Keskosa interpolatsioonil põhinevad meetodid nõrgalt singulaarsete integraalvõrrandite lahendamiseks

Paljud keemia, polümeeride füüsika, matemaatilise füüsika jt teadusalade probleemid on formuleeritavad integraalvõrrandite kujul ning nende probleemide käsitlus taandub integraalvõrrandite lahendamisele või kvalitatiivsele uurimisele. Integraalvõrrandeid, mida saab täpselt lahendada, on suhteliselt vähe, seega on väga olulised meetodid võrrandite numbriliseks lahendamiseks. Käesolevas doktoritöös pakume välja kaks kõrget järku numbrilist meetodit, mis ei kasuta lineaarse teist liiki singulaarsustega Fredholmi integraalvõrrandi lahendamiseks ebaühtlast võrku. Need meetodid on kollokatsioonimeetod ja korrutise integreerimise meetod. Nimetatud meetodid põhinevad keskosa interpolatsioonil polünoomidega ühtlasel võrgul ja silendaval muutujate vahetusel. Lõigu keskosas on interpolatsioonivea hinnang ligikaudu 2m korda täpsem kui kogu lõigul. Lisaks on interpolatsiooniprotsess ühtlasel võrgul lõigu keskosas m-i kasvades stabiilne. Muutujate vahetuse abil parendame me võrrandi täpse lahendi käitumist. Doktoritöös on kirjeldatud toodud meetodite koondumist ja koondumiskiirustThere are a number of problems from many different fields, for example chemistry, physics of polymers and mathematical physics, which are directly formulated in terms of integral equations; and there are problems that are represented in terms of differential equations with auxiliary conditions, but which can be reduced to integral equations. There are relatively few integral equations which can be solved exactly, hence, numerical schemes are required for dealing with these equations in a proper manner. In this thesis we propose two new classes of high order numerical methods, which do not need graded grids for solving linear Fredholm integral equations of the second kind with singularities. The methods are developed by means of the 'central part' interpolation by polynomials on the uniform grid and smoothing change of variables. In the central parts of the interval, the estimates of interpolation error are approximately 2m times more precise than on the whole interval. In the central parts of the interval, the interpolation process on the uniform grid also has good stability properties as m increases. With the help of a change of variables we improve the boundary behaviour of the exact solution of the problem. The convergence and the convergence order of methods is studied

On the numerical solution of Volterra integral equations on equispaced nodes

In the present paper, a Nystrom-type method for second kind Volterra integral equations is introduced and studied. The method makes use of generalized Bernstein polynomials, defined for continuous functions and based on equally spaced points. Stability and convergence are studied in the space of continuous functions. Numerical tests illustrate the performance of the proposed approach

Product integration rules by the constrained mock-Chebyshev least squares operator

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates

Regularized System Identification

This open access book provides a comprehensive treatment of recent developments in kernel-based identification that are of interest to anyone engaged in learning dynamic systems from data. The reader is led step by step into understanding of a novel paradigm that leverages the power of machine learning without losing sight of the system-theoretical principles of black-box identification. The authors’ reformulation of the identification problem in the light of regularization theory not only offers new insight on classical questions, but paves the way to new and powerful algorithms for a variety of linear and nonlinear problems. Regression methods such as regularization networks and support vector machines are the basis of techniques that extend the function-estimation problem to the estimation of dynamic models. Many examples, also from real-world applications, illustrate the comparative advantages of the new nonparametric approach with respect to classic parametric prediction error methods. The challenges it addresses lie at the intersection of several disciplines so Regularized System Identification will be of interest to a variety of researchers and practitioners in the areas of control systems, machine learning, statistics, and data science. This is an open access book

The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics

The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods