Multigrid waveform relaxation for the time-fractional heat equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(N M \log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented

Quasinormal ringing of acoustic black holes in Laval nozzles: Numerical simulations

Quasinormal ringing of acoustic black holes in Laval nozzles is discussed. The equation for sounds in a transonic flow is written into a Schr\"{o}dinger-type equation with a potential barrier, and the quasinormal frequencies are calculated semianalytically. From the results of numerical simulations, it is shown that the quasinormal modes are actually excited when the transonic flow is formed or slightly perturbed, as well as in the real black hole case. In an actual experiment, however, the purely-outgoing boundary condition will not be satisfied at late times due to the wave reflection at the end of the apparatus, and a late-time ringing will be expressed as a superposition of "boxed" quasinormal modes. It is shown that the late-time ringing damps more slowly than the ordinary quasinormal ringing, while its central frequency is not greatly different from that of the ordinary one. Using this fact, an efficient way for experimentally detecting the quasinormal ringing of an acoustic black hole is discussed.Comment: 9 pages, 8 figures, accepted for publication in Physical Review

Compressive Parameter Estimation for Sparse Translation-Invariant Signals Using Polar Interpolation

We propose new compressive parameter estimation algorithms that make use of polar interpolation to improve the estimator precision. Our work extends previous approaches involving polar interpolation for compressive parameter estimation in two aspects: (i) we extend the formulation from real non-negative amplitude parameters to arbitrary complex ones, and (ii) we allow for mismatch between the manifold described by the parameters and its polar approximation. To quantify the improvements afforded by the proposed extensions, we evaluate six algorithms for estimation of parameters in sparse translation-invariant signals, exemplified with the time delay estimation problem. The evaluation is based on three performance metrics: estimator precision, sampling rate and computational complexity. We use compressive sensing with all the algorithms to lower the necessary sampling rate and show that it is still possible to attain good estimation precision and keep the computational complexity low. Our numerical experiments show that the proposed algorithms outperform existing approaches that either leverage polynomial interpolation or are based on a conversion to a frequency-estimation problem followed by a super-resolution algorithm. The algorithms studied here provide various tradeoffs between computational complexity, estimation precision, and necessary sampling rate. The work shows that compressive sensing for the class of sparse translation-invariant signals allows for a decrease in sampling rate and that the use of polar interpolation increases the estimation precision.Comment: 13 pages, 5 figures, to appear in IEEE Transactions on Signal Processing; minor edits and correction

Detectability of gravitational wave events by spherical resonant-mass antennas

We have calculated signal-to-noise ratios for eight spherical resonant-mass antennas interacting with gravitational radiation from inspiralling and coalescing binary neutron stars and from the dynamical and secular bar-mode instability of a rapidly rotating star. We find that by using technology that could be available in the next several years, spherical antennas can detect neutron star inspiral and coalescence at a distance of 15 Mpc and the dynamical bar-mode instability at a distance of 2 Mpc.Comment: 39 pages, 4 EPS Figures, some additional SNRs for secular instabilities, some changes to LIGO SNRs, Appendix added on the asymptotic expansion of energy sensitivity, corrected supernova rates. Results available at http://www.physics.umd.edu/rgroups/gen_rel_exp/snr.html Submitted to Phys. Rev.

Gravitational wave bursts from cusps and kinks on cosmic strings

The strong beams of high-frequency gravitational waves (GW) emitted by cusps and kinks of cosmic strings are studied in detail. As a consequence of these beams, the stochastic ensemble of GW's generated by a cosmological network of oscillating loops is strongly non Gaussian, and includes occasional sharp bursts that stand above the ``confusion'' GW noise made of many smaller overlapping bursts. Even if only 10% of all string loops have cusps these bursts might be detectable by the planned GW detectors LIGO/VIRGO and LISA for string tensions as small as $G \mu \sim 10^{-13}$. In the implausible case where the average cusp number per loop oscillation is extremely small, the smaller bursts emitted by the ubiquitous kinks will be detectable by LISA for string tensions as small as $G \mu \sim 10^{-12}$. We show that the strongly non Gaussian nature of the stochastic GW's generated by strings modifies the usual derivation of constraints on $G \mu$ from pulsar timing experiments. In particular the usually considered ``rms GW background'' is, when G \mu \gaq 10^{-7}, an overestimate of the more relevant confusion GW noise because it includes rare, intense bursts. The consideration of the confusion GW noise suggests that a Grand Unified Theory (GUT) value $G \mu \sim 10^{-6}$ is compatible with existing pulsar data, and that a modest improvement in pulsar timing accuracy could detect the confusion noise coming from a network of cuspy string loops down to $G \mu \sim 10^{-11}$. The GW bursts discussed here might be accompanied by Gamma Ray Bursts.Comment: 24 pages, 3 figures, Revtex, submitted to Phys. Rev.

Optimisation of Quantum Trajectories Driven by Strong-field Waveforms

Quasi-free field-driven electron trajectories are a key element of strong-field dynamics. Upon recollision with the parent ion, the energy transferred from the field to the electron may be released as attosecond duration XUV emission in the process of high harmonic generation (HHG). The conventional sinusoidal driver fields set limitations on the maximum value of this energy transfer, and it has been predicted that this limit can be significantly exceeded by an appropriately ramped-up cycleshape. Here, we present an experimental realization of such cycle-shaped waveforms and demonstrate control of the HHG process on the single-atom quantum level via attosecond steering of the electron trajectories. With our optimized optical cycles, we boost the field-ionization launching the electron trajectories, increase the subsequent field-to-electron energy transfer, and reduce the trajectory duration. We demonstrate, in realistic experimental conditions, two orders of magnitude enhancement of the generated XUV flux together with an increased spectral cutoff. This application, which is only one example of what can be achieved with cycle-shaped high-field light-waves, has farreaching implications for attosecond spectroscopy and molecular self-probing