arXiv subjects

Minghua Chen

Publications and source records attributed to Minghua Chen.

At least 55 records · Page 3Linked to original sources

arXiv2021

Correction of high-order $L_k$ approximation for subdiffusion

The subdiffusion equations with a Caputo fractional derivative of order $α\in (0,1)$ arise in a wide variety of practical problems, which is describing the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree $k$ ($k\leq 6$) convolution quadrature, called $L_k$ approximation, for the subdiffusion, which are easy to implement on variable grids. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of $L_k$ approximation by the polylogarithm function or Bose-Einstein integral. To construct a $τ_8$ approximation of Bose-Einstein integral, the desired $(k+1-α)$th-order convergence rate can be proved for the correction $L_k$ scheme with nonsmooth data, which is higher than $k$th-order BDF$k$ method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129--A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.

math.NA↗

arXiv2021

Efficient and Robust Equilibrium Strategies of Utilities in Day-ahead Market with Load Uncertainty

We consider the scenario where $N$ utilities strategically bid for electricity in the day-ahead market and balance the mismatch between the committed supply and actual demand in the real-time market, with uncertainty in demand and local renewable generation in consideration. We model the interactions among utilities as a non-cooperative game, in which each utility aims at minimizing its per-unit electricity cost. We investigate utilities' optimal bidding strategies and show that all utilities bidding according to (net load) prediction is a unique pure strategy Nash Equilibrium with two salient properties. First, it incurs no loss of efficiency; hence, competition among utilities does not increase the social cost. Second, it is robust and (0, $N-1$) fault immune. That is, fault behaviors of irrational utilities only help to reduce other rational utilities' costs. The expected market supply-demand mismatch is minimized simultaneously, which improves the planning and supply-and-demand matching efficiency of the electricity supply chain. We prove the results hold under the settings of correlated prediction errors and a general class of real-time spot pricing models, which capture the relationship between the spot price, the day-ahead clearing price, and the market-level mismatch. Simulations based on real-world traces corroborate our theoretical findings. Our study adds new insights to market mechanism design. In particular, we derive a set of fairly general sufficient conditions for the market operator to design real-time pricing schemes so that the interactions among utilities admit the desired equilibrium.

eess.SY↗

arXiv2021

Principle-driven Fiber Transmission Model based on PINN Neural Network

In this paper, a novel principle-driven fiber transmission model based on physical induced neural network (PINN) is proposed. Unlike data-driven models which regard fiber transmission problem as data regression tasks, this model views it as an equation solving problem. Instead of adopting input signals and output signals which are calculated by SSFM algorithm in advance before training, this principle-driven PINN based fiber model adopts frames of time and distance as its inputs and the corresponding real and imaginary parts of NLSE solutions as its outputs. By taking into account of pulses and signals before transmission as initial conditions and fiber physical principles as NLSE in the design of loss functions, this model will progressively learn the transmission rules. Therefore, it can be effectively trained without the data labels, referred as the pre-calculated signals after transmission in data-driven models. Due to this advantage, SSFM algorithm is no longer needed before the training of principle-driven fiber model which can save considerable time consumption. Through numerical demonstration, the results show that this principle-driven PINN based fiber model can handle the prediction tasks of pulse evolution, signal transmission and fiber birefringence for different transmission parameters of fiber telecommunications.

eess.SP↗

arXiv2021

Weighted and shifted BDF2 methods on variable grids

Variable steps implicit-explicit multistep methods for PDEs have been presented in [17], where the zero-stability is studied for ODEs; however, the stability analysis still remains an open question for PDEs. Based on the idea of linear multistep methods, we present a simple weighted and shifted BDF2 methods with variable steps for the parabolic problems, which serve as a bridge between BDF2 and Crank-Nicolson scheme. The contributions of this paper are as follows: we first prove that the optimal adjacent time-step ratios for the weighted and shifted BDF2, which greatly improve the maximum time-step ratios for BDF2 in [11,15]. Moreover, the unconditional stability and optimal convergence are rigorous proved, which make up for the vacancy of the theory for PDEs in [17]. Finally, numerical experiments are given to illustrate theoretical results.

math.NA↗

arXiv2021

DeepOPF-V: Solving AC-OPF Problems Efficiently

AC optimal power flow (AC-OPF) problems need to be solved more frequently in the future to maintain stable and economic power system operation. To tackle this challenge, a deep neural network-based voltage-constrained approach (DeepOPF-V) is proposed to solve AC-OPF problems with high computational efficiency. Its unique design predicts voltages of all buses and then uses them to reconstruct the remaining variables without solving non-linear AC power flow equations. A fast post-processing process is developed to enforce the box constraints. The effectiveness of DeepOPF-V is validated by simulations on IEEE 118/300-bus systems and a 2000-bus test system. Compared with existing studies, DeepOPF-V achieves decent computation speedup up to four orders of magnitude and comparable performance in optimality gap and preserving the feasibility of the solution.

eess.SY↗

arXiv2021

Optimal Online Algorithms for Peak-Demand Reduction Maximization with Energy Storage

The high proportions of demand charges in electric bills motivate large-power customers to leverage energy storage for reducing the peak procurement from the outer grid. Given limited energy storage, we expect to maximize the peak-demand reduction in an online fashion, challenged by the highly uncertain demands and renewable injections, the non-cumulative nature of peak consumption, and the coupling of online decisions. In this paper, we propose an optimal online algorithm that achieves the best competitive ratio, following the idea of maintaining a constant ratio between the online and the optimal offline peak-reduction performance. We further show that the optimal competitive ratio can be computed by solving a linear number of linear-fractional programs. Moreover, we extend the algorithm to adaptively maintain the best competitive ratio given the revealed inputs and actions at each decision-making round. The adaptive algorithm retains the optimal worst-case guarantee and attains improved average-case performance. We evaluate our proposed algorithms using real-world traces and show that they obtain up to 81% peak reduction of the optimal offline benchmark. Additionally, the adaptive algorithm achieves at least 20% more peak reduction against baseline alternatives.

eess.SY↗

arXiv2021

BDF$6$ SAV schemes for time-fractional Allen-Cahn dissipative systems

Recently, the error analysis of BDF$k$ $(1\leqslant k\leqslant5)$ SAV (scalar auxiliary variable) schemes are given in \cite{Huangg:20} for the classical Allen-Cahn equation. However, it remains unavailable for BDF$6$ SAV schemes. In this paper, we construct and analyze BDF$6$ SAV schemes for the time-fractional dissipative systems. We carry out a rigorous error analysis for the time-fractional Allen-Cahn equation, which also fills up a gap for the classical case. Finally, numerical experiment is shown to illustrate the effectiveness of the presented methods. As far as we know, this is the first SAV schemes for the time-fractional dissipative systems.

math.NA↗

arXiv2021

Hybrid integrated low noise linearly chirped Frequency Modulated Continuous Wave laser source based on self-injection to external cavity

A low noise linearly frequency modulated continuous wave laser source with a wide frequency bandwidth is demonstrated. By two-dimensional thermal tuning, the laser source shows 42 GHz continuous frequency tuning with 49.86 Hz intrinsic linewidth under static operation. For dynamical FMCW, the laser source has 10.25 GHz frequency bandwidth at 100 Hz chirped frequency and 5.56 GHz at 1 kHz chirped frequency. With pre-distortion linearization, it can distinguish 3 m length difference at 45 km distance in the fibre length measured experiment, which demonstrate a potential for application in ultra-long fibre sensing and FMCW LiDAR.

physics.optics↗

arXiv2020

FourierCam:A camera for video spectrum acquisition in single-shot

The novel camera architecture facilitates the development of machine vision. Instead of capturing frame sequences in the temporal domain as traditional video cameras, FourierCam directly measures the pixel-wise temporal spectrum of the video in single-shot through optical coding. Compared with the classic video cameras and time-frequency transformation pipeline, this programmable frequency-domain sampling strategy has an attractive combination of characteristics for low detection bandwidth, low-light imaging, low computational burden and low data volume. Based on the various temporal filter kernel designed by FourierCam, we demonstrated a series of exciting machine vision functions, such as video compression, background subtraction, object extraction, and trajectory tracking.

eess.IV↗

arXiv2020

Backward difference formula: The energy technique for subdiffusion equation

Based on the equivalence of A-stability and G-stability, the energy technique of the six-step BDF method for the heat equation has been discussed in [Akrivis, Chen, Yu, Zhou, Math. Comp., Revised]. Unfortunately, this theory is hard to extend the time-fractional PDEs. In this work, we consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations. We present a novel and concise stability analysis of time stepping schemes generated by $k$-step backward difference formula (BDF$k$), for approximately solving the subdiffusion equation. The analysis mainly relies on the energy technique by applying Grenander-Szegö theorem. This kind of argument has been widely used to confirm the stability of various $A$-stable schemes (e.g., $k=1,2$). However, it is not an easy task for the higher-order BDF methods, due to the loss the $A$-stability. The core object of this paper is to fill in this gap.

math.NA↗

arXiv2020

DeepOPF: Deep Neural Network for DC Optimal Power Flow

We develop DeepOPF as a Deep Neural Network (DNN) approach for solving direct current optimal power flow (DC-OPF) problems. DeepOPF is inspired by the observation that solving DC-OPF for a given power network is equivalent to characterizing a high-dimensional mapping between the load inputs and the dispatch and transmission decisions. We construct and train a DNN model to learn such mapping, then we apply it to obtain optimized operating decisions upon arbitrary load inputs. We adopt uniform sampling to address the over-fitting problem common in generic DNN approaches. We leverage on a useful structure in DC-OPF to significantly reduce the mapping dimension, subsequently cutting down the size of our DNN model and the amount of training data/time needed. We also design a post-processing procedure to ensure the feasibility of the obtained solution. Simulation results of IEEE test cases show that DeepOPF always generates feasible solutions with negligible optimality loss, while speeding up the computing time by two orders of magnitude as compared to conventional approaches implemented in a state-of-the-art solver.

eess.SY↗

arXiv2020

DeepOPF: A Deep Neural Network Approach for Security-Constrained DC Optimal Power Flow

We develop DeepOPF as a Deep Neural Network (DNN) approach for solving security-constrained direct current optimal power flow (SC-DCOPF) problems, which are critical for reliable and cost-effective power system operation.DeepOPF is inspired by the observation that solving SC-DCOPF problems for a given power network is equivalent to depicting a high-dimensional mapping from the load inputs to the generation and phase angle outputs. We first train a DNN to learn the mapping and predict the generations from the load inputs. We then directly reconstruct the phase angles from the generations and loads by using the power flow equations. Such a predict-and-reconstruct approach reduces the dimension of the mapping to learn, subsequently cutting down the size of the DNN and the amount of training data needed. We further derive a condition for tuning the size of the DNN according to the desired approximation accuracy of the load-generation mapping. We develop a post-processing procedure based on $\ell_1$-projection to ensure the feasibility of the obtained solution, which can be of independent interest. Simulation results for IEEE test cases show that DeepOPF generates feasible solutions with less than 0.2% optimality loss, while speeding up the computation time by up to two orders of magnitude as compared to a state-of-the-art solver.

eess.SY↗

arXiv2020

DeepOPF+: A Deep Neural Network Approach for DC Optimal Power Flow for Ensuring Feasibility

Deep Neural Networks (DNNs) approaches for the Optimal Power Flow (OPF) problem received considerable attention recently. A key challenge of these approaches lies in ensuring the feasibility of the predicted solutions to physical system constraints. Due to the inherent approximation errors, the solutions predicted by DNNs may violate the operating constraints, e.g., the transmission line capacities, limiting their applicability in practice. To address this challenge, we develop DeepOPF+ as a DNN approach based on the so-called "preventive" framework. Specifically, we calibrate the generation and transmission line limits used in the DNN training, thereby anticipating approximation errors and ensuring that the resulting predicted solutions remain feasible. We theoretically characterize the calibration magnitude necessary for ensuring universal feasibility. Our DeepOPF+ approach improves over existing DNN-based schemes in that it ensures feasibility and achieves a consistent speed up performance in both light-load and heavy-load regimes. Detailed simulation results on a range of test instances show that the proposed DeepOPF+ generates 100% feasible solutions with minor optimality loss. Meanwhile, it achieves a computational speedup of two orders of magnitude compared to state-of-the-art solvers.

eess.SY↗

arXiv2020

High order algorithms for Fokker-Planck equation with Caputo-Fabrizio fractional derivative

Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures with different scales. Extending the discretizations for fractional substantial calculus [Chen and Deng, \emph{ ESAIM: M2AN.} \textbf{49}, (2015), 373--394], we first provide the numerical discretizations of the Caputo-Fabrizio fractional derivative with the global truncation error $\mathcal{O}(τ^ν)$ $ (ν=1,2,3,4)$. Then we use the derived schemes to solve the Caputo-Fabrizio fractional diffusion equation. By analysing the positive definiteness of the stiffness matrices of the discretized Caputo-Fabrizio operator, the unconditional stability and the convergence with the global truncation error $\mathcal{O}(τ^2+h^2)$ are theoretically proved and numerical verified.

math.NA↗

arXiv2020

Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation

The main contribution of this work is to construct and analyze stable and high order schemes to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite difference method in time and spectral methods in space, the proposed scheme is unconditionally stable and has the global truncation error $\mathcal{O}(τ^3+N^{-m})$, where $τ$, $N$ and $m$ are the time step size, polynomial degree and regularity in the space variable of the exact solution, respectively. It should be noted that the global truncation error $\mathcal{O}(τ^2+N^{-m})$ is well established in [ Li, Lv and Xu, {\em Numer. Methods Partial Differ. Equ}. (2019)]. Finally, some numerical experiments are carried out to verify the theoretical analysis. To the best of our knowledge, this is the first proof for the stability of the third-order scheme for the Caputo-Fabrizio fractional operator.

math.NA↗

arXiv2020

The energy technique for the six-step BDF method

In combination with the Grenander--Szegö theorem, we observe that a relaxed positivity condition on multipliers, milder than the basic %fundamental requirement of the Nevanlinna--Odeh multipliers that the sum of the absolute values of their components is strictly less than $1$, makes the energy technique applicable to the stability analysis of BDF methods for parabolic equations with selfadjoint elliptic part. This is particularly useful for the six-step BDF method for which no Nevanlinna--Odeh multiplier exists. We introduce multipliers satisfying the positivity property for the six-step BDF method and establish stability of the method for parabolic equations.

math.NA↗

arXiv2020

Correction of BDFk for fractional Feynman-Kac equation with Lévy flight

In this work, we present the correction formulas of the $k$-step BDF convolution quadrature at the starting $k-1$ steps for the fractional Feynman-Kac equation with Lévy flight. The desired $k$th-order convergence rate can be achieved with nonsmooth data. Based on the idea of [{\sc Jin, Li, and Zhou}, SIAM J. Sci. Comput., 39 (2017), A3129--A3152], we provide a detailed convergence analysis for the correction BDF$k$ scheme. The numerical experiments with spectral method are given to illustrate the effectiveness of the presented method. To the best of our knowledge, this is the first proof of the convergence analysis and numerical verified the sapce fractional evolution equation with correction BDF$k$.

math.NA↗

arXiv2020

Fourier temporal ghost imaging

Ghost imaging is a fascinating framework which constructs the image of an object by correlating measurements between received beams and reference beams, none of which carries the structure information of the object independently. Recently, by taking into account space-time duality in optics, computational temporal ghost imaging has attracted attentions. Here, we propose a novel Fourier temporal ghost imaging (FTGI) scheme to achieve single-shot non-reproducible temporal signals. By sinusoidal coded modulation, ghost images are obtained and recovered by applying Fourier transformation. For demonstration, non-repeating events are detected with single-shot exposure architecture. It's shown in results that the peak signal-to-noise ratio (PSNR) of FTGI is significantly better (13dB increase) than traditional temporal ghost imaging in the same condition. In addition, by using the obvious physical meaning of Fourier spectrum, we show some potential applications of FTGI, such as frequency division multiplexing demodulation in the visible light communications.

eess.SP↗