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Register a new userComputing the Sound of the Sea in a Seashell
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Added by
Frank R\\\"osler
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $mathcal C^2$. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.
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A simple iterative scheme is proposed for locating the parameter values for which a 2-parameter family of real symmetric matrices has a double eigenvalue. The convergence is proved to be quadratic. An extension of the scheme to complex Hermitian matrices (with 3 parameters) and to location of triple eigenvalues (5 parameters for real symmetric matrices) is also described. Algorithm convergence is illustrated in several examples: a real symmetric family, a complex Hermitian family, a family of matrices with an avoided crossing (no covergence) and a 5-parameter family of real symmetric matrices with a triple eigenvalue.
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix $M$ is $s$-sparse, the complexity is $widetilde{O}(srho^2 kappa^2/epsilon^2)$ for diagonalizable matrices that only have real eigenvalues, and $widetilde{O}(srho|M|_{max} /epsilon^2)$ for normal matrices. Here $rho$ is an upper bound of the eigenvalues, $kappa$ is the conditioning of the eigenvalue problem, and $epsilon$ is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.
In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichm{u}ller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM.
Modelling neutral beam injection (NBI) in fusion reactors requires computing the trajectories of large ensembles of particles. Slowing down times of up to one second combined with nanosecond time steps make these simulations computationally very costly. This paper explores the performance of BGSDC, a new numerical time stepping method, for tracking ions generated by NBI in the DIII-D and JET reactors. BGSDC is a high-order generalisation of the Boris method, combining it with spectral deferred corrections and the Generalized Minimal Residual method GMRES. Without collision modelling, where numerical drift can be quantified accurately, we find that BGSDC can deliver higher quality particle distributions than the standard Boris integrator at comparable cost or comparable distributions at lower cost. With collision models, quantifying accuracy is difficult but we show that BGSDC produces stable distributions at larger time steps than Boris.
The Fourier spectrum at a fractional period is often examined when extracting features from biological sequences and time series. It reflects the inner information structure of the sequences. A fractional period is not uncommon in time series. A typical example is the 3.6 period in protein sequences, which determines the $alpha$-helix secondary structure. Computing the spectrum of a fractional period offers a high-resolution insight into a time series. It has thus become an important approach in genomic analysis. However, computing Fourier spectra of fractional periods by the traditional Fourier transform is computationally expensive. In this paper, we present a novel, fast algorithm for directly computing the fractional period spectrum (FPS) of time series. The algorithm is based on the periodic distribution of signal strength at periodic positions of the time series. We provide theoretical analysis, deduction, and special techniques for reducing the computational costs of the algorithm. The analysis of the computational complexity of the algorithm shows that the algorithm is much faster than traditional Fourier transform. Our algorithm can be applied directly in computing fractional periods in time series from a broad of research fields. The computer programs of the FPS algorithm are available at https://github.com/cyinbox/FPS.
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