A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.Comment: 33 pages, 9 figure

Quantum algorithms for problems in number theory, algebraic geometry, and group theory

Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in Quantum Computation/Information at Kinki Universit

Computing Nearly Singular Solutions Using Pseudo-Spectral Methods

In this paper, we investigate the performance of pseudo-spectral methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing errors in a pseudo-spectral method. The first one is the traditional 2/3 dealiasing rule. The second one is a high (36th) order Fourier smoothing which keeps a significant portion of the Fourier modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3 dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler equations are considered. We demonstrate that the pseudo-spectral method with the high order Fourier smoothing gives a much better performance than the pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the high order Fourier smoothing method captures about $12 \sim 15$ more effective Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D Euler equations, the gain in the effective Fourier codes for the high order Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method. Another interesting observation is that the error produced by the high order Fourier smoothing method is highly localized near the region where the solution is most singular, while the 2/3 dealiasing method tends to produce oscillations in the entire domain. The high order Fourier smoothing method is also found be very stable dynamically. No high frequency instability has been observed.Comment: 26 pages, 23 figure