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Register a new userA randomised trapezoidal quadrature
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Added by
Yue Wu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Yue Wu
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No Arabic abstract
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.
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A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the fast algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (1999).
In this paper we propose a method for computing the Faddeeva function $w(z) := e^{-z^2}mathrm{erfc}(-i z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel (Math. Comp. 25 (1971), pp. 339-344) and Hunter and Regan (Math. Comp. 26 (1972), pp. 339-541). Addressing shortcomings flagged by Weideman (SIAM. J. Numer. Anal. 31 (1994), pp. 1497-1518), we construct approximations which we prove are exponentially convergent as a function of $N+1$, the number of quadrature points, obtaining error bounds which show that accuracies of $2times 10^{-15}$ in the computation of $w(z)$ throughout the complex plane are achieved with $N = 11$, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where $w(z)$ is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing $w(z)$ for complex $z$.
There is growing awareness that errors in the model equations cannot be ignored in data assimilation methods such as four-dimensional variational assimilation (4D-Var). If allowed for, more information can be extracted from observations, longer time windows are possible, and the minimisation process is easier, at least in principle. Weak constraint 4D-Var estimates the model error and minimises a series of linear least-squares cost functionsfunctions, which can be achieved using the conjugate gradient (CG) method; minimising each cost function is called an inner loop. CG needs preconditioning to improve its performance. In previous work, limited memory preconditioners (LMPs) have been constructed using approximations of the eigenvalues and eigenvectors of the Hessian in the previous inner loop. If the Hessian changes significantly in consecutive inner loops, the LMP may be of limited usefulness. To circumvent this, we propose using randomised methods for low rank eigenvalue decomposition and use these approximations to cheaply construct LMPs using information from the current inner loop. Three randomised methods are compared. Numerical experiments in idealized systems show that the resulting LMPs perform better than the existing LMPs. Using these methods may allow more efficient and robust implementations of incremental weak constraint 4D-Var.
We examine sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere, in the case of worst case quadrature error for rules with arbitrary quadrature weights. We describe a dimension adaptive quadrature algorithm based on an algorithm of Hegland (2003), and also formulate a version of Wasilkowski and Wozniakowskis WTP algorithm (1999), here called the WW algorithm. We prove that the dimension adaptive algorithm is optimal in the sense of Dantzig (1957) and therefore no greater in cost than the WW algorithm. Both algorithms therefore have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wozniakowski (1999). A numerical example shows that, even though the asymptotic convergence rate is optimal, if the dimension weights decay slowly enough, and the dimensionality of the problem is large enough, the initial convergence of the dimension adaptive algorithm can be slow.
The numerical integration of an analytical function $f(x)$ using a finite set of equidistant points can be performed by quadrature formulas like the Newton-Cotes. Unlike Gaussian quadrature formulas however, higher-order Newton-Cotes formulas are not stable, limiting the usable order of such formulas. Existing work showed that by the use of orthogonal polynomials, stable high-order quadrature formulas with equidistant points can be developed. We improve upon such work by making use of (orthogonal) Gram polynomials and deriving an iterative algorithm, together allowing us to reduce the space-complexity of the original algorithm significantly.
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