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Dynamical low-rank approximation allows for solving large-scale matrix differential equations (MDEs) with significantly fewer degrees of freedom and has been applied to a growing number of applications. However, most existing techniques rely on explicit time integration schemes. In this work, we introduce a cost-effective Newton's method for the implicit time integration of stiff, nonlinear MDEs on low-rank matrix manifolds. Our methodology is focused on MDEs resulting from the discretization of random partial differential equations (PDEs). Cost-effectiveness is achieved by solving the MDE at the minimum number of entries required for a rank-$r$ approximation. We present a novel CUR low-rank approximation that requires solving the parametric PDE at $r$ strategically selected parameters and $\mathcal{O}(r)$ grid points using Newton's method. The selected random samples and grid points adaptively vary over time and are chosen using the discrete empirical interpolation method or similar techniques. The proposed methodology is developed for high-order implicit multistep and Runge-Kutta schemes and incorporates rank adaptivity, allowing for dynamic rank adjustment over time to control error. Several analytical and PDE examples, including the stochastic Burgers' and Gray-Scott equations, demonstrate the accuracy and efficiency of the presented methodology.

We propose a feasible experimental scheme to improve the few-photon optomechanical effects, including photon blockade and mechanical-Schrodinger cat-state generation, as well as photon-phonon entanglement in a tripartite microwave optomechanical circuit. The system under consideration is formed by a single-Cooper-pair transistor, a microwave LC resonator, and a micromechanical resonator. Our scheme is based on an additional higher-order (generalized) nonlinear cross-Kerr type of coupling, linearly dependent on photon number while quadratically dependent on mechanical phonon one, which can be realized via adjusting the gate charge of the Cooper-pair transistor. We show, both analytically and numerically, that the presence of both cross-Kerr and generalized cross-Kerr nonlinearities not only may give rise to the enhancement of one- and two-photon blockades as well as photon induced tunneling but can also provide more controllability over them. Furthermore, it is shown that in the regime of zero optomechanical coupling, with the aid of generalized cross-Kerr nonlinearity, one can generate multi-components mechanical superposition states which exhibit robustness against system dissipations. We also study the steady-state entanglement between the microwave and mechanical modes, the results of which signify the role of generalized cross-Kerr nonlinearity in enhancing the entanglement in the regime of large-red detuning. The proposed generalized cross-Kerr optomechanical system can be found potential applications in microwave quantum sensing, quantum telecommunication, and quantum information protocols.

Image retargeting aims at altering an image size while preserving important content and minimizing noticeable distortions. However, previous image retargeting methods create outputs that suffer from artifacts and distortions. Besides, most previous works attempt to retarget the background and foreground of the input image simultaneously. Simultaneous resizing of the foreground and background causes changes in the aspect ratios of the objects. The change in the aspect ratio is specifically not desirable for human objects. We propose a retargeting method that overcomes these problems. The proposed approach consists of the following steps. Firstly, an inpainting method uses the input image and the binary mask of foreground objects to produce a background image without any foreground objects. Secondly, the seam carving method resizes the background image to the target size. Then, a super-resolution method increases the input image quality, and we then extract the foreground objects. Finally, the retargeted background and the extracted super-resolued objects are fed into a particle swarm optimization algorithm (PSO). The PSO algorithm uses aesthetic quality assessment as its objective function to identify the best location and size for the objects to be placed in the background. We used image quality assessment and aesthetic quality assessment measures to show our superior results compared to popular image retargeting techniques., Comment: 12 pages, 9 figures

Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-$r$ reduced-order model (ROM) based on time-dependent basis, a.k.a. TDB-ROM, is roughly equal to that of solving the full-order model for $r$ random samples. As of now, this nominal performance can only be achieved for linear or quadratic SPDEs -- at the expense of a highly intrusive process. On the other hand, for problems with non-polynomial nonlinearity, the computational cost of solving the TDB evolution equations is the same as solving the full-order model. In this work, we present an adaptive sparse interpolation algorithm that enables stochastic TDB-ROMs to achieve nominal computational cost for generic nonlinear SPDEs. Our algorithm constructs a low-rank approximation for the right hand side of the SPDE using the discrete empirical interpolation method (DEIM). The presented algorithm does not require any offline computation and as a result the low-rank approximation can adapt to any transient changes of the dynamics on the fly. We also propose a rank-adaptive strategy to control the error of the sparse interpolation. Our algorithm achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers' equation, and (2) two-dimensional compressible Navier-Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, the presented algorithm results in orders of magnitude reduction in the computational cost., Comment: 31 pages, 14 figures

Unsteady fluid systems are nonlinear high-dimensional dynamical systems that may exhibit multiple complex phenomena both in time and space. Reduced Order Modeling (ROM) of fluid flows has been an active research topic in the recent decade with the primary goal to decompose complex flows to a set of features most important for future state prediction and control, typically using a dimensionality reduction technique. In this work, a novel data-driven technique based on the power of deep neural networks for reduced order modeling of the unsteady fluid flows is introduced. An autoencoder network is used for nonlinear dimension reduction and feature extraction as an alternative for singular value decomposition (SVD). Then, the extracted features are used as an input for long short-term memory network (LSTM) to predict the velocity field at future time instances. The proposed autoencoder-LSTM method is compared with non-intrusive reduced order models based on dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD). Moreover, an autoencoder-DMD algorithm is introduced for reduced order modeling, which uses the autoencoder network for dimensionality reduction rather than SVD rank truncation. Results show that the autoencoder-LSTM method is considerably capable of predicting fluid flow evolution, where higher values for coefficient of determination $R^{2}$ are obtained using autoencoder-LSTM compared to other models., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Phys. Fluids 32, 105104 (2020), and may be found at https://doi.org/10.1063/5.0020526

Photon-number statistics of the emitted photons from a quantum dot placed in the vicinity of a metallic nanoparticle (with either shell or solid-sphere geometry) in the non-Markovian regime is investigated theoretically. In the model scheme, the quantum dot is considered as a InAs three- level system in Lambda-type configuration with two transition channels. One of channels is driven by a polarized classical field while the two channels are coupled to the plasmon modes. Plasmon resonance modes of a nanoshell, in contrast of a nanosphere, are tunable at demand frequency by controlling the thickness and the materials of the core and the embedding media. The results reveal that the emitted photons from the hybrid system under consideration are antibunched. Moreover, the anti- bunching behavior of the emitted photons can be controlled by the geometrical parameters of the system, namely, the quantum dot-metal nano particle separation distance, as well as the system's physical parameters including the detuning frequency of the quantum dot transitions with respect to the surface plasmon modes, and the Rabi frequency of the polarized driving field. Additionally, the studied system has the potential to be a highly controllable single-photon source.

The entanglement between photon pairs generated from the biexciton cascade transition in a semi- conductor quantum dot located in the vicinity of a metal nanoparticle is theoretically investigated. In the model scheme, the biexciton-exciton and exciton-ground state transitions are assumed to be coupled to two principal plasmon modes of orthogonal polarizations. For a broad spectral window, because the horizontal and vertical spectra are overlapped, the biexciton and exciton photons are degenerate in energy. This allows us to overcome the natural splitting between the intermediate exciton states. Moreover, the degree of entanglement depends on the geometrical parameters of the system. i.e., the radius of the metal nanoparticle and the distance between the quantum dot and the nanoparticle. The results reveal that such a hybrid system profoundly modifies the photon entanglement even in the absence of strong coupling between the emitter and the metal nanosphere., Comment: 13 pages, 17 figures