• Corpus ID: 238744138

Computing semigroups with error control

@article{Colbrook2021ComputingSW,
  title={Computing semigroups with error control},
  author={Matthew J. Colbrook},
  journal={ArXiv},
  year={2021},
  volume={abs/2110.06350}
}
  • M. Colbrook
  • Published 12 October 2021
  • Computer Science, Mathematics
  • ArXiv
We develop an algorithm that computes strongly continuous semigroups on infinitedimensional Hilbert spaces with explicit error control. Given a generator A, a time t > 0, an arbitrary initial vector u0 and an error tolerance > 0, the algorithm computes exp(tA)u0 with error bounded by . The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules, and the adaptive computation of resolvents in infinite dimensions. As a particular case, we show… 
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References

SHOWING 1-10 OF 101 REFERENCES
THE HOLOMORPHIC FUNCTIONAL CALCULUS APPROACH TO OPERATOR SEMIGROUPS
In this article we construct a holomorphic functional calculus for operators of half-plane type and show how key facts of semigroup theory (Hille- Yosida and Gomilko-Shi-Feng generation theorems,
Computing solutions of Schrödinger equations on unbounded domains- On the brink of numerical algorithms
TLDR
The results provide classifications of which mathematical problems may be solved by computer assisted proofs and are a part of the Solvability Complexity Index (SCI) hierarchy and Smale's program on the foundations of computational mathematics.
Resolvent Krylov subspace approximation to operator functions
We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation of
Validated and Numerically Efficient Chebyshev Spectral Methods for Linear Ordinary Differential Equations
TLDR
This work develops a validated numeric method for the solution of linear ordinary differential equations (LODEs), based on both the theoretical and practical complexity analysis of a so-called a posteriori quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map.
Talbot quadratures and rational approximations
Many computational problems can be solved with the aid of contour integrals containing ez in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and
Spectral methods using rational basis functions on an infinite interval
ON RATIONAL APPROXIMATIONS OF SEMIGROUPS
We show that if $r^n (hA),nh = t$, is an A-acceptable rational approximation of a strongly continuous semigroup $e^{tA} $ on a Banach space, then for t bounded, $\| {r^n (hA)} \| \leqq Cn^{{1/2}} $,
Time discretization of an evolution equation via Laplace transforms
Following earlier work by Sheen, Sloan, and Thomee concerning parabolic equations we study the discretization in time of a Volterra type integro-differential equation in which the integral operator
Computing highly oscillatory integrals
TLDR
Two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points are developed and results show that the proposed methods outperform methods published most recently.
ON THE COMPUTATION OF GEOMETRIC FEATURES OF SPECTRA OF LINEAR OPERATORS ON HILBERT SPACES
Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum
...
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