32,713 research outputs found
Approximation of Elements of Exponentials of Differential Operators With Rational Quadrature
We explore the possibility of improving the accuracy of approximations of elements of exponentials of differential operators, by using a rational function, instead of a polynomial function, as the interpolating function. Since a rational function behaves more like a decaying exponential function, it seems logical that the approximation should be more accurate. Through the use of high accuracy rational interpolants, we experiment with a numerical integration method to determine explicitly whether the error produced by a rational type approximation will indeed be less than that produced by a polynomial type approximation
ON SUBDIAGONAL RATIONAL PADE APPROXIMATIONS AND THE BRENNER-THOMEE APPROXIMATION THEOREM FOR OPERATOR SEMIGROUPS
The computational powers of Mathematica are used to prove polynomial identities that are essential to obtain growth estimates for subdiagonal rational Pade approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thomee Approximation Theorem of Semigroup Theory
Representation of conformal maps by rational functions
The traditional view in numerical conformal mapping is that once the boundary
correspondence function has been found, the map and its inverse can be
evaluated by contour integrals. We propose that it is much simpler, and 10-1000
times faster, to represent the maps by rational functions computed by the AAA
algorithm. To justify this claim, first we prove a theorem establishing
root-exponential convergence of rational approximations near corners in a
conformal map, generalizing a result of D. J. Newman in 1964. This leads to the
new algorithm for approximating conformal maps of polygons. Then we turn to
smooth domains and prove a sequence of four theorems establishing that in any
conformal map of the unit circle onto a region with a long and slender part,
there must be a singularity or loss of univalence exponentially close to the
boundary, and polynomial approximations cannot be accurate unless of
exponentially high degree. This motivates the application of the new algorithm
to smooth domains, where it is again found to be highly effective
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of , where is a negative definite matrix and is the exponential function or one of the related `` functions'' such as . Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as , where is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour
Approximation of SPDE covariance operators by finite elements: A semigroup approach
The problem of approximating the covariance operator of the mild solution to
a linear stochastic partial differential equation is considered. An integral
equation involving the semigroup of the mild solution is derived and a general
error decomposition is proven. This formula is applied to approximations of the
covariance operator of a stochastic advection-diffusion equation and a
stochastic wave equation, both on bounded domains. The approximations are based
on finite element discretizations in space and rational approximations of the
exponential function in time. Convergence rates are derived in the trace class
and Hilbert--Schmidt norms with numerical simulations illustrating the results.Comment: 31 pages, 8 figures; added derivation of trace error formula; to
appear in IMA Journal of Numerical Analysi
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