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Solving the three-dimensional (3D) Bratu equation is highly challenging due to the presence of multiple and sharp solutions. Research on this equation began in the late 1990s, but there are no satisfactory results to date. To address this issue, we introduce a symmetric finite difference method (SFDM) which embeds the symmetry properties of the solutions into a finite difference method (FDM). This SFDM is primarily used to obtain more accurate solutions and bifurcation diagrams for the 3D Bratu equation. Additionally, we propose modifying the Bratu equation by incorporating a new constraint that facilitates the construction of bifurcation diagrams and simplifies handling the turning points. The proposed method, combined with the use of sparse matrix representation, successfully solves the 3D Bratu equation on grids of up to $301^3$ points. The results demonstrate that SFDM outperforms all previously employed methods for the 3D Bratu equation. Furthermore, we provide bifurcation diagrams for the 1D, 2D, 4D, and 5D cases, and accurately identify the first turning points in all dimensions. All simulations indicate that the bifurcation diagrams of the Bratu equation on the cube domains closely resemble the well-established behavior on the ball domains described by Joseph and Lundgren [1]. Furthermore, when SFDM is applied to linear stability analysis, it yields the same largest real eigenvalue as the standard FDM despite having fewer equations and variables in the nonlinear system., Comment: Accepted for publication in Applied Mathematics and Computation

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis., Comment: Accepted for publication in Applied Mathematics and Computation

We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency., Comment: Accepted for publication in Computers and Mathematics with Applications, 19 pages, 6 figures

We consider the multi-user detection (MUD) problem in uplink grant-free non-orthogonal multiple access (NOMA), where the access point has to identify the total number and correct identity of the active Internet of Things (IoT) devices and decode their transmitted data. We assume that IoT devices use complex spreading sequences and transmit information in a random-access manner following the burst-sparsity model, where some IoT devices transmit their data in multiple adjacent time slots with a high probability, while others transmit only once during a frame. Exploiting the temporal correlation, we propose an attention-based bidirectional long short-term memory (BiLSTM) network to solve the MUD problem. The BiLSTM network creates a pattern of the device activation history using forward and reverse pass LSTMs, whereas the attention mechanism provides essential context to the device activation points. By doing so, a hierarchical pathway is followed for detecting active devices in a grant-free scenario. Then, by utilising the complex spreading sequences, blind data detection for the estimated active devices is performed. The proposed framework does not require prior knowledge of device sparsity levels and channels for performing MUD. The results show that the proposed network achieves better performance compared to existing benchmark schemes.

This paper studies the joint channel estimation and signal detection for the uplink power-domain non-orthogonal multiple access. The proposed technique performs both detection and estimation without the need of pilot symbols by using a clustering technique. We apply rotational-invariant coding to assist signal detection at the receiver without sending pilot symbols. We utilize Gaussian mixture model (GMM) to automatically cluster the received signals without supervision and optimize decision boundaries to improve the bit error rate (BER) performance. Simulation results show that the proposed scheme without using any pilot symbol achieves almost the same BER performance as that for the conventional maximum likelihood receiver with full channel state information., Comment: arXiv admin note: text overlap with arXiv:2010.03091

We propose a joint channel estimation and signal detection approach for the uplink non-orthogonal multiple access (NOMA) using unsupervised machine learning. We apply a Gaussian mixture model (GMM) to cluster the received signals, and accordingly optimize the decision regions to enhance the symbol error rate (SER) performance. We show that, when the received powers of the users are sufficiently different, the proposed clustering-based approach achieves an SER performance on a par with that of the conventional maximum-likelihood detector (MLD) with full channel state information (CSI). We study the tradeoff between the accuracy of the proposed approach and the blocklength, as the accuracy of the utilized clustering algorithm depends on the number of symbols available at the receiver. We provide a comprehensive performance analysis of the proposed approach and derive a theoretical bound on its SER performance. Our simulation results corroborate the effectiveness of the proposed approach and verify that the calculated theoretical bound can predict the SER performance of the proposed approach well. We further explore the application of the proposed approach to a practical grant-free NOMA scenario, and show that its performance is very close to that of the optimal MLD with full CSI, which usually requires long pilot sequences.

Due to an increasing number of unmanned aerial vehicles (UAVs) with generally limited battery power, energy efficient data transmission schemes with massive connectivity capabilities are required for future wireless networks. Non-orthogonal multiple access (NOMA) is one of the promising techniques that provide such massive connectivity by allowing superposed data transmission of multiple users over the same resource block. Unlike existing literature, this paper presents a user pairing and power allocation technique for energy efficient and quality of service (QoS) aware NOMA transmission for cellular-connected mobile UAVs. The aim is to minimize the power consumption of the mobile UAVs during uplink data transmission and guarantee their required transmission rate by jointly optimizing the UAV pairing and power allocation. Furthermore, the performance and pairing complexity for mobile UAVs is analyzed to show that the proposed pairing technique efficiently fulfills their QoS requirements. Numerical and simulation results show the performance gain of the proposed technique compared to the conventional NOMA techniques.

We propose a joint channel estimation and signal detection technique for the uplink non-orthogonal multiple access using an unsupervised clustering approach. We apply the Gaussian mixture model to cluster received signals and accordingly optimize the decision regions to enhance the symbol error rate (SER). We show that when the received powers of the users are sufficiently different, the proposed clustering-based approach with no channel state information (CSI) at the receiver achieves an SER performance similar to that of the conventional maximum likelihood detector with full CSI. Since the accuracy of the utilized clustering algorithm depends on the number of the data points available at the receiver, the proposed technique delivers a tradeoff between the accuracy and block length., Comment: GC 2020 Workshop - FIIoT

In this letter, role switching and power allocation schemes are proposed to tackle user mobility in non-orthogonal multiple access (NOMA) systems. When cell center user (CCU) and cell edge user (CEU) come very close or even cross each other in NOMA pairing, channel gains of paired users violate the basic NOMA conditions. This article refers to such condition as NOMA principle violation problem (NPVP). To solve this NPVP, optimized power role switching-NOMA (OPRS-NOMA) technique is proposed. Role switching technique is used where roles of mobile users are switched on the basis of their channel gains. Furthermore, a power allocation scheme based on bisection search power optimization is presented to maximize the average sum capacity of mobile NOMA users. Random way point mobility model is considered for user mobility. Individual and sum capacity are used for performance evaluation. Simulation results show that OPRS-NOMA outperforms the conventional NOMA and orthogonal multiple access (OMA).

Massive machine-type communications (mMTC) is one of the main three focus areas in the 5th generation (5G) of mobile standards to enable connectivity of a massive number of internet of things (IoT) devices with little or no human intervention. In conventional human-type communications (HTC), due to the limited number of available radio resources and orthogonal/non-overlapping nature of existing resource allocation techniques, users need to compete for connectivity through a random access (RA) process, which may turn into a performance bottleneck in mMTC. In this context, non-orthogonal multiple access (NOMA) has emerged as a potential technology that allows overlapping of multiple users over a radio resource, thereby creating an opportunity to enable more autonomous and grant-free communication, where devices can transmit data whenever they need. The existing literature on NOMA schemes majorly considers centralized scheduling based HTC, where users are already connected, and various system parameters like spreading sequences, interleaving patterns, power control, etc., are predefined. Contrary to HTC, mMTC traffic is different with mostly uplink communication, small data size per device, diverse quality of service, autonomous nature, and massive number of devices. Hence, the signaling overhead and latency of centralized scheduling becomes a potential performance bottleneck. To tackle this, grant-free access is needed, where mMTC devices can autonomously transmit their data over randomly chosen radio resources. This article, in contrast to existing surveys, comprehensively discusses the recent advances in NOMA from a grant-free connectivity perspective. Moreover, related practical challenges and future directions are discussed., Comment: Survey Paper

In this paper, joint transmission coordinated multipoint (JT-CoMP) is exploited by using virtual user pairing non-orthogonal multiple access (VP-NOMA), termed as JT-CoMP VP-NOMA. The technique combines both VP-NOMA for enhancing ergodic sum capacity (ESC) and JT-CoMP for inter-cell interference mitigation. To show the performance gains, ESC of a three-cell scenario is analyzed as a key performance metric. The analytical and simulation results of JT-CoMP VP-NOMA are compared with orthogonal multiple access (OMA), non-orthogonal multiple access (NOMA), and VP-NOMA. It is shown that the proposed JT-CoMP VP-NOMA outperforms the other schemes in the viewpoint of ESC.