An efficient iterative method based on two-stage splitting methods to solve weakly nonlinear systems

[EN] In this paper, an iterative method for solving large, sparse systems of weakly nonlinear equations is presented. This method is based on Hermitian/skew-Hermitian splitting (HSS) scheme. Under suitable assumptions, we establish the convergence theorem for this method. In addition, it is shown that any faster and less time-consuming two-stage splitting method that satisfies the convergence theorem can be replaced instead of the HSS inner iterations. Numerical results, such as CPU time, show the robustness of our new method. This method is easy, fast and convenient with an accurate solution.The third and fourth authors have been partially supported by the Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Amiri, A.; Darvishi, MT.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). An efficient iterative method based on two-stage splitting methods to solve weakly nonlinear systems. Mathematics. 7(9):1-17. https://doi.org/10.3390/math7090815S1177

Remarks on Solving Methods of Nonlinear Equations

Abstract: In the field of mechanical engineering, many practical problems can be converted into nonlinear problems, such as the meshing problem of mechanical transmission. So the solution of nonlinear equations has important theoretical research and practical application significance. Whether the traditional Newton iteration method or the intelligent optimization algorithm after the popularization of computers, both them have been greatly enriched and developed through the continuous in-depth research of scholars at home and abroad, and a series of improved algorithms have emerged. This paper mainly reviews the research status of solving nonlinear equations from two aspects of traditional iterative method and intelligent optimization algorithm, systematically reviews the research achievements of domestic and foreign scholars, and puts forward prospects for future research directions

A fast algorithm to solve systems of nonlinear equations

[EN] A new HSS-based algorithm for solving systems of nonlinear equations is presented and its semilocal convergence is proved. Spectral properties of the new method are investigated. Performance profile for the new scheme is computed and compared with HSS algorithm. Besides, by a numerical example in which a two-dimensional nonlinear convection diffusion equation is solved, we compare the new method and the Newton-HSS method. Numerical results show that the new scheme solves the problem faster than the NewtonHSS scheme in terms of CPU -time and number of iterations. Moreover, the application of the new method is found to be fast, reliable, flexible, accurate, and has small CPU time.This research was partially supported by Ministerio de Economia y Competitividad, Spain under grants MTM2014-52016-C2-2-P and Generalitat Valenciana, Spain PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2019). A fast algorithm to solve systems of nonlinear equations. Journal of Computational and Applied Mathematics. 354:242-258. https://doi.org/10.1016/j.cam.2018.03.048S24225835

Searching for Gravitational Waves from Scorpius X-1 with a Cross-correlation Method: from Mock Data to Advanced LIGO

Gravitational waves (GWs) are propagating ripples of space-time predicted by general relativity. 100 years after Albert Einstein published his theory of GR, the Laser Interferometer Gravitational-Wave Observatory (LIGO) found the first direct detection of GW in the first Advanced LIGO observing run. The GW signal known as GW150914 (Abbott et al., 2016), was the first of a series of binary black hole mergers observed by LIGO. These detections marked the beginning of gravitational-wave astronomy. The continuous wave (CW) signal emitted by fast spinning neutron stars (NSs) is an another interesting source for a detector like LIGO. The low-mass X-ray binary (LMXB) Scorpius X-1 (Sco X-1) is considered to be one of the most promising CW sources. With improving sensitivity of advanced detectors and improving methods, we are getting closer to being able to detect an astrophysically feasible GW signal from Sco X-1 in the coming few years. Searching for CWs from NSs of unknown phase evolution is computationally intensive. For a target with large uncertainty in its parameters such as Sco X-1, the fully coherent search is computationally impractical, while faster algorithms have limited sensitivity. The cross-correlation method combines all data-pairs in a maximum time offset from same and different detectors coherently based on the signal model. We can adjust the maximum coherence time to trade off computing cost and sensitivity. The cross-correlation method is flexible and so far the most sensitive. In this dissertation I will present the implementation of Cross-correlation method for Sco X-1, its test on a Sco X-1 mock-data challenge (MDC) data set and the Advanced LIGO O1 observations. This search gave the best results in the Sco X-1 mock data challenge and recent LIGO Sco X-1 search. In the O1 run, the Cross-correlation search managed to improve the upper limit on GW strain strength from Sco X-1 closer than ever before to the level estimated from a torque balance argument

Bayesian Modelling Approaches for Quantum States -- The Ultimate Gaussian Process States Handbook

Capturing the correlation emerging between constituents of many-body systems accurately is one of the key challenges for the appropriate description of various systems whose properties are underpinned by quantum mechanical fundamentals. This thesis discusses novel tools and techniques for the (classical) modelling of quantum many-body wavefunctions with the ultimate goal to introduce a universal framework for finding accurate representations from which system properties can be extracted efficiently. It is outlined how synergies with standard machine learning approaches can be exploited to enable an automated inference of the most relevant intrinsic characteristics through rigorous Bayesian regression techniques. Based on the probabilistic framework forming the foundation of the introduced ansatz, coined the Gaussian Process State, different compression techniques are explored to extract numerically feasible representations of relevant target states within stochastic schemes. By following intuitively motivated design principles, the resulting model carries a high degree of interpretability and offers an easily applicable tool for the numerical study of quantum systems, including ones which are notoriously difficult to simulate due to a strong intrinsic correlation. The practical applicability of the Gaussian Process States framework is demonstrated within several benchmark applications, in particular, ground state approximations for prototypical quantum lattice models, Fermi-Hubbard models and $J_1-J_2$ models, as well as simple ab-initio quantum chemical systems.Comment: PhD Thesis, King's College London, 202 page

Crop development monitoring from Synthetic Aperture Radar (SAR) imagery

Satellite remote sensing plays a vital role in providing large-scale and timely data to stakeholders of the agricultural supply chain. This allows for informed decision-making that promotes sustainable and cost-effective crop management practices. In particular, data derived from satellite-based Synthetic Aperture Radar (SAR) systems, provide opportunities for continuous crop monitoring, taking advantage of its ability to acquire images during day or night and under almost all weather conditions. Moreover, an abundance of SAR data can be anticipated in the next 5 years with the launch of several international SAR missions. However, research on crop development monitoring with data from SAR satellites has not been as widely studied as with data derived from passive multi-spectral satellites and contributions can be made to the current state-of-the-art techniques. This thesis aims at improving the current knowledge on the use of satellite-based SAR imagery for crop development monitoring. This is approached by developing novel methodologies and detailed interpretations of multitemporal SAR and Polarimetric SAR (PolSAR) responses to crop growth in three different test sites. Chapter two presents a detailed analysis of the Sentinel-1 SAR satellite response to asparagus crop development in Peru, investigating the capabilities of the sensor to capture seasonality effects as well as providing an interpretation of the temporal backscatter signature. This is complemented with a case study where a multiple-output random forest regression algorithm is used to successfully retrieve crop growth stage from Sentinel-1 data and temperature measurements. Following the limitations identified with this approach, a methodology that builds upon ideas of Bayesian Filtering Frameworks (BFFs) for crop monitoring is proposed in chapter three. It incorporates Gaussian processes to model crop dynamics as well as to model the remote sensing response to the crop state. Using this approach, it is possible to derive daily predictions with the associated uncertainties, to combine in near-real-time data from active and passive satellites as well as to estimate past and future crop key events that are of strategic importance for different stakeholders. The final section of this thesis looks at the new developments of the SAR technology considering that future open access missions will provide Quad Polarimetric SAR data. An algorithm based on multitemporal PolSAR change detection is introduced in chapter four. It defines a Change Matrix to encode an interpretable representation of the crop dynamics as captured by the evolution of the scattering mechanisms over time. We use rice fields in Spain and multiple cereal crops in Canada to test the use of the algorithm for crop monitoring. A supervised learning-based crop type classification methodology is then proposed with the same method by using the encoded scattering mechanisms as input for a neural-network-based classifier, achieving comparable performances to state-of-the-art classifiers. The results obtained in this thesis represent novel additions to the literature that contribute to our understanding and successful use of SAR imagery for agricultural monitoring. For the first time, a detailed analysis of asparagus crops is presented. It is a key crop for agricultural exports of Peru, the largest exporter of asparagus in the world. Secondly, two key contributions to the state of the art BFFs for crop monitoring are presented: a) A better exploitation of the SAR temporal dimension and an application with freely available data and b) given that it is a learning-based approach, it overcomes current limitations of transferability among crop types and regions. Finally, the PolSAR change detection approach presented in the last thesis chapter, provides a novel and easy-to-interpret tool for both crop monitoring and crop type mapping applications