Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u \in C^{\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^\epsilon)$, with $\epsilon > 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.Comment: 24 pages, 6 figures, and 6 table

Application of a high density ratio lattice-Boltzmann model for the droplet impingement on flat and spherical surfaces

In the current study, a 3-dimensional lattice Boltzmann model which can tolerate high density ratios is employed to simulate the impingement of a liquid droplet onto a flat and a spherical target. The four phases of droplet impact on a flat surface, namely, the kinematic, spreading, relaxation and equilibrium phase, have been obtained for a range of Weber and Reynolds numbers. The predicted maximum spread factor is in good agreement with experimental data published in the literature. For the impact of the liquid droplet onto a spherical target, the temporal variation of the film thickness on the target surface is investigated. The three different temporal phases of the film dynamics, namely, the initial drop deformation phase, the inertia dominated phase and the viscosity dominated phase are reproduced and studied. The effect of the droplet Reynolds number and the target-to-drop size ratio on the film flow dynamics is investigated

Deep learning-based flow disaggregation for short-term hydropower plant operations

High temporal resolution data plays a vital role in effective short-term hydropower plant operations. In the majority of the Norwegian hydropower system, inflow data is predominantly collected at daily resolutions through measurement installations. However, for enhanced precision in managerial decision-making within hydropower plants, hydrological data with intraday resolutions, such as hourly data, are often indispensable. To address this gap, time series disaggregation utilizing deep learning emerges as a promising tool. In this study, we propose a deep learning-based time series disaggregation model to derive hourly inflow data from daily inflow data for short-term hydropower plant operations. Our preliminary results demonstrate the applicability of our method, with scope for further improvements

Bayesian analysis for the intraclass model and for the quantile semiparametric mixed-effects double regression models

This dissertation consists of three distinct but related research projects. The first two projects focus on objective Bayesian hypothesis testing and estimation for the intraclass correlation coefficient in linear models. The third project deals with Bayesian quantile inference for the semiparametric mixed-effects double regression models. In the first project, we derive the Bayes factors based on the divergence-based priors for testing the intraclass correlation coefficient (ICC). The hypothesis testing of the ICC is used to test the uncorrelatedness in multilevel modeling, and it has not well been studied from an objective Bayesian perspective. Simulation results show that the two sorts of Bayes factors have good performance in the hypothesis testing. Moreover, the Bayes factors can be easily implemented due to their unidimensional integral expressions. In the second project, we consider objective Bayesian analysis for the ICC in the context of normal linear regression model. We first derive two objective priors for the unknown parameters and show that both result in proper posterior distributions. Within a Bayesian decision-theoretic framework, we then propose an objective Bayesian solution to the problems of hypothesis testing and point estimation of the ICC based on a combined use of the intrinsic discrepancy loss function and objective priors. The proposed solution has an appealing invariance property under one-to-one reparameterization of the quantity of interest. Simulation studies are conducted to investigate the performance the proposed solution. Finally, a real data application is provided for illustrative purposes. In the third project, we study Bayesian quantile regression for semiparametric mixed effects model, which includes both linear and nonlinear parts. We adopt the popular cubic spline functions for the nonlinear part and model the variance of the random effect as a function of the explanatory variables. An efficient Gibbs sampler with the Metropolis-Hastings algorithm is proposed to generate posterior samples of the unknown parameters from their posterior distributions. Simulation studies and a real data example are used to illustrate the performance of the proposed methodology

Overløpskontroll i avløpsnett med forskjellige modelleringsteknikker og internet of things

Increased urbanization and extreme rainfall events are causing more frequent instances of sewer overflow, leading to the pollution of water resources and negative environmental, health, and fiscal impacts. At the same time, the treatment capacity of wastewater treatment plants is seriously affected. The main aim of this Ph.D. thesis is to use the Internet of Things and various modeling techniques to investigate the use of real-time control on existing sewer systems to mitigate overflow. The role of the Internet of Things is to provide continuous monitoring and real-time control of sewer systems. Data collected by the Internet of Things are also useful for model development and calibration. Models are useful for various purposes in real-time control, and they can be distinguished as those suitable for simulation and those suitable for prediction. Models that are suitable for a simulation, which describes the important phenomena of a system in a deterministic way, are useful for developing and analyzing different control strategies. Meanwhile, models suitable for prediction are usually employed to predict future system states. They use measurement information about the system and must have a high computational speed. To demonstrate how real-time control can be used to manage sewer systems, a case study was conducted for this thesis in Drammen, Norway. In this study, a hydraulic model was used as a model suitable for simulation to test the feasibility of different control strategies. Considering the recent advances in artificial intelligence and the large amount of data collected through the Internet of Things, the study also explored the possibility of using artificial intelligence as a model suitable for prediction. A summary of the results of this work is presented through five papers. Paper I demonstrates that one mainstream artificial intelligence technique, long short-term memory, can precisely predict the time series data from the Internet of Things. Indeed, the Internet of Things and long short-term memory can be powerful tools for sewer system managers or engineers, who can take advantage of real-time data and predictions to improve decision-making. In Paper II, a hydraulic model and artificial intelligence are used to investigate an optimal in-line storage control strategy that uses the temporal storage volumes in pipes to reduce overflow. Simulation results indicate that during heavy rainfall events, the response behavior of the sewer system differs with respect to location. Overflows at a wastewater treatment plant under different control scenarios were simulated and compared. The results from the hydraulic model show that overflows were reduced dramatically through the intentional control of pipes with in-line storage capacity. To determine available in-line storage capacity, recurrent neural networks were employed to predict the upcoming flow coming into the pipes that were to be controlled. Paper III and Paper IV describe a novel inter-catchment wastewater transfer solution. The inter-catchment wastewater transfer method aims at redistributing spatially mismatched sewer flows by transferring wastewater from a wastewater treatment plant to its neighboring catchment. In Paper III, the hydraulic behaviors of the sewer system under different control scenarios are assessed using the hydraulic model. Based on the simulations, inter-catchment wastewater transfer could efficiently reduce total overflow from a sewer system and wastewater treatment plant. Artificial intelligence was used to predict inflow to the wastewater treatment plant to improve inter-catchment wastewater transfer functioning. The results from Paper IV indicate that inter-catchment wastewater transfer might result in an extra burden for a pump station. To enhance the operation of the pump station, long short-term memory was employed to provide multi-step-ahead water level predictions. Paper V proposes a DeepCSO model based on large and high-resolution sensors and multi-task learning techniques. Experiments demonstrated that the multi-task approach is generally better than single-task approaches. Furthermore, the gated recurrent unit and long short-term memory-based multi-task learning models are especially suitable for capturing the temporal and spatial evolution of combined sewer overflow events and are superior to other methods. The DeepCSO model could help guide the real-time operation of sewer systems at a citywide level.publishedVersio

Transmit design for MIMO wiretap channel with a malicious jammer

In this paper, we consider the transmit design for multi-input multi-output (MIMO) wiretap channel including a malicious jammer. We first transform the system model into the traditional three-node wiretap channel by whitening the interference at the legitimate user. Additionally, the eavesdropper channel state information (ECSI) may be fully or statistically known, even unknown to the transmitter. Hence, some strategies are proposed in terms of different levels of ECSI available to the transmitter in our paper. For the case of unknown ECSI, a target rate for the legitimate user is first specified. And then an inverse water-filling algorithm is put forward to find the optimal power allocation for each information symbol, with a stepwise search being used to adjust the spatial dimension allocated to artificial noise (AN) such that the target rate is achievable. As for the case of statistical ECSI, several simulated channels are randomly generated according to the distribution of ECSI. We show that the ergodic secrecy capacity can be approximated as the average secrecy capacity of these simulated channels. Through maximizing this average secrecy capacity, we can obtain a feasible power and spatial dimension allocation scheme by using one dimension search. Finally, numerical results reveal the effectiveness and computational efficiency of our algorithms.Comment: 2015 IEEE 81st Vehicular Technology Conference (VTC Spring