Iterative Krylov subspace methods have proven to be efficient tools for solving linear systems of equations. In the context of ill-posed inverse problems, they tend to exhibit semiconvergence behavior making it difficult detect ``inverted noise" and stop iterations before solutions become contaminated. Regularization methods such as spectral filtering methods use the singular value decomposition (SVD) and are effective at filtering inverted noise from solutions, but are computationally prohibitive on large problems. Hybrid methods apply regularization techniques to the smaller ``projected problem" that is inherent to iterative Krylov methods at each iteration, thereby overcoming the semiconvergence behavior. Commonly, the Golub-Kahan bidiagonalization is used to construct a set of orthonormal basis vectors that span the Krylov subspaces from which solutions will be chosen, but seeking a solution in the orthonormal basis generated by the Arnoldi process (which is fundamental to the popular iterative method GMRES) has been of renewed interest recently. We discuss some of the positive and negative aspects of each process and use example problems to examine some qualities of the bases they produce. Computing optimal solutions in a given basis gives some insight into the performance of the corresponding iterative methods and how hybrid methods can contribute. / Master of Science

In this work we study, in details, the Cauchy problem of the nonlinear Schrödinger equation, with initial datas in periodic Sobolev spaces. Specifically, we prove that this problem is locally well posed for datas in Hsper, with s ≥ 0. Particularly, for initial datas in L2 the problem is globally well posed, due to the conservation law of the equation in this space. Moreover, we prove the this result is the best one, seeing we expose examples that show that the equation flow is not locally uniformly continuous for initial datas with regularity less than L2. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, fazemos um estudo detalhado do problema de Cauchy para a equação não-linear cúbica de Schrödinger, com dados iniciais em espaços de Sobolev no toro. Especificamente, provaremos que este modelo é localmente bem posto para dados em Hsper, com s ≥ 0. Em particular, para dados iniciais em L2 o modelo é globalmente bem posto, devido à lei de conservação da equação neste espaço. Além disso, provaremos que os resultados obtidos são os melhores possíveis, visto que exibiremos exemplos que mostram que o fluxo da equação não é localmente uniformemente contínuo para dados iniciais com regularidade menor que L2.

In this thesis we consider problems Ax b arising from the discretization of ill-posed problems, where the right-hand side b is polluted by (unknown) noise. It was shown in [29] that under some natural assumptions, using the Golub-Kahan iterative bidiagonalization the noise level in the data can be estimated at a negligible cost. Such information can be further used in solving ill-posed problems. Here we suggest criteria for detecting the noise revealing iteration in the Golub-Kahan iterative bidiagonalization. We discuss the presence of noise of different colors. We study how the loss of orthogonality affects the noise revealing property of the bidiagonalization.

Pass-by noise testing is an important step in vehicle design and regulation compliance. Finite element analysis simulations have been used to cut costs on prototyping and testing, but the high computational cost of simulating surface vibrations from complex geometries and the resulting airborne noise propagation is making the switch to digital twin methods not viable. This paper aims at investigating the use of equivalent source methods as an alternative to the before mentioned simulations. Through the use of a simple 2D model, the difficulties such as ill-conditioning of the transfer matrix and the required regularisation techniques such as TSVD and the Tikhonov L-curve method are tested and then applied to a mesh of a 3D engine model. Source and pressure field errors are measured and their origins are explained. A heavy emphasis is put on the model geometry as a source of error. Finally, rules of thumb based on the regularisation balance and the wavelength dependent pressure sampling positions are formulated in order to achieve usable results. / Bullerprovning vid passage är ett viktigt steg i fordonsdesign och regelefterlevnad. Simuleringar med finita elementanalyser har använts för att minska kostnaderna för prototypframtagning och provning, men de höga beräkningskostnaderna för att simulera ytvibrationer från komplexa geometrier och den resulterande luftburna bullerspridningen gör att övergången till digitala tvillingmetoder inte är genomförbar. Denna uppsats syftar till att undersöka användningen av ekvivalenta källmetoder som ett alternativ till de tidigare nämnda simuleringarna. Genom att använda en enkel 2D-modell testas svårigheterna som dålig konditionering av överföringsmatrisen och de nödvändiga regulariseringsteknikerna som TSVD och Tikhonov L-kurvmetoden och tillämpas sedan på ett nät av en 3D-motormodell. Käll- och tryckfältsfel mäts och deras ursprung förklaras. Stor vikt läggs vid modellgeometrin som en felkälla. Slutligen formuleras tumregler baserade på regulariseringsbalansen och de våglängdsberoende tryckprovtagningspositionerna för att uppnå användbara resultat.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In Chapter 1 a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. We also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme. In Chapter 2 we give a convergence analysis of the following iterative scheme: u[subscript]n[superscript]delta=q u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) K[superscript]*f[subscript]delta, u[subscript]0[superscript]delta=0, where T:=K[superscript]* K, T[subscript]a :=T+aI, q in the interval (0,1),\quad a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional approximations of T and K[superscript]* for solving stably Fredholm integral equations of the first kind with noisy data. In Chapter 3 a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f(t) is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f(t), are proposed in this chapter.

<p>Agrarna politika vlasti u periodu od 1945-1950. godine je<br />shodno ideolo&scaron;kim određenjima, bila usmerena na promenu<br />dru&scaron;tvenih odnosa na selu. Međutim, osim ideolo&scaron;kih<br />razloga ne treba zanemariti činjenicu da je usitnjen zemlji&scaron;ni<br />posed već ispoljio mnoge slabosti i da su u uslovima<br />uvećane potražnje za poljoprivrednim proizvodima ti problemi<br />dodatno ubedili vlast u neophodnost koncentracije<br />zemlje i formiranja krupnih, pre svega, zadružnih gazdinstava.<br />U cilju realizacije tog cilja, primenjene su rigidne<br />ekonomske i političke mere, naročito na prostoru Vojvodine<br />među kojima je bilo i brutalnih metoda izrazito oprečnih<br />tradicionalnim vrednostima i etičkom kodeksu očuvanom u<br />seoskim sredinama. Ovakva agrarna politika izazvala je<br />veliko nezadovoljstvo selja&scaron;tva i veliki pad poljoprivredne<br />proizvodnje. Razni vidovi opstrukcije velikog dela stanovni&scaron;tva<br />prema državi i strah od gladi u jeku sukoba sa SSSRom,<br />ubedili su partijsko rukovodstvo zemlje u neophodnost<br />promene dotada&scaron;njeg odnosa prema poljoprivredi i selja&scaron;tvu.<br />Mada se nije lako odustajalo od politike kolektivizacije, gore<br />navedeni razlozi doveli su do njenog napu&scaron;tanja već<br />1950. godine, a zadružni sistem od 1953. godine transformisan<br />je u ekonomski racionalnije organizacije. Ove promene,<br />nisu značile i kraj primene revolucionarnih mera na<br />selu. One su posebno nagla&scaron;ene 1953. godine ograničenjem<br />privatnog poseda na 10 hektara po domaćinstvu.<br />Ipak, agrarna politika KPJ od tada se razvija na principima<br />većeg uvažavanja selja&scaron;tva i uspostavljanja ekonomske<br />saradnje sa privatnim posedom. Novu politiku karakteri&scaron;e<br />dobrovoljnost kooperacije i značajnije ulaganje u prerađivačku<br />industriju i poljoprivredu proizvodnju.</p> / <p>Agrarian policy of the Government in the period from 1945<br />to 1950, according to ideological determinations, was<br />pointed to the changes of social relations in a village. However,<br />beside ideological reasons, it should not be neglected<br />the fact that fragmented land property had already shown<br />many weaknesses and that in the circumstances of higher<br />requests for agricultural products, those issues persuaded<br />the Government in the necessity of gathering land and<br />forming large collective economies. In order to realize that<br />goal, rigid economic and political measures were taken,<br />especially in the territory of Vojvodina, some of them very<br />brutal, totally opposite to traditional values and ethical codex<br />preserved in rural areas. This agrarian policy caused<br />great discontent among peasantry and huge decline of agricultural<br />production. Different ways of obstruction by people<br />towards the state and the fear of hunger in the middle<br />of the conflict with Soviet Union (USSR), persuaded the<br />Party leader-ship of the State in the necessity of changing<br />the relation towards agriculture and peasantry.<br />Even though it was not easy to give up the policy of collectivization,<br />reasons mentioned above brought to it&#39;s abandonment<br />in 1950, and collective economy sys-tem was<br />transformed in 1953 into economically more rational organizations.<br />Those changes did not led to the end of use<br />revolutionary measures in a village. They were even emphasized<br />in 1953 by limiting the private properties to 10<br />hectares per house-holds. However, agrarian policy since<br />that period was developing on the principles of higher respect<br />of the peasantry and establishing cooperation with<br />private property. New policy was characterized by willingly<br />cooperation and significant investment in manufacturing<br />industry and agrarian production.</p>

The determination of the atmospheric aerosol size distribution is an inverse illposed problem. The shape and the material composition of the air-carried particles are two substantial model parameters. Present evaluation algorithms only used an approximation with spherical homogeneous particles. In this paper we propose a new numerically efficient recursive algorithm for inhomogeneous multilayered coated and absorbing particles. Numerical results of real existing particles show that the influence of the two parameters on the model is very important and therefore cannot be ignored.

This thesis presents a robust method for estimating the relaxations of a metallic object from its electromagnetic induction (EMI) response. The EMI response of a metallic object can be accurately modeled by a sum of real decaying exponentials. However, it is difficult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown or the terms are strongly correlated. Traditionally, the relaxation constants are estimated by nonlinear iterative search that often leads to unsatisfactory results. An effective EMI modeling technique is developed by first linearizing the problem through enumeration and then solving the linearized model using a sparsity-regularized minimization. This approach overcomes several long-standing challenges in EMI signal modeling, including finding the unknown model order as well as handling the ill-posed nature of the problem. The resulting algorithm does not require a good initial guess to converge to a satisfactory solution. This new modeling technique is extended to incorporate multiple measurements in a single parameter estimation step. More accurate estimates are obtained by exploiting an invariance property of the EMI response, which states that the relaxation frequencies do not change for different locations and orientations of a metallic object. Using tests on synthetic data and laboratory measurement of known targets, the proposed multiple-measurement method is shown to provide accurate and stable estimates of the model parameters. The ability to estimate the relaxation constants of targets enables more robust subsurface target discrimination using the relaxations. A simple relaxation-based subsurface target detection algorithm is developed to demonstrate the potential of the estimated relaxations.

This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y.

Magnetic induction tomography (MIT) is a non-invasive, soft field imaging modality that has the potential to map the electrical conductivity (σ) distribution inside an object under investigation. In MIT, a number of exciter and receiver coils are distributed around the periphery of the object. A primary magnetic field is emitted by each exciter, and interacts with the object. This induces eddy currents in the object, which in turn create a secondary field. This latter is coupled to the receiver coils and voltages are induced. An image reconstruction algorithm is then used to infer the conductivity map of the object. In this thesis, the application of MIT for volumetric imaging of objects with low conductivity materials (< 5 Sm-1) and dimensions < 1 m is investigated. In particular, two low conductivity applications are approached: imaging cerebral stroke and imaging the saline water in multiphase flows. In low conductivity applications, the measured signals are small and the spatial sensitivity is critically compromised making the associated inverse problem severely non-linear and ill-posed.The main contribution from this study is to investigate three non-linear optimisation techniques for solving the MIT inverse problem. The first two methods, namely regularised Levenberg Marquardt method and trust region Powell's Dog Leg method, employ damping and trust region strategies respectively. The third method is a modification of the Gauss Newton method and utilises a damping regularisation technique. An optimisation in the convergence and stability of the inverse solution was observed with these methods compared to standard Gauss Newton method. For such non linear treatment, re-evaluation of the forward problem is also required. The forward problem is solved numerically using the impedance method and a weakly coupled field approximation is employed to reduce the computation time and memory requirements. For treating the ill-posedness, different regularisation methods are investigated. Results show that the subspace regularisation technique is suitable for absolute imaging of the stroke in a real head model with synthetic data. Tikhonov based smoothing and edge preserving regularisation methods also produced successful results from simulations of oil/water. However, in a practical setup, still large geometrical and positioning noise causes a major problem and only difference imaging was viable to achieve a reasonable reconstruction.