A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.Research of the first and third authors supported in part by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by MTM2015-64382-P. Research of the fourth and fifth authors supported by Ministerio de Economía y Competitividad under grant MTM2014-52016-C2-1P. This research received no external funding

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

Subdivision schemes for curve design and image analysis

Subdivision schemes are able to produce functions, which are smooth up to pixel accuracy, in a few steps through an iterative process. They take as input a coarse control polygon and iteratively generate new points using some algebraic or geometric rules. Therefore, they are a powerful tool for creating and displaying functions, in particular in computer graphics, computer-aided design, and signal analysis. A lot of research on univariate subdivision schemes is concerned with the convergence and the smoothness of the limit curve, especially for schemes where the new points are a linear combination of points from the previous iteration. Much less is known for non-linear schemes: in many cases there are only ad hoc proofs or numerical evidence about the regularity of these schemes. For schemes that use a geometric construction, it could be interesting to study the continuity of geometric entities. Dyn and Hormann propose sufficient conditions such that the subdivision process converges and the limit curve is tangent continuous. These conditions can be satisfied by any interpolatory scheme and they depend only on edge lengths and angles. The goal of my work is to generalize these conditions and to find a sufficient constraint, which guarantees that a generic interpolatory subdivision scheme gives limit curves with continuous curvature. To require the continuity of the curvature it seems natural to come up with a condition that depends on the difference of curvatures of neighbouring circles. The proof of the proposed condition is not completed, but we give a numerical evidence of it. A key feature of subdivision schemes is that they can be used in different fields of approximation theory. Due to their well-known relation with multiresolution analysis they can be exploited also in image analysis. In fact, subdivision schemes allow for an efficient computation of the wavelet transform using the filterbank. One current issue in signal processing is the analysis of anisotropic signals. Shearlet transforms allow to do it using the concept of multiple subdivision schemes. One drawback, however, is the big number of filters needed for analysing the signal given. The number of filters is related to the determinant of the expanding matrix considered. Therefore, a part of my work is devoted to find expanding matrices that give a smaller number of filters compared to the shearlet case. We present a family of anisotropic matrices for any dimension d with smaller determinant than shearlets. At the same time, these matrices allow for the definition of a valid directional transform and associated multiple subdivision schemes

Behaviour near extinction for the Fast Diffusion Equation on bounded domains

AbstractWe consider the Fast Diffusion Equation ut=Δum, m<1, posed in a bounded smooth domain Ω⊂Rd with homogeneous Dirichlet conditions. It is known that in the exponent range ms=(d−2)+/(d+2)<m<1 all bounded positive solutions u(t,x) of such problem extinguish in a finite time T=T(u), and also that such solutions approach a separate variable solution u(t,x)∼(T−t)1/(1−m)S(x), as t→T−.Here, we are interested in describing the behaviour of the solutions near the extinction time in that range of exponents. We first show that the convergence v(x,t)=u(t,x)(T−t)−1/(1−m) to S(x) takes place uniformly in the relative error norm. Then, we study the question of rates of convergence of the rescaled flow, i.e., v→S. For m close to 1 we get such rates by means of entropy methods and weighted Poincaré inequalities. The analysis of the latter point makes an essential use of fine properties of the associated stationary elliptic problem −ΔSm=cS in the limit m→1, and such a study has an independent interest

Robust Procedures for Estimating and Testing in the Framework of Divergence Measures

The scope of the contributions to this book will be to present new and original research papers based on MPHIE, MHD, and MDPDE, as well as test statistics based on these estimators from a theoretical and applied point of view in different statistical problems with special emphasis on robustness. Manuscripts given solutions to different statistical problems as model selection criteria based on divergence measures or in statistics for high-dimensional data with divergence measures as loss function are considered. Reviews making emphasis in the most recent state-of-the art in relation to the solution of statistical problems base on divergence measures are also presented

Modelling high intensity laser pulse propagation in air using the modified Korteweg-de Vries equation

Ultrafast laser pulse experiments and applications are entering a phase that challenges the validity of mathematical models utilised to model longer pulses in nonlinear optics. This thesis aims to propose a possible mathematical model for high intensity laser pulse propagation in air through a multiple scales expansion of Maxwell’s equations and discuss a method on how to solve the corresponding differential equation, known as the modified Korteweg-de Vries equation in to the small dispersion regime. This equation is solvable using a technique named the scattering transform and due to weak dispersion the equation can be solved asymptotically. The method is based on using the asymptotic WKB approximation for the forward scattering problem and reformulating the inverse scattering as a Riemann-Hilbert problem. Both analytical steps and numerical procedures needed to use the method is discussed and implemented. A full example calculation using a particular initial condition is performed and some challenges using the method for more general initial conditions are discussed

Advanced laboratory testing for offshore pile foundations under monotonic and cyclic loading

Laboratory experimental research is described that supported the PISA Joint Industry monopile design project and a smaller scale international investigation into the ageing behaviour of micro-piles driven in sands. The study focused on the soils encountered at the PISA test sites, including the stiff high OCR, low plasticity, glacial till sampled at Cowden (Humberside, UK), and a silica-dominated fine marine sand retrieved from Dunkirk (Northern France). Tests were also conducted on sands from two other sites at Blessington (Ireland) and Larvik (Norway). Working in conjunction with parallel research by Ushev (2018), four important aspects of the PISA test sites’ soil conditions were studied: (i) Stress-strain-stiffness behaviour from the initial linear elastic range (Y1) over the full non-linear range and up to final critical states; (ii) Stiffness and shear strength anisotropy; (iii) Response to cyclic loading; and (iv) Soil-soil/steel interface shearing. Hollow Cylinder and fully-instrumented Bishop-Wesley triaxial apparatuses were employed in the Author’s tests that revealed the Cowden till’s shear strength and stiffness anisotropy, both of which impact on the interpretation of the lateral loading behaviour observed in the large scale PISA field tests. Extensive monotonic drained triaxial tests were conducted on reconstituted Dunkirk sand to characterise its highly non-linear small-strain stiffness characteristics, considering the effects of state as well as stress history (OCR), effective stress level and evolving anisotropy. The sand’s Y1 and Y2 kinematic surfaces were depicted and tracked in effective stress space. The effects of end restraint in triaxial testing were also examined carefully, before exploring the large strain behaviour of Dunkirk sand and interpreting the results with a critical state and state parameter approach. The monotonic testing was followed by an extensive programme of drained cyclic triaxial tests on Dunkirk sand that applied up to 104 cycles to demonstrate how the samples’ state, stress history, mean cyclic stress ratio and cyclic amplitude affected the sand’s response to repetitive loading. Cyclic strain accumulation flow and dilation characteristics were found to vary significantly with all of the above factors. The detailed response could not be depicted accurately by simple cyclic dilation models that consider only the mean cyclic stress ratio. Cyclic threshold conditions were characterised in terms of two kinematic yielding surfaces that correlated with those seen under monotonic loading conditions. Additional experiments on pre-cycled samples demonstrated that the repetitive loading enhanced the sand’s monotonic shear strength and stiffness characteristics. The associated study of sand-steel interface shearing employed Bishop ring shear tests to impose large shear displacements that revealed a notable dependence of the shear resistances and dilation angles on normal stress levels and ageing times, and so provide significant insights into the ageing mechanisms that boost the axial capacities of steel piles driven in sandy soils. Information from the Author’s test programmes contributed to the pile ageing JIP test interpretation and was central to the successful modelling of the PISA field tests undertaken by other members of the Imperial College team.Open Acces

Thermomechanical analysis of rock asperity in fractures of enhanced geothermal systems

Enhanced Geothermal Systems (EGS) offer great potential for dramatically expanding the use of geothermal energy and become a promising supplement for fossil energy. The EGS is to extract heat by creating a subsurface system to which cold water can be added through injection wells. Injected water is heated by contact with rock and returns to the surface through production well. Fracture provides the primary conduit for fluid flow and heat transfer in natural rock. Fracture is propped by fracture roughness with varying heights which is called asperity. The stability of asperity determines fracture aperture and hence imposes substantial effect on hydraulic conductivity and heat transfer efficiency in EGS. Firstly, two rough fracture surfaces are characterized by statistical method and fractal analysis. The asperity heights and enclosed aperture heights are described by probability density function before cold water is pumped into fracture. Secondly, when water injection and induced cooling occurs, the thermomechanical analysis of single asperity is studied by establishing an un-symmetric damage mechanics model. The deformation curve of asperity under thermal stress is determined. Thirdly, deformation of fracture with various asperities on it in response to thermal stress is analyzed by a new stratified continuum percolation model. This model incorporates the fracture surface characteristics and preceding deformation curve of asperity. The fracture closure and fracture stiffness can be accurately quantified by this model. In addition, the scaling invariance and multifractal parameters in this process are identified and validated with Monte Carlo simulation --Abstract, page iii