Iterative algorithms for solutions of nonlinear equations in Banach spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF

A study of optimization problems and fixed point iterations in Banach spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF

Matematiske aspekter ved lokalisert aktivitet i nevrofeltmodeller

Neural field models assume the form of integral and integro-differential equations, and describe non-linear interactions between neuron populations. Such models reduce the dimensionality and complexity of the microscopic neural-network dynamics and allow for mathematical treatment, efficient simulation and intuitive understanding. Since the seminal studies byWilson and Cowan (1973) and Amari (1977) neural field models have been used to describe phenomena like persistent neuronal activity, waves and pattern formation in the cortex. In the present thesis we focus on mathematical aspects of localized activity which is described by stationary solutions of a neural field model, so called bumps. While neural field models represent a considerable simplification of the neural dynamics in a large network, they are often studied under further simplifying assumptions, e.g., approximating the firing-rate function with a unit step function. In some cases these assumptions may not change essential features of the model, but in other cases they may cause some properties of the model to vary significantly or even break down. The work presented in the thesis aims at studying properties of bump solutions in one- and two-population models relaxing on the common simplifications. Numerical approaches used in mathematical neuroscience sometimes lack mathematical justification. This may lead to numerical instabilities, ill-conditioning or even divergence. Moreover, there are some methods which have not been used in neuroscience community but might be beneficial. We have initiated a work in this direction by studying advantages and disadvantages of a wavelet-Galerkin algorithm applied to a simplified framework of a one-population neural field model. We also focus on rigorous justification of iteration methods for constructing bumps. We use the theory of monotone operators in ordered Banach spaces, the theory of Sobolev spaces in unbounded domains, degree theory, and other functional analytical methods, which are still not very well developed in neuroscience, for analysis of the models.Nevrofeltmodeller formuleres som integral og integro-differensiallikninger. De beskriver ikke-lineære vekselvirkninger mellom populasjoner av nevroner. Slike modeller reduserer dimensjonalitet og kompleksitet til den mikroskopiske nevrale nettverksdynamikken og tillater matematisk behandling, effektiv simulering og intuitiv forståelse. Siden pionerarbeidene til Wilson og Cowan (1973) og Amari (1977), har nevrofeltmodeller blitt brukt til å beskrive fenomener som vedvarende nevroaktivitet, bølger og mønsterdannelse i hjernebarken. I denne avhandlingen vil vi fokusere på matematiske aspekter ved lokalisert aktivitet som beskrives ved stasjonære løsninger til nevrofeltmodeller, såkalte bumps. Mens nevrofeltmodeller innebærer en betydelig forenkling av den nevrale dynamikken i et større nettverk, så blir de ofte studert ved å gjøre forenklende tilleggsantakelser, som for eksempel å approksimere fyringratefunksjonen med en Heaviside-funksjon. I noen tilfeller vil disse forenklingene ikke endre vesentlige trekk ved modellen, mens i andre tilfeller kan de forårsake at modellegenskapene endres betydelig eller at de bryter sammen. Arbeidene presentert i denne avhandlingen har som mål å studere egenskapene til bump-løsninger i en- og to-populasjonsmodeller når en lemper på de vanlige antakelsene. Numeriske teknikker som brukes i matematisk nevrovitenskap mangler i noen tilfeller matematisk begrunnelse. Dette kan lede til numeriske instabiliteter, dårlig kondisjonering, og til og med divergens. I tillegg finnes det metoder som ikke er blitt brukt i nevrovitenskap, men som kunne være fordelaktige å bruke. Vi har startet et arbeid i denne retningen ved å studere fordeler og ulemper ved en wavelet-Galerkin algoritme anvendt på et forenklet rammeverk for en en-populasjons nevrofelt modell. Vi fokuserer også på rigorøs begrunnelse for iterasjonsmetoder for konstruksjon av bumps. Vi bruker teorien for monotone operatorer i ordnede Banachrom, teorien for Sobolevrom for ubegrensede domener, gradteori, og andre funksjonalanalytiske metoder, som for tiden ikke er vel utviklet i nevrovitenskap, for analyse av modellene

Expanding the applicability of Newton-Tikhonov method for ill-posed equations

We present a new semilocal convergence analysis of Newton- Tikhonov methods for solving ill-posed operator equations in a Hilbert space setting. Using more precise majorizing sequences and under the same computational cost as in earlier studies such as [13]-[20], we provide: weaker sufficient convergence criteria; tighter error estimates on the distances involved and an at least as precise information on the location of the solution. Applications include Hammertein nonlinear integral equations

Hägusad teist liiki integraalvõrrandid

Käesolevas doktoritöös on uuritud hägusaid teist liiki integraalvõrrandeid. Need võrrandid sisaldavad hägusaid funktsioone, s.t. funktsioone, mille väärtused on hägusad arvud. Me tõestasime tulemuse sileda tuumaga hägusate Volterra integraalvõrrandite lahendite sileduse kohta. Kui integraalvõrrandi tuum muudab märki, siis integraalvõrrandi lahend pole üldiselt sile. Nende võrrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tükiti lineaarsete ja tükiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tõestasime meetodite koonduvuskiiruse. Me vaatlesime ka nõrgalt singulaarse tuumaga hägusaid Volterra integraalvõrrandeid. Uurisime lahendi olemasolu, ühesust, siledust ja hägusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tükiti polünoomide ruumis. Tõestasime meetodite koonduvuskiiruse ning uurisime lähislahendi hägusust. Nii analüüs kui ka numbrilised eksperimendid näitavad, et gradueeritud võrke kasutades saame parema koonduvuskiiruse kui ühtlase võrgu korral. Teist liiki hägusate Fredholmi integraalvõrrandite lahendamiseks pakkusime uue lahendusmeetodi, mis põhineb kõigi võrrandis esinevate funktsioonide lähendamisel Tšebõšovi polünoomidega. Uurisime nii täpse kui ka ligikaudse lahendi olemasolu ja ühesust. Tõestasime meetodi koonduvuse ja lähislahendi hägususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers. We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods. We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems. We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

Analysis of a Nonlinear Boundary Value Problem with Application to Heat Transfer in Electric Cables

In der vorliegenden Dissertation werden Verfahren zur kontrollierten Modellreduktion des Wärmetransports in elektrischen Leitern entwickelt. Eine typische Reduktionsmethode besteht darin, das zeitabhängige Wärmeleitungsproblem für große Zeiten durch ein stationäres zu ersetzen. Eine weitere Methode vereinfacht das dreidimensionale Randwertproblem in einem zylindrischen Leiter zu einem Problem auf dem zweidimensionalen Querschnitt des Leiters. Diese Reduktionsmethoden werden jedoch oft ohne eine Kenntnis des auftretenden Fehlers angewendet. Daher untersuchen wir die Konvergenz der Lösung des vollen Wärmeleitungsproblems gegen die Lösung eines stationären auf dem Leiterquerschnitt definierten Randwertproblems. Diese Abschätzungen wenden wir auf ein elektrisches Kabel an und identifizieren die zunächst abstrakt bestimmten Approximationsfehler mit konkreten physikalischen Größen. Danach verwenden wir nichtlineare Randintegralmethoden auf mehrfach zusammenhängenden Gebieten um das reduzierte Modell auszuwerten. Zusätzlich zur kontrollierten Modellreduktion liefern die theoretischen Untersuchungen Ergebnisse von praktischer Relevanz. So implizieren z.B. die Bedingungen für die Existenz und Eindeutigkeit des vollen Wärmeleitungsproblems, dass ab einer hinreichen hohen Stromstärke keine endliche Temperatur mehr erreicht wird. Dies wird durch die Unterscheidung subresonanter und resonanter Zustände semilinearer elliptischer Gleichungen beschrieben. Die Analyse des Querschnittsproblems durch Randintegralgleichungen liefert wiederum eine geometrische Eigenschaft von mehrfach zusammenhängenden Gebieten - die Dämpfungseigenschaft. Diese Eigenschaft kann als eine natürliche Eigenschaft von Isolierungen interpretiert werden und ist wesentlich für die Konvergenz der Fixpunktiteration im mehrfach zusammenhängenden Fall

Variational regularization theory for sparsity promoting wavelet regularization

In many scientific and industrial applications, the quantity of interest is not what is directly observed, but is instead a parameter which has a causal effect on experimental measurements. To obtain the desired unknown quantity, one must use an inverse transform on the data. The main challenge in such an inverse problem is that these unknowns may not continuously depend on the observations, and as a result, the effects of noise in data are magnified in the inverted results. To obtain stable approximations of the desired parameters from noisy observations, regularization methods are used. This thesis contributes to the mathematical analysis of generalized Tikhonov regularization, and in particular sparsity promoting Tikhonov regularization, which are popular examples of regularization methods. Using variational source conditions as an intermediate step, order optimal upper bounds on the reconstruction error are shown for sparsity promoting wavelet regularization under smoothness assumptions given by Besov spaces. The framework includes practically relevant forward operators, such as the Radon transform, and some nonlinear inverse problems in differential equations with distributed measurements. In numerical simulations for a parameter identification problem in a differential equation it is demonstrated that these theoretical results correctly predict convergence rates for piecewise smooth unknown coefficients.2022-02-0