scholarly journals Worst-case errors in a Sobolev space setting for cubature over the sphere S2

2005 ◽  
Vol 71 (1) ◽  
pp. 81-105 ◽  
Author(s):  
Kerstin Hesse ◽  
Ian H. Sloan
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Numerical Integration ◽  
Universal Constant ◽  
Regularity Property ◽  
Local Regularity ◽  
Worst Case ◽  
Positive Weight ◽  
Spherical Polynomials ◽  
Integration Rules

This paper studies the problem of numerical integration over the unit sphere S2 ⊆ ℝ3 for functions in the Sobolev space H3/2(S2). We consider sequences Qm(n), n ∈ ℕ, of cubature (or numerical integration) rules, where Qm(n) is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and to use m = m(n) values of f. The cubature weights of all rules Qm(n) are assumed to be positive, or alternatively the sequence Qm(n), n ∈ ℕ, is assumed to have a certain local regularity property which involves the weights and the points of the rules Qm(n), n ∈ ℕ. Under these conditions it is shown that the worst-case (cubature) error, denoted by E3/2 (Qm(n)), for all functions in the unit ball of the Hilbert space H3/2 (S2) satisfies the estimate E3/2 (Qm(n)) ≤ c n−3/2, where the constant c is a universal constant for all sequences of positive weight cubature rules. For a sequence Qm(n), n ∈ ℕ, of cubature rules that satisfies the alternative local regularity property the constant c may depend on the sequence Qm(n), n ∈ ℕ. Examples of cubature rules that satisfy the assumptions are discussed.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

scholarly journals Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

2018 ◽  
Vol 16 (05) ◽  
pp. 693-715 ◽  
Author(s):  
Erich Novak ◽  
Mario Ullrich ◽  
Henryk Woźniakowski ◽  
Shun Zhang
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Absolute Error ◽  
Reproducing Kernels ◽  
Worst Case ◽  
Reproducing Kernel Formula ◽  
Positive Integers ◽  
Integration Errors ◽  
Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

scholarly journals Optimal quadrature formula in the sense of Sard in K2(P3) space

2014 ◽  
Vol 95 (109) ◽  
pp. 29-47 ◽  
Author(s):  
Abdullo Hayotov ◽  
Gradimir Milovanovic ◽  
Kholmat Shadimetov
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Quadrature Formula ◽  
Asymptotic Optimality ◽  
Trigonometric Functions ◽  
Numerical Examples ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Formula

We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K2(P3). Using Sobolev?s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L (3)2 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms

2016 ◽  
Vol 23 (4) ◽  
pp. 615-622 ◽  
Author(s):  
Armen Sergeev
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Noncommutative Geometry ◽  
Function Theory ◽  
Banach Algebras ◽  
Linear Operators ◽  
Boundary Values ◽  
Differentiable Functions ◽  
Quasisymmetric Homeomorphisms

AbstractIn this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group ${\operatorname{QS}(S^{1})}$ with respect to composition. This group acts on the Sobolev space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group ${\operatorname{QS}(S^{1})}$ and the space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ in terms of noncommutative geometry.

scholarly journals Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401881990
Author(s):  
Chigbogu Godwin Ozoegwu
Keyword(s):  
Numerical Integration ◽  
Distributed Delay ◽  
Original Method ◽  
Higher Order ◽  
Trapezoidal Rule ◽  
Point Of View ◽  
Engineering Systems ◽  
Two Stage ◽  
Integration Rule ◽  
Integration Rules

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

scholarly journals Decay of correlations in suspension semi-flows of angle-multiplying maps

2008 ◽  
Vol 28 (1) ◽  
pp. 291-317 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Spectral Radius ◽  
Decay Of Correlations ◽  
Square Root ◽  
Precise Description ◽  
Anisotropic Sobolev Space ◽  
Essential Spectral Radius ◽  
Frobenius Operator ◽  
Dense Set

AbstractWe consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

Porosity and the bounded linear regularity property

2014 ◽  
Vol 20 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Simeon Reich ◽  
Alexander J. Zaslavski
Keyword(s):  
Hilbert Space ◽  
Convex Sets ◽  
Regularity Property ◽  
Nonempty Intersection ◽  
Normed Spaces ◽  
Infinite Products ◽  
Bounded Linear ◽  
Linear Regularity ◽  
Nearest Point ◽  
Convex Subsets

Abstract.H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

scholarly journals Function Values Are Enough for $$L_2$$-Approximation

2020 ◽  
Author(s):  
David Krieg ◽  
Mario Ullrich
Keyword(s):  
Hilbert Space ◽  
Fourier Coefficients ◽  
Approximation Number ◽  
Approximation Numbers ◽  
Approximation Of Functions ◽  
Worst Case ◽  
Linear Information ◽  
Grid Algorithm ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

AbstractWe study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

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