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Register a new userComputational efficiency of symplectic integration schemes: Application to multidimensional disordered Klein-Gordon lattices
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We implement several symplectic integrators, which are based on two part splitting, for studying the chaotic behavior of one- and two-dimensional disordered Klein-Gordon lattices with many degrees of freedom and investigate their numerical performance. For this purpose, we perform extensive numerical simulations by considering many different initial energy excitations and following the evolution of the created wave packets in the various dynamical regimes exhibited by these models. We compare the efficiency of the considered integrators by checking their ability to correctly reproduce several features of the wave packets propagation, like the characteristics of the created energy distribution and the time evolution of the systems maximum Lyapunov exponent estimator. Among the tested integrators the fourth order $ABA864$ scheme cite{BCFLMM13} showed the best performance as it needed the least CPU time for capturing the correct dynamical behavior of all considered cases when a moderate accuracy in conserving the systems total energy value was required. Among the higher order schemes used to achieve a better accuracy in the energy conservation, the sixth order scheme $s11ABA82_6$ exhibited the best performance.
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We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrodinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the systems Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to $Napprox;$70 sites. An advantage of the latter methods is the better conservation of the systems second integral, i.e. the wave packets norm.
Simulations of systems with quenched disorder are extremely demanding, suffering from the combined effect of slow relaxation and the need of performing the disorder average. As a consequence, new algorithms, improved implementations, and alternative and even purpose-built hardware are often instrumental for conducting meaningful studies of such systems. The ensuing demands regarding hardware availability and code complexity are substantial and sometimes prohibitive. We demonstrate how with a moderate coding effort leaving the overall structure of the simulation code unaltered as compared to a CPU implementation, very significant speed-ups can be achieved from a parallel code on GPU by mainly exploiting the trivial parallelism of the disorder samples and the near-trivial parallelism of the parallel tempering replicas. A combination of this massively parallel implementation with a careful choice of the temperature protocol for parallel tempering as well as efficient cluster updates allows us to equilibrate comparatively large systems with moderate computational resources.
Restricted solid on solid surface growth models can be mapped onto binary lattice gases. We show that efficient simulation algorithms can be realized on GPUs either by CUDA or by OpenCL programming. We consider a deposition/evaporation model following Kardar-Parisi-Zhang growth in 1+1 dimensions related to the Asymmetric Simple Exclusion Process and show that for sizes, that fit into the shared memory of GPUs one can achieve the maximum parallelization speedup ~ x100 for a Quadro FX 5800 graphics card with respect to a single CPU of 2.67 GHz). This permits us to study the effect of quenched columnar disorder, requiring extremely long simulation times. We compare the CUDA realization with an OpenCL implementation designed for processor clusters via MPI. A two-lane traffic model with randomized turning points is also realized and the dynamical behavior has been investigated.
We introduce a modified version of the disordered Klein-Gordon lattice model, having two parameters for controlling the disorder strength: $D$, which determines the range of the coefficients of the on-site potentials, and $W$, which defines the strength of the nearest-neighbor interactions. We fix $W=4$ and investigate how the properties of the systems normal modes change as we approach its ordered version, i.e. $Drightarrow 0$. We show that the probability density distribution of the normal modes frequencies takes a `U-shaped profile as $D$ decreases. Furthermore, we use two quantities for estimating the modes spatial extent, the so-called localization volume $V$ (which is related to the modes second moment) and the modes participation number $P$. We show that both quantities scale as $propto D^{-2}$ when $D$ approaches zero and we numerically verify a proportionality relation between them as $V/P approx 2.6$.
Differentiable programming has emerged as a key programming paradigm empowering rapid developments of deep learning while its applications to important computational methods such as Monte Carlo remain largely unexplored. Here we present the general theory enabling infinite-order automatic differentiation on expectations computed by Monte Carlo with unnormalized probability distributions, which we call automatic differentiable Monte Carlo (ADMC). By implementing ADMC algorithms on computational graphs, one can also leverage state-of-the-art machine learning frameworks and techniques to traditional Monte Carlo applications in statistics and physics. We illustrate the versatility of ADMC by showing some applications: fast search of phase transitions and accurately finding ground states of interacting many-body models in two dimensions. ADMC paves a promising way to innovate Monte Carlo in various aspects to achieve higher accuracy and efficiency, e.g. easing or solving the sign problem of quantum many-body models through ADMC.
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