob)\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}}). This implies that H(KΞ±,Ξ³,skor+cos)\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}}) is strictly smaller than H(KΞ±,Ξ³,ssob)\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}}) as sets for Ξ±β‰₯2\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 H(K2,Ξ³,ssob)\mathcal{H}(K_{2,\boldsymbol{\gamma},s}^{\mathrm{sob}}) and also the worst-case error of symmetrized lattice rules in an intermediate space between H(KΞ±,Ξ³,skor+cos)\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}}) and H(KΞ±,Ξ³,ssob)\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

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    The paper studies Sard's problem on construction of optimal quadrature formulas in the space W2(m,0)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 W2(m,0)W_2^{(m,0)} is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue Dm(hΞ²)D_m(h\beta) of the differential operator d2mdx2mβˆ’1\frac{d^{2m}}{d x^{2m}}-1 is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space W2(m,0)W_2^{(m,0)}, which based on the discrete analogue Dm(hΞ²)D_m(h\beta), is described. Finally, for m=1m=1 and m=3m=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

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    We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces Sp,qrB(Td)S^r_{p,q}B(\mathbb{T}^d) with dominating mixed smoothness 1/p<r<21/p<r<2. We show that order 2 digital nets achieve the optimal rate of convergence Nβˆ’r(log⁑N)(dβˆ’1)(1βˆ’1/q)N^{-r} (\log N)^{(d-1)(1-1/q)}. The logarithmic term does not depend on rr and hence improves the known bound provided by J. Dick for the special case of Sobolev spaces Hmixr(Td)H^r_{\text{mix}}(\mathbb{T}^d). Secondly, the rate of convergence is independent of the integrability pp 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=2r=2.Comment: 7 figures in Num. Mathem., online 201
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