arXiv subjects

Minghua Chen

Publications and source records attributed to Minghua Chen.

At least 73 records · Page 4Linked to original sources

arXiv2020

Fast and High-order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in $\mathbb{R}^2

In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(τ^2+h^{4-γ}\right)$ with $0<γ<1$, where $τ$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in non-local problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $O(M \log M)$ operations where $M$ is the number of collocation points.

math.NA↗

arXiv2019

A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels

As is well known, using piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC), respectively, to approximate the weakly singular integral $$I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^γ}dy, \quad x \in (a,b) ,\quad 0< γ<1,$$ have the local truncation error $\mathcal{O}\left(h^2\right)$ and $\mathcal{O}\left(h^{4-γ}\right)$. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $λu(x)- I(a,b,x) =f(x)$ with $ λ\neq 0$, also have global convergence rate $\mathcal{O}\left(h^2\right)$ and $\mathcal{O}\left(h^{4-γ}\right)$ in [Atkinson and Han, Theoretical Numerical Analysis, Springer, 2009]. Formally, following nonlocal models can be viewed as Fredholm weakly singular integral equations $$\int^b_a \frac{u(x)-u(y)}{|x-y|^γ}dy =f(x), \quad x \in (a,b) ,\quad 0< γ<1.$$ However, there are still some significant differences for the models in these two fields. In the first part of this paper we prove that the weakly singular integral by PQC have an optimal local truncation error $\mathcal{O}\left(h^4η_i^{-γ}\right)$, where $η_i=\min\left\{x_i-a,b-x_i\right\}$ and $x_i$ coincides with an element junction point. Then a sharp global convergence estimate with $\mathcal{O}\left(h\right)$ and $\mathcal{O}\left(h^3\right)$ by PLC and PQC, respectively, are established for nonlocal problems. Finally, the numerical experiments including two-dimensional case are given to illustrate the effectiveness of the presented method.

math.NA↗

arXiv2019

Electro-optical Neural Networks based on Time-stretch Method

In this paper, a novel architecture of electro-optical neural networks based on the time-stretch method is proposed and numerically simulated. By stretching time-domain ultrashort pulses, multiplications of large scale weight matrices and vectors can be implemented on light and multiple-layer of feedforward neural network operations can be easily implemented with fiber loops. Via simulation, the performance of a three-layer electro-optical neural network is tested by the handwriting digit recognition task and the accuracy reaches 88% under considerable noise.

eess.SP↗

arXiv2019

A Probabilistic Approach for Demand-Aware Ride-Sharing Optimization

Ride-sharing is a modern urban-mobility paradigm with tremendous potential in reducing congestion and pollution. Demand-aware design is a promising avenue for addressing a critical challenge in ride-sharing systems, namely joint optimization of request-vehicle assignment and routing for a fleet of vehicles. In this paper, we develop a probabilistic demand-aware framework to tackle the challenge. We focus on maximizing the expected number of passenger pickups, given the probability distributions of future demands. The key idea of our approach is to assign requests to vehicles in a probabilistic manner. It differentiates our work from existing ones and allows us to explore a richer design space to tackle the request-vehicle assignment puzzle with a performance guarantee but still keeping the final solution practically implementable. The optimization problem is non-convex, combinatorial, and NP-hard in nature. As a key contribution, we explore the problem structure and propose an elegant approximation of the objective function to develop a dual-subgradient heuristic. We characterize a condition under which the heuristic generates a $\left(1-1/e\right)$ approximation solution. Our solution is simple and scalable, amendable for practical implementation. Results of numerical experiments based on real-world traces in Manhattan show that, as compared to a conventional demand-oblivious scheme, our demand-aware solution improves the passenger pickups by up to 46%. The results also show that joint optimization at the fleet level leads to 19% more pickups than that by separate optimizations at individual vehicles.

eess.SY↗

arXiv2019

Minimizing Age-of-Information with Throughput Requirements in Multi-Path Network Communication

We consider the scenario where a sender periodically sends a batch of data to a receiver over a multi-hop network, possibly using multiple paths. Our objective is to minimize peak/average Age-of-Information (AoI) subject to throughput requirements. The consideration of batch generation and multi-path communication differentiates our AoI study from existing ones. We first show that our AoI minimization problems are NP-hard, but only in the weak sense, as we develop an optimal algorithm with a pseudo-polynomial time complexity. We then prove that minimizing AoI and minimizing maximum delay are "roughly" equivalent, in the sense that any optimal solution of the latter is an approximate solution of the former with bounded optimality loss. We leverage this understanding to design a general approximation framework for our problems. It can build upon any $α$-approximation algorithm of the maximum delay minimization problem, to construct an $(α+c)$-approximate solution for minimizing AoI. Here $c$ is a constant depending on the throughput requirements. Simulations over various network topologies validate the effectiveness of our approach.

cs.NI↗

arXiv2019

A second-order accurate scheme for two-dimensional space fractional diffusion equations with time Caputo-Fabrizio fractional derivative

We provide and analyze a second order scheme for the model describing the functional distributions of particles performing anomalous motion with exponential Debye pattern and no-time-taking jumps eliminated, and power-law jump length. The model is derived in [M. Chen, J. Shi, W. Deng, arXiv:1809.03263], being called the space fractional diffusion equation with the time Caputo-Fabrizio fractional derivative. The designed schemes are unconditionally stable and have the second order global truncation error with the nonzero initial condition, being theoretically proved and numerically verified by two methods (a prior estimate with $L^2$-norm and mathematical induction with $l_\infty$ norm). Moreover, the optimal estimates are obtained.

math.NA↗

arXiv2019

Competitive Online Optimization under Inventory Constraints

This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization that is a generalization of the classic one-way trading problem and has a wide range of applications. We present a new algorithmic framework, \textsf{CR-Pursuit}, and prove that it achieves the minimal competitive ratio among all deterministic algorithms (up to a problem-dependent constant factor) for inventory-constrained online optimization. Our algorithm and its analysis not only simplify and unify the state-of-the-art results for the standard one-way trading problem, but they also establish novel bounds for generalizations including concave revenue functions. For example, for one-way trading with price elasticity, the \textsf{CR-Pursuit} algorithm achieves a competitive ratio that is within a small additive constant (i.e., 1/3) to the lower bound of $\ln θ+1$, where $θ$ is the ratio between the maximum and minimum base prices.

cs.PF↗

arXiv2019

Balancing Cost and Dissatisfaction in Online EV Charging under Real-time Pricing

We consider an increasingly popular demand-response scenario where a user schedules the flexible electric vehicle (EV) charging load in response to real-time electricity prices. The objective is to minimize the total charging cost with user dissatisfaction taken into account. We focus on the online setting where neither accurate prediction nor distribution of future real-time prices is available to the user when making irrevocable charging decision in each time slot. The emphasis on considering user dissatisfaction and achieving optimal competitive ratio differentiates our work from existing ones and makes our study uniquely challenging. Our key contribution is two simple online algorithms with the best possible competitive ratio among all deterministic algorithms. The optimal competitive ratio is upper-bounded by \min\left\{ \sqrt{α/p_{\min}},p_{\max}/p_{\min}\right\} and the bound is asymptotically tight with respect to α, where p_{\max} and p_{\min} are the upper and lower bounds of real-time prices and α\geq p_{\min} captures the consideration of user dissatisfaction. The bound under small and large values of αsuggests the fundamental difference of the problems with considering user dissatisfaction (αtakes small values) and without (αtakes large values). Along the way we also develop a general technique for designing online algorithms from an idea hinted in One-Way-Trading Problem, which can be of independent interest. Simulation results based on real-world traces corroborate our theoretical findings and show that our algorithms achieve substantial performance gain as compared to conceivable alternatives. The results also suggest that increasing EV charging rate limit decreases overall cost almost linearly.

eess.SY↗

arXiv2019

Time-encoded single-pixel 3D imaging

Recently three-dimensional (3D) imaging achieves tremendous success in consumer and industrial manufacturing. However, current 3D imaging methods rarely observe dynamical events happening in a short time interval due to the imaging speed limitation. Here we propose a time-encoded single-pixel 3D (TESP-3D) imaging method based on active infrared imaging with high-speed structured illumination, which can acquire the 3D information of dynamic events at an ultra-high frame rate. This method uses single-pixel photodetectors instead of conventional CCD or CMOS devices as the imaging sensors to overcome the image capturing speed limitation. As a proof of concept, we perform continuous real-time 3D scanning imaging at a record 500,000 fps and acquire 3D information of objects moving at a speed of up to 25 m/s. We anticipate that this technology will have important applications in industrial on-line inspection and high throughput 3D screening.

physics.optics↗

arXiv2018

Network Utility Maximization under Maximum Delay Constraints and Throughput Requirements

We consider the problem of maximizing aggregate user utilities over a multi-hop network, subject to link capacity constraints, maximum end-to-end delay constraints, and user throughput requirements. A user's utility is a concave function of the achieved throughput or the experienced maximum delay. The problem is important for supporting real-time multimedia traffic, and is uniquely challenging due to the need of simultaneously considering maximum delay constraints and throughput requirements. We first show that it is NP-complete either (i) to construct a feasible solution strictly meeting all constraints, or (ii) to obtain an optimal solution after we relax maximum delay constraints or throughput requirements up to constant ratios. We then develop a polynomial-time approximation algorithm named PASS. The design of PASS leverages a novel understanding between non-convex maximum-delay-aware problems and their convex average-delay-aware counterparts, which can be of independent interest and suggest a new avenue for solving maximum-delay-aware network optimization problems. Under realistic conditions, PASS achieves constant or problem-dependent approximation ratios, at the cost of violating maximum delay constraints or throughput requirements by up to constant or problem-dependent ratios. PASS is practically useful since the conditions for PASS are satisfied in many popular application scenarios. We empirically evaluate PASS using extensive simulations of supporting video-conferencing traffic across Amazon EC2 datacenters. Compared to existing algorithms and a conceivable baseline, PASS obtains up to $100\%$ improvement of utilities, by meeting the throughput requirements but relaxing the maximum delay constraints that are acceptable for practical video conferencing applications.

cs.NI↗

arXiv2018

Device-to-Device Load Balancing for Cellular Networks

Small-cell architecture is widely adopted by cellular network operators to increase network capacity. By reducing the size of cells, operators can pack more (low-power) base stations in an area to better serve the growing demands, without causing extra interference. However, this approach suffers from low spectrum temporal efficiency. When a cell becomes smaller and covers fewer users, its total traffic fluctuates significantly due to insufficient traffic aggregation and exhibiting a large "peak-to-mean" ratio. As operators customarily provision spectrum for peak traffic, large traffic temporal fluctuation inevitably leads to low spectrum temporal efficiency. In this paper, we advocate device-to-device (D2D) load-balancing as a useful mechanism to address the fundamental drawback of small-cell architecture. The idea is to shift traffic from a congested cell to its adjacent under-utilized cells by leveraging inter-cell D2D communication, so that the traffic can be served without using extra spectrum, effectively improving the spectrum temporal efficiency. We provide theoretical modeling and analysis to characterize the benefit of D2D load balancing, in terms of total spectrum requirements of all individual cells. We also derive the corresponding cost, in terms of incurred D2D traffic overhead. We carry out empirical evaluations based on real-world 4G data traces to gauge the benefit and cost of D2D load balancing under practical settings. The results show that D2D load balancing can reduce the spectrum requirement by 25% as compared to the standard scenario without D2D load balancing, at the expense of negligible 0.7% D2D traffic overhead.

cs.NI↗

arXiv2018

A Multigrid method for nonlocal problems: non-diagonally dominant Toeplitz-plus-tridiagonal systems

The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the Lévy processes and stochastic interfaces. This paper deals with the nonlocal problems on a bounded domain, where the stiffness matrices of the resulting systems are Toeplitz-plus-tridiagonal and far from being diagonally dominant, as it occurs when dealing with linear finite element approximations. By exploiting a weakly diagonally dominant Toeplitz property of the stiffness matrices, the optimal convergence of the two-grid method is well established [Fiorentino and Serra-Capizzano, {\em SIAM J. Sci. Comput.}, {17} (1996), pp. 1068--1081; Chen and Deng, {\em SIAM J. Matrix Anal. Appl.}, {38} (2017), pp. 869--890]; and there are still questions about best ways to define coarsening and interpolation operator when the stiffness matrix is far from being weakly diagonally dominant [Stüben, {\em J. Comput. Appl. Math.}, {128} (2001), pp. 281--309]. In this work, using spectral indications from our analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are {\em neither Toeplitz nor weakly diagonally dominant} corresponding to the fractional Laplacian kernel and the constant kernel, respectively. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such situations. Moreover, the convergence of the full multigrid is also discussed with the constant kernel. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform, where $N$ is the number of the grid points.

math.NA↗

arXiv2018

Learning-aided Stochastic Network Optimization with Imperfect State Prediction

We investigate the problem of stochastic network optimization in the presence of imperfect state prediction and non-stationarity. Based on a novel distribution-accuracy curve prediction model, we develop the predictive learning-aided control (PLC) algorithm, which jointly utilizes historic and predicted network state information for decision making. PLC is an online algorithm that requires zero a-prior system statistical information, and consists of three key components, namely sequential distribution estimation and change detection, dual learning, and online queue-based control. Specifically, we show that PLC simultaneously achieves good long-term performance, short-term queue size reduction, accurate change detection, and fast algorithm convergence. In particular, for stationary networks, PLC achieves a near-optimal $[O(ε)$, $O(\log(1/ε)^2)]$ utility-delay tradeoff. For non-stationary networks, \plc{} obtains an $[O(ε), O(\log^2(1/ε)$ $+ \min(ε^{c/2-1}, e_w/ε))]$ utility-backlog tradeoff for distributions that last $Θ(\frac{\max(ε^{-c}, e_w^{-2})}{ε^{1+a}})$ time, where $e_w$ is the prediction accuracy and $a=Θ(1)>0$ is a constant (the Backpressue algorithm \cite{neelynowbook} requires an $O(ε^{-2})$ length for the same utility performance with a larger backlog). Moreover, PLC detects distribution change $O(w)$ slots faster with high probability ($w$ is the prediction size) and achieves an $O(\min(ε^{-1+c/2}, e_w/ε)+\log^2(1/ε))$ convergence time. Our results demonstrate that state prediction (even imperfect) can help (i) achieve faster detection and convergence, and (ii) obtain better utility-delay tradeoffs.

math.OC↗

arXiv2018

On the Min-Max-Delay Problem: NP-completeness, Algorithm, and Integrality Gap

We study a delay-sensitive information flow problem where a source streams information to a sink over a directed graph G(V,E) at a fixed rate R possibly using multiple paths to minimize the maximum end-to-end delay, denoted as the Min-Max-Delay problem. Transmission over an edge incurs a constant delay within the capacity. We prove that Min-Max-Delay is weakly NP-complete, and demonstrate that it becomes strongly NP-complete if we require integer flow solution. We propose an optimal pseudo-polynomial time algorithm for Min-Max-Delay, with time complexity O(\log (Nd_{\max}) (N^5d_{\max}^{2.5})(\log R+N^2d_{\max}\log(N^2d_{\max}))), where N = \max\{|V|,|E|\} and d_{\max} is the maximum edge delay. Besides, we show that the integrality gap, which is defined as the ratio of the maximum delay of an optimal integer flow to the maximum delay of an optimal fractional flow, could be arbitrarily large.

cs.DS↗

arXiv2018

Uniform convergence of multigrid finite element method for time-dependent Riesz tempered fractional problem

In this article a theoretical framework for the Galerkin finite element approximation to the time-dependent Riesz tempered fractional problem is provided without the fractional regularity assumption. Because the time-dependent problems should become easier to solve as the time step $τ\rightarrow 0$, which correspond to the mass matrix dominant [R. E. Bank and T. Dupont, {\em Math. Comp.}, 153 (1981), pp. 35--51]. Based on the introduced and analysis of the fractional $τ$-norm, the uniform convergence estimates of the V-cycle multigrid method with the time-dependent fractional problem is strictly proved, which means that the convergence rate of the V-cycle MGM is independent of the mesh size $h$ and the time step $τ$. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform method, where $N$ is the number of the grid points. To the best of our knowledge, this is the first proof for the convergence rate of the V-cycle multigrid finite element method with $τ\rightarrow 0$.

math.NA↗

arXiv2017

Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation

In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the contributions of this paper are as follows: (1) It tackles the Toeplitz systems directly for the elliptic PDEs. (2) Simple (traditional) restriction operator and prolongation operator are employed in order to handle general Toeplitz systems at each level of the recursion. Such a technique is then applied to systems of algebraic equations generated by the difference scheme of the two-dimensional fractional Feynman-Kac equation, which describes the joint probability density function of non-Brownian motion. In particular, we consider the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and Galerkin approach (algebraic MGM), which lead to the distinct coarsening stiffness matrices in the general case: however, several numerical experiments show that the two algorithms produce almost the same error behaviour.

math.NA↗

arXiv2017

Energy estimates for two-dimensional space-Riesz fractional wave equation

The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic viscoelasticity. In this paper, we first develop the energy method to estimate the one-dimensional space-Riesz fractional wave equation. For two-dimensional cases with the variable coefficients, the discretized matrices are proved to be commutative, which ensures to carry out of the priori error estimates. The unconditional stability and convergence with the global truncation error $\mathcal{O}(τ^2+h^2)$ are theoretically proved and numerically verified. In particulary, the framework of the priori error estimates and convergence analysis are still valid for the compact finite difference scheme and the nonlocal wave equation.

math.NA↗

arXiv2017

High order algorithm for the time-tempered fractional Feynman-Kac equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the time-tempered fractional Feynman-Kac equation. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as $${^S\!}D_t^{γ,\widetildeλ} G(x,p,t)\!=\!D_t^{γ,\widetildeλ} G(x,p,t)\!-\!λ^γG(x,p,t) ~{\rm with}~\widetildeλ=λ+ pU(x),\, p=ρ+Jη,\, J=\sqrt{-1},$$ where $$D_t^{γ,\widetildeλ} G(x,p,t) =\frac{1}{Γ(1-γ)} \left[\frac{\partial}{\partial t}+\widetildeλ \right] \int_{0}^t{\left(t-z\right)^{-γ}}e^{-\widetildeλ\cdot(t-z)}{G(x,p,z)}dz,$$ and $λ\ge 0$, $0<γ<1$, $ρ>0$, and $η$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(τ^2+h^2)$, being theoretically proved and numerically verified in {\em complex} space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the `physical' equation (without artificial source term).

math.NA↗