Long-time and unitary properties of semiclassical initial value representations.

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

Full-chip statistical leakage power analysis typically requires quadratic time complexity in the presence of spatial correlation. When spatial correlation are strong (with large spatial correlation length), efficient linear time complexity analysis can be attained as the number of variational variables can be significantly reduced. However this is not the case for circuits where gate leakage currents are weakly correlated. In this paper, we present a linear time algorithm for statistical leakage power analysis in the presence of weak spatial correlation. The new algorithm exploits the fact that gate leakage current can be efficiently computed locally when correlation is weak. We adopt a newly proposed spatial correlation model where a new set of location-dependent uncorrelated variables are defined over virtual grids to represent the original physical random variables via fitting. To compute the leakage current of a gate on the new set of variables, the new method uses the orthogonal polynomials based collocation method, which can be applied to any gate leakage models. The total leakage currents are then computed by simply summing up the resulting orthogonal polynomials (their coefficients) on the new set of variables for all gates. Experimental results show that the proposed method is about two orders of magnitude faster than the recently proposed grid-based method [3] with similar accuracy and many orders of magnitude times over the Monte Carlo method.

Cost-effective waveform modeling is the key to practical reverse time migration (RTM) and full-waveform inversion (FWI) implementations. We evaluated an explicit time evolution (ETE) method to efficiently simulate wave propagation in acoustic media with high temporal accuracy. We started from the constant-density acoustic wave equation and obtained an analytical time-marching scheme in the wavenumber domain. We then formulated an ETE scheme in the time-space domain by introducing a cosine function approximation. Although the ETE operator appears to be similar to the second-order temporal finite-difference (FD) operator, the exact nature of the ETE formula ensures high accuracy in time. We further introduced a set of optimum stencils and coefficients by minimizing evolution errors in a least-squares sense. Our numerical tests indicated that ETE can achieve similar waveform accuracy as FD with four times larger time steps. Meanwhile, the compact ETE operator keeps the computation efficient. The efficiency and capability to handle complex velocity field make ETE an attractive engine in acoustic RTM and FWI.

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

Fourth-order compact finite difference schemes for solving the time fractional diffusion equation with nonlocal boundary conditions are considered in this paper. The second-order derivative in one space variable is approximated by a fourth-order compact finite difference. Two different time derivatives, the Caputo derivative and Riemann–Liouville derivative are considered in the equations that we discuss. The Caputo time derivative is discretized by the L1-approximation and a second-order approximation is used for the Riemann–Liouville derivative. Stability analysis using the discrete energy method for Scheme I is given, and error estimation of this scheme is derived by conducting the local truncation error analysis, using the stability result. Numerical results are provided to show that both two fully discrete schemes have fourth-order accuracy in space, and $$2-\gamma $$ order of accuracy in time for Scheme I with $$\gamma $$ being the order of the fractional derivative, while Scheme II has second-order accuracy in time when the theoretical solutions are smooth with zero initial values.

An efficient method, the self-consistent hybrid method, is proposed for accurately simulating time-dependent quantum dynamics in complex systems. The method is based on an iterative convergence procedure for a dynamical hybrid approach. In this approach, the overall system is first partitioned into a “core” and a “reservoir” (an initial guess). The former is treated via an accurate quantum mechanical method, namely, the time-dependent multiconfiguration self-consistent field or multiconfiguration time-dependent Hartree approach, and the latter is treated via a more approximate method, e.g., classical mechanics, semiclassical initial value representations, quantum perturbation theories, etc. Next, the number of “core” degrees of freedom, as well as other variational parameters, is systematically increased to achieve numerical convergence for the overall quantum dynamics. The method is applied to two examples of quantum dissipative dynamics in the condensed phase: the spin-boson problem and the electronic resonance decay in the presence of a vibrational bath. It is demonstrated that the method provides a practical way of obtaining accurate quantum dynamical results for complex systems.

The traditional high-order staggered-grid finite-difference (SGFD) method has high-order accuracy in space, but only the second-order accuracy in time, which makes the traditional SGFD method suffer from a large temporal dispersion error during long-distance wave propagation. This paper develops temporal fourth- and sixth-order and spatial arbitrary evenorder SGFD schemes to model isotropic elastic wave propagation. The temporal high-order SGFD schemes have smaller temporal dispersion than the traditional temporal second-order scheme, and thus allow larger time steps to attain a similar accuracy. The developed temporal high-order SGFD schemes are applied to simulate a quasi-stress–velocity wave equation (QWE) that is derived in the framework of a $k$ -space approach. A split QWE (SQWE) is further developed, and numerical simulation of SQWE results in separated P (compressional)-wave and S (shear)-wave. Theoretical computational cost analysis verifies that the numerical simulation of QWE using the temporal fourthand sixth-order SGFD schemes is more efficient than the numerical simulation of the traditional stress–velocity wave equation using the traditional temporal second-order SGFD scheme in 2-D. In 3-D, the temporal fourth-order SGFD scheme still runs faster than the traditional temporal second-order scheme; however, the temporal sixth-order scheme is more efficient only when a longer stencil length than 12 is adopted. Numerical examples confirm the correctness of the developed elastic wave modeling schemes.

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.

Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-$N$ Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an $O(1)$ timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the $1/N$ correction, we demonstrate that the system attains complete thermalization on an $O(N)$ timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-$N$ quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.

Fast operator splitting methods for obstacle problems

ABSTRACTThe biggest challenge of the traditional 3D orthogonal matching pursuit (OMP) method for direction-of-departure (DOD), direction-of-arrival (DOA) and Doppler frequency estimation in bistatic multiple-input multiple-output (MIMO) radar is the heavy computational burden due to a large number of atoms in the overcomplete dictionary. In this paper, low complexity 3D-OMP algorithms are investigated. First, the traditional 3D-OMP algorithm is given. Then, two-dimensionality reduced OMP-based algorithms are proposed exploiting the property of Khatri-Rao product and proper sparse representation. Also, the multiple measurement vectors (MMV) model is introduced to our OMP algorithms to guarantee the estimation performance. The simulation results show that the DOD, DOA and Doppler frequency can be effectively estimated with a small number of pulses and low computation cost. With similar accuracy compared with the traditional 3D-OMP method, much lower computational burden can be achieved by using the proposed methods.

Throughout recent years, dynamic time warping (DTW) has remained as a robust similarity measure in time series classification (TSC). 1-nearest neighbor (1-NN) algorithm with DTW is the most widely used classification method on time series serving as a benchmark. With the increasing demand for TSC on low-resource devices and the widespread of wearable devices, the need for a efficient and accurate time series classifier has never been higher. Although 1-NN DTW attains accurate results, it highly falls back on efficiency due to its quadratic complexity in the length of time series. In this paper, we propose a new approximation method for reducing the length of the time series as the input of DTW. We call it control chart approximation (CCA), after a similar concept used in statistical quality control processing. CCA representation approximates raw time series by transforming them into a set of segments with aggregated values and durations forming a reduced 3-D vector. We also propose an adaptation of DTW in 3-D space as a distance measure for 1-NN classifier, and denote the method as 1-NN 3-D DTW. Our experiments on 85 datasets from UCR archive—including 28 long-length (>500 points) time series datasets—show up to two orders of magnitude performance gain in running time compared to the state-of-the-art 1-NN DTW implementation. Moreover, it shows similar or better accuracy on the long time series in the experiment.

Internal conversion is an inherently quantum mechanical process. To date, "ab initio" computation of internal conversion rates was limited to harmonic based approximations. These are questionable since the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the Semiclassical Initial Value Representation (SCIVR) approach which is in principle amenable to "on the fly" studies even with "many" degrees of freedom. In this work we apply the Herman-Kluk-SCIVR methodology to compute the internal conversion rates for formaldehyde for a variety of initial vibronic states. The SCIVR computation gives reasonable agreement with experiment, while the harmonic approximation typically gives rates that are too high.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.