Esta tese versa sobre os produtos de Stratonovich e de Berezin de funções na esfera \'S POT. 2\'. Cada um destes produtos é definido atravéz de uma correspondência de símbolos, que é uma aplicação linear bijetiva entre operadores lineares num espaço de Hilbert complexo de dimensão n + 1, ou seja matrizes complexas (n + 1) × (n + 1), e polinômios complexos de grau próprio n definidos na 2-esfera, PolyC(\'S POT. 2\')n, satisfazendo algumas propriedades básicas, como equivariância pela ação do grupo de rotações SO(3), preservação das estruturas reais e normalização [12]. Mais geralmente, toda correspondência define um produto associativo mas não comutativo em PolyC(\'S POT. 2\')n induzido do produto de operadores, chamado de produto twisted em PolyC(\'S POT. \'2)n. Cada um destes produtos twisted, por sua vez, pode ser escrito na forma integral f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, onde f, g PolyC(\'S POT. 2)n, \'n IND. 1\',\' n IND. 2\', n \'S POT. 2\'. Em tal representação integral, todas as propriedades do produto twisted são convertidas em propriedades do trikernel integral L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' C. Os produtos twisted estudados nesta tese são os produtos induzidos pela correspondência padrão de Stratonovich e a correspondência padrão de Berezin, respectivamente, que num certo limite assintótico 2j = n definem deformações estritas da álgebra de Poisson de \'S POT. 2\' [12]. Para cada um deste dois produtos, denotados por \'n SOB. 1\' e \'n SOB. b\' respectivamente, seu trikernel integral é denotado por L \'SOB. j 1\' e L \'j SOB. ~b, respectivamente. A primeira parte desta tese consistiu em desenvolver fórmulas mais tratáveis para L \'j SOB. 1\' e L j SOB. b\' nos casos de número de spin j = 1/2, 1, 3/2, 2, fórmulas estas escritas em termos de funções de dois e de três pontos, invariantes por SO(3), como produtos escalares e determinantes. Nossa esperança inicial era de que pudéssemos encontar padrões que nos permitissem inferir fórmulas fechadas para cada um destes trikernels, válidas para qualquer j, ou pelo menos que nos permitissem inferir fórmulas assintóticas para estes trikernels quando 2j = n . Porém, o grau de complexidade das fórmulas desenvolvidas se mostrou fortemente crescente com j, frustrando nossas expectativas iniciais. Partimos então para uma exploração preliminar de um tipo de aproximação assintótica destes produtos de certas funções oscilatórias na esfera. Mais precisamente, na segunda parte desta tese, preparamos e estudamos preliminarmente o produto de Stratonovich e o produto de Berezin (assim como o produto pontual) de dois harmônicos esféricos, \'Y POT. m1 INF. l1\' e \'Y POT. m2 INF. l2 PolyC(\'S POT. 2\')n, no limite assintótico quando tanto \'l IND. 1\' como \'l IND. 2\' tendem a infinito linearmente com n (mantendo \'l IND. i\' n). Este tipo de assintótica para tais produtos, que faz parte do que chamamos mais geralmente de high-l asymptotics, difere muito do tipo de assintótica estudada de forma detalhada em [12], na qual n , mas \'l IND. 1\' e \'l IND. 2\' são mantidos finitos. Então, a partir de um exemplo particular para nossa exploração preliminar, levantamos uma conjectura sobre como estes produtos se comparam no limite assintótico quando \'l IND. 1\' e \'l IND. 2\' tendem para infinito linearmente com o número de spin j / This thesis is about the Stratonovich and the Berezin products of functions on the 2-sphere. Each one of these products is defined via a spin-j symbol correspondence, a linear bijective map from the space of operators on an (n + 1)-dimensional complex Hilbert Space, i.e. (n + 1) × (n + 1) complex matrices, and the space of complex polynomials on \'S POT. 2\' of proper degree n, denoted PolyC(\'S POT. 2\')n, satisfying certain basic properties like equivariance under the action of SO(3), preservation of real structures and normalization [12]. More generally, every spin-j symbol correspondence defines an associative noncommutative product on PolyC(\'S POT. 2\')n induced from the operator product, which is called a twisted product on PolyC(\'S POT. 2\')n. Each twisted product can be written in integral form as f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, where f, g PolyC(\'S POT. 2) \' > OR =\' n, \'n IND. 1\',\' n IND. 2\', n \'IT BELONGS\' \'S POT. 2\'. In such and an integral representation, all properties of the twisted product are translated to properties of its integral trikernel L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' \'ARROW\' C. The twisted products studied in this thesis are the ones obtained via the standard Stratonovich-Weyl and the standard Berezin symbol correspondences, respectively, which in a certain asymptotic limit 2j = n \'ARROW\' \'INFINITY\' define strict deformation quantizations of \'S POT. 2\', i.e. strict deformations of the Poisson algebra of \'S POT. 2\' [12]. For 10 each of these two products on PolyC(\'S POT. 2\') \'< OR =\' n, denoted by * \'n SOB. 1\' and \'n SOB. b\' respectively, its integral trikernel is denoted by L\'j SOB. 1\' and L\'j SOB. b\' , respectively. In the first part of this thesis, we obtained better formulae for these trikernels, for values of spin number j = 1/2, 1/3/1, 2. These formulas are written in terms of SO(3)-invariant functions of two and three points on \'S POT. 2\', like scalar products and determinants. We initially hoped to be able to obtain closed formulae for these trikernels which would be valid for every j, or at least be able to infer asymptotic formulas for these trikernels when 2j = n \'ARROW\' \' INFINITY\' . However, the degree of complexity of the formulae we have obtained increases strongly with j, frustrating our initial expectations. We thus started on a preliminary investigation of a kind of asymptotic approximation for these products of certain oscillatory functions on the sphere. More precisely, in the second part of this thesis, we prepared and preliminarily studied the Stratonovich product and the Berezin product (as well as the pointwise product) of spherical harmonics Y \'m1 SOB. l1\' and Y \'m2 SOB. l2\' PolyC(\'S POT. 2\')n, in the asymptotic limit when l1 and l2 tend to infinity linearly with n (keeping \'l IND. i\' \'< OR =\' n). This kind of asymptotics for these products, belonging to what we more generally callhigh-l asymptotics, differs drastically from the kind of asymptotics studied in detail in [12], in which n \'ARROW\' \'INFINITY\' but \'l IND. 1\' and \'l IND. 2\' are kept finite. Then, based on a particular example of our preliminary exploration, we advanced a conjecture on how these products behave and compare with each other, in the asymptotic limit when \'l IND. \'1\' and \'l IND. 2\' tend to infinity linearly with the spin number j

Esta tese versa sobre os produtos de Stratonovich e de Berezin de funções na esfera \'S POT. 2\'. Cada um destes produtos é definido atravéz de uma correspondência de símbolos, que é uma aplicação linear bijetiva entre operadores lineares num espaço de Hilbert complexo de dimensão n + 1, ou seja matrizes complexas (n + 1) × (n + 1), e polinômios complexos de grau próprio n definidos na 2-esfera, PolyC(\'S POT. 2\')n, satisfazendo algumas propriedades básicas, como equivariância pela ação do grupo de rotações SO(3), preservação das estruturas reais e normalização [12]. Mais geralmente, toda correspondência define um produto associativo mas não comutativo em PolyC(\'S POT. 2\')n induzido do produto de operadores, chamado de produto twisted em PolyC(\'S POT. \'2)n. Cada um destes produtos twisted, por sua vez, pode ser escrito na forma integral f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, onde f, g PolyC(\'S POT. 2)n, \'n IND. 1\',\' n IND. 2\', n \'S POT. 2\'. Em tal representação integral, todas as propriedades do produto twisted são convertidas em propriedades do trikernel integral L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' C. Os produtos twisted estudados nesta tese são os produtos induzidos pela correspondência padrão de Stratonovich e a correspondência padrão de Berezin, respectivamente, que num certo limite assintótico 2j = n definem deformações estritas da álgebra de Poisson de \'S POT. 2\' [12]. Para cada um deste dois produtos, denotados por \'n SOB. 1\' e \'n SOB. b\' respectivamente, seu trikernel integral é denotado por L \'SOB. j 1\' e L \'j SOB. ~b, respectivamente. A primeira parte desta tese consistiu em desenvolver fórmulas mais tratáveis para L \'j SOB. 1\' e L j SOB. b\' nos casos de número de spin j = 1/2, 1, 3/2, 2, fórmulas estas escritas em termos de funções de dois e de três pontos, invariantes por SO(3), como produtos escalares e determinantes. Nossa esperança inicial era de que pudéssemos encontar padrões que nos permitissem inferir fórmulas fechadas para cada um destes trikernels, válidas para qualquer j, ou pelo menos que nos permitissem inferir fórmulas assintóticas para estes trikernels quando 2j = n . Porém, o grau de complexidade das fórmulas desenvolvidas se mostrou fortemente crescente com j, frustrando nossas expectativas iniciais. Partimos então para uma exploração preliminar de um tipo de aproximação assintótica destes produtos de certas funções oscilatórias na esfera. Mais precisamente, na segunda parte desta tese, preparamos e estudamos preliminarmente o produto de Stratonovich e o produto de Berezin (assim como o produto pontual) de dois harmônicos esféricos, \'Y POT. m1 INF. l1\' e \'Y POT. m2 INF. l2 PolyC(\'S POT. 2\')n, no limite assintótico quando tanto \'l IND. 1\' como \'l IND. 2\' tendem a infinito linearmente com n (mantendo \'l IND. i\' n). Este tipo de assintótica para tais produtos, que faz parte do que chamamos mais geralmente de high-l asymptotics, difere muito do tipo de assintótica estudada de forma detalhada em [12], na qual n , mas \'l IND. 1\' e \'l IND. 2\' são mantidos finitos. Então, a partir de um exemplo particular para nossa exploração preliminar, levantamos uma conjectura sobre como estes produtos se comparam no limite assintótico quando \'l IND. 1\' e \'l IND. 2\' tendem para infinito linearmente com o número de spin j / This thesis is about the Stratonovich and the Berezin products of functions on the 2-sphere. Each one of these products is defined via a spin-j symbol correspondence, a linear bijective map from the space of operators on an (n + 1)-dimensional complex Hilbert Space, i.e. (n + 1) × (n + 1) complex matrices, and the space of complex polynomials on \'S POT. 2\' of proper degree n, denoted PolyC(\'S POT. 2\')n, satisfying certain basic properties like equivariance under the action of SO(3), preservation of real structures and normalization [12]. More generally, every spin-j symbol correspondence defines an associative noncommutative product on PolyC(\'S POT. 2\')n induced from the operator product, which is called a twisted product on PolyC(\'S POT. 2\')n. Each twisted product can be written in integral form as f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, where f, g PolyC(\'S POT. 2) \' > OR =\' n, \'n IND. 1\',\' n IND. 2\', n \'IT BELONGS\' \'S POT. 2\'. In such and an integral representation, all properties of the twisted product are translated to properties of its integral trikernel L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' \'ARROW\' C. The twisted products studied in this thesis are the ones obtained via the standard Stratonovich-Weyl and the standard Berezin symbol correspondences, respectively, which in a certain asymptotic limit 2j = n \'ARROW\' \'INFINITY\' define strict deformation quantizations of \'S POT. 2\', i.e. strict deformations of the Poisson algebra of \'S POT. 2\' [12]. For 10 each of these two products on PolyC(\'S POT. 2\') \'< OR =\' n, denoted by * \'n SOB. 1\' and \'n SOB. b\' respectively, its integral trikernel is denoted by L\'j SOB. 1\' and L\'j SOB. b\' , respectively. In the first part of this thesis, we obtained better formulae for these trikernels, for values of spin number j = 1/2, 1/3/1, 2. These formulas are written in terms of SO(3)-invariant functions of two and three points on \'S POT. 2\', like scalar products and determinants. We initially hoped to be able to obtain closed formulae for these trikernels which would be valid for every j, or at least be able to infer asymptotic formulas for these trikernels when 2j = n \'ARROW\' \' INFINITY\' . However, the degree of complexity of the formulae we have obtained increases strongly with j, frustrating our initial expectations. We thus started on a preliminary investigation of a kind of asymptotic approximation for these products of certain oscillatory functions on the sphere. More precisely, in the second part of this thesis, we prepared and preliminarily studied the Stratonovich product and the Berezin product (as well as the pointwise product) of spherical harmonics Y \'m1 SOB. l1\' and Y \'m2 SOB. l2\' PolyC(\'S POT. 2\')n, in the asymptotic limit when l1 and l2 tend to infinity linearly with n (keeping \'l IND. i\' \'< OR =\' n). This kind of asymptotics for these products, belonging to what we more generally callhigh-l asymptotics, differs drastically from the kind of asymptotics studied in detail in [12], in which n \'ARROW\' \'INFINITY\' but \'l IND. 1\' and \'l IND. 2\' are kept finite. Then, based on a particular example of our preliminary exploration, we advanced a conjecture on how these products behave and compare with each other, in the asymptotic limit when \'l IND. \'1\' and \'l IND. 2\' tend to infinity linearly with the spin number j

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2016-02-22T12:27:05Z No. of bitstreams: 2 Dissertação - Luiz Eduardo de Sousa Freire - 2015.pdf: 1113525 bytes, checksum: 2604d25df22a0fce378e1f9840262c72 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-02-22T12:28:32Z (GMT) No. of bitstreams: 2 Dissertação - Luiz Eduardo de Sousa Freire - 2015.pdf: 1113525 bytes, checksum: 2604d25df22a0fce378e1f9840262c72 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2016-02-22T12:28:32Z (GMT). No. of bitstreams: 2 Dissertação - Luiz Eduardo de Sousa Freire - 2015.pdf: 1113525 bytes, checksum: 2604d25df22a0fce378e1f9840262c72 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-20 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this paper we investigate two-dimensional membranes, such as graphene, using Quantum Field Theory, more specifically by the path integral formalism and using Hubbard- Stratonovich transformations. With graphene as a motivation, we use models for electrons in a graphene layer, when they interact with phonons, Kekule deformation and gauge fields. We start by considering a theory in which bosons and fermions interact via a Yukawa type coupling. We present a method already known to eliminate the degrees of freedom of the fermionic system and use the Hubbard-Stratonovich transformations to derive an effective bosonic theory. For this, we introduce an auxiliary field in the model and show that the effective theory for this field is equivalent to the effective theory for the physical bosonic field. Thus, we calculated and obtained the gap equations for this system in 1+1, 2+1 and 3+1 dimensions and then compared with the Gross-Neveu model for quartic interactions between fermions in 1+1 dimensions. We see that for a particular coupling constant the massive electrons lose all their mass by interacting with bosons, an effect caused by symmetry breaking. We then present the chiral gauge model of Jackiw-Pi for graphene, where Yukawa type interactions are present. However, this theory is a particular case of a more general model proposed by Frederico et al. where bosonic self-interactions at higher orders and bosons/fermions with more general interactions are considered. Again, we use the Hubbard- Stratonovich transformations to derive effective models for fermions and the gap equations. We identified chiral invariance transformations of the group U(1) for the limiting case of Jackiw-Pi model. Finally, we investigated a model for phonons in the graphene background, more specifically building on the papers by Katsnelson et al. Guinea et al.. At this point, anharmonic terms are included in the Lagrangian of the system in an attempt to describe changes in the background structure. We eliminate the degrees of freedom of the scalar system, which are responsible for describing power modes on and off plane and thus obtained an effective theory for the electronic part. As in the previous model, we see a change in the effective potential and derive their gap equations. Finally, we present the Coulomb potential to derive an effective theory for fermions when they interact with a gauge field. Thus, we compare this result with the studied models and analyzed the effective fermionic interactions present in the obtained Lagrangians. / Neste trabalho investigamos membranas bidimensionais, tal como o grafeno, usando Teoria Quântica de Campos, mais especificamente pelo formalismo de integrais de trajetória e usando transformações de Hubbard-Stratonovich. Com o grafeno como motivação, usamos modelos para os elétrons em uma camada de grafeno, quando estes interagem com fônons da rede, deformações de Kekulé e campos de gauge. Começamos analisando uma teoria em que bósons e férmions interagem entre si por meio de um acoplamento tipo Yukawa. Apresentamos um método já conhecido para eliminar os graus de liberdade fermiônicos do sistema e usamos as transformações de Hubbard-Stratonovich para derivar uma teoria efetiva bosônica. Para isto, inserimos um campo auxiliar no modelo e mostramos que a teoria efetiva pra este campo é equivalente à teoria efetiva para o campo bosônico físico. Assim, calculamos e simulamos as equações de gap para este sistema em 1+1, 2+1 e 3+1 dimensões e comparamos com o modelo de Gross-Neveu para interações quárticas entre férmions em 1+1 dimensões. Vemos que para dada constante de acoplamento, férmions massivos perdem toda sua massa ao interagir com os bósons, efeito causado por quebra de simetria. Em seguida, apresentamos o modelo de gauge quiral de Jackiw-Pi para o grafeno, onde interações tipo Yukawa estão presentes. Contudo, uma generalização é proposta por Frederico et al. em que auto-interações bosônicas de ordens mais altas e interações bósons/férmions mais gerais são consideradas. Novamente, usamos as transformações de Hubbard-Stratonovich para derivar modelos efetivos para os férmions e as equações de gap. Identificamos a invariância por transformações quirais do grupo U(1) para o caso limite do modelo de Jackiw-Pi. Em seguida, investigamos um modelo para os fônons da rede no grafeno, mais especificamente, a partir dos trabalhos de Katsnelson et al e Guinea et al. Neste ponto, termos anarmônicos são incluídos na lagrangiana do sistema no intento de descrever modificações na estrutura da rede. Eliminamos os graus de liberdade escalares do sistema, que são aqueles responsáveis por descrever os modos de energia dentro e fora do plano e, assim, obtivemos uma teoria efetiva para a parte eletrônica. Como no modelo anterior, inferimos uma mudança no potencial efetivo e derivamos as respectivas equações de gap. Por fim, apresentamos o potencial de Coulomb ao derivar uma teoria efetiva para férmions quando estes interagem com um campo de gauge. Assim, comparamos este resultado com os modelos estudados e analisamos as interações fermiônicas efetivas presentes nas lagrangianas obtidas.

This thesis is on an advanced method for pricing financial derivatives in a market model,which comprises two stochastic volatilities. Financial derivatives are instruments whosethat is related to any financial asset. Underlying assets in derivatives are mostly financialinstruments; such as security, currency or a commodity. Stochastic volatilities are used infinancial mathematics to assess financial derivative securities; such as contingent claims andoptions for valuation of the derivatives, at the expiration of the contract. This study examinedtheoretical frameworks that evolve around the pricing of financial deriv- atives in a marketmodel and it mainly examines two stochastic volatilities: cubature formula and splittingmethod by analysing how these volatilities affect the pricing of financial derivatives. The studydeveloped an approximation approach with a double stochastic volatilities model in termsof Stratonovich integrals to evaluate the contingent claim, examined the similarities betweenNinomiya–Ninomiya scheme and Ninomiya–Victoir scheme, and rewrite the system of doublestochastic volatility model in terms of the standard Brownian motion.

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The focus of the thesis is on the development of a few stochastic search schemes for inverse problems and their applications in medical imaging. After the introduction in Chapter 1 that motivates and puts in perspective the work done in later chapters, the main body of the thesis may be viewed as composed of two parts: while the first part concerns the development of stochastic search algorithms for inverse problems (Chapters 2 and 3), the second part elucidates on the applicability of search schemes to inverse problems of interest in tomographic imaging (Chapters 4 and 5). The chapter-wise contributions of the thesis are summarized below. Chapter 2 proposes a Monte Carlo stochastic filtering algorithm for the recursive estimation of diffusive processes in linear/nonlinear dynamical systems that modulate the instantaneous rates of Poisson measurements. The same scheme is applicable when the set of partial and noisy measurements are of a diffusive nature. A key aspect of our development here is the filter-update scheme, derived from an ensemble approximation of the time-discretized nonlinear Kushner Stratonovich equation, that is modified to account for Poisson-type measurements. Specifically, the additive update through a gain-like correction term, empirically approximated from the innovation integral in the filtering equation, eliminates the problem of particle collapse encountered in many conventional particle filters that adopt weight-based updates. Through a few numerical demonstrations, the versatility of the proposed filter is brought forth, first with application to filtering problems with diffusive or Poisson-type measurements and then to an automatic control problem wherein the exterminations of the associated cost functional is achieved simply by an appropriate redefinition of the innovation process. The aim of one of the numerical examples in Chapter 2 is to minimize the structural response of a duffing oscillator under external forcing. We pose this problem of active control within a filtering framework wherein the goal is to estimate the control force that minimizes an appropriately chosen performance index. We employ the proposed filtering algorithm to estimate the control force and the oscillator displacements and velocities that are minimized as a result of the application of the control force. While Fig. 1 shows the time histories of the uncontrolled and controlled displacements and velocities of the oscillator, a plot of the estimated control force against the external force applied is given in Fig. 2. (a) (b) Fig. 1. A plot of the time histories of the uncontrolled and controlled (a) displacements and (b) velocities. Fig. 2. A plot of the time histories of the external force and the estimated control force Stochastic filtering, despite its numerous applications, amounts only to a directed search and is best suited for inverse problems and optimization problems with unimodal solutions. In view of general optimization problems involving multimodal objective functions with a priori unknown optima, filtering, similar to a regularized Gauss-Newton (GN) method, may only serve as a local (or quasi-local) search. In Chapter 3, therefore, we propose a stochastic search (SS) scheme that whilst maintaining the basic structure of a filtered martingale problem, also incorporates randomization techniques such as scrambling and blending, which are meant to aid in avoiding the so-called local traps. The key contribution of this chapter is the introduction of yet another technique, termed as the state space splitting (3S) which is a paradigm based on the principle of divide-and-conquer. The 3S technique, incorporated within the optimization scheme, offers a better assimilation of measurements and is found to outperform filtering in the context of quantitative photoacoustic tomography (PAT) to recover the optical absorption field from sparsely available PAT data using a bare minimum ensemble. Other than that, the proposed scheme is numerically shown to be better than or at least as good as CMA-ES (covariance matrix adaptation evolution strategies), one of the best performing optimization schemes in minimizing a set of benchmark functions. Table 1 gives the comparative performance of the proposed scheme and CMA-ES in minimizing a set of 40-dimensional functions (F1-F20), all of which have their global minimum at 0, using an ensemble size of 20. Here, 10 5 is the tolerance limit to be attained for the objective function value and MAX is the maximum number of iterations permissible to the optimization scheme to arrive at the global minimum. Table 1. Performance of the SS scheme and Chapter 4 gathers numerical and experimental evidence to support our conjecture in the previous chapters that even a quasi-local search (afforded, for instance, by the filtered martingale problem) is generally superior to a regularized GN method in solving inverse problems. Specifically, in this chapter, we solve the inverse problems of ultrasound modulated optical tomography (UMOT) and diffraction tomography (DT). In UMOT, we perform a spatially resolved recovery of the mean-squared displacements, p r of the scattering centres in a diffusive object by measuring the modulation depth in the decaying autocorrelation of the incident coherent light. This modulation is induced by the input ultrasound focussed to a specific region referred to as the region of interest (ROI) in the object. Since the ultrasound-induced displacements are a measure of the material stiffness, in principle, UMOT can be applied for the early diagnosis of cancer in soft tissues. In DT, on the other hand, we recover the real refractive index distribution, n r of an optical fiber from experimentally acquired transmitted intensity of light traversing through it. In both cases, the filtering step encoded within the optimization scheme recovers superior reconstruction images vis-à-vis the GN method in terms of quantitative accuracies. Fig. 3 gives a comparative cross-sectional plot through the centre of the reference and reconstructed p r images in UMOT when the ROI is at the centre of the object. Here, the anomaly is presented as an increase in the displacements and is at the centre of the ROI. Fig. 4 shows the comparative cross-sectional plot of the reference and reconstructed refractive index distributions, n r of the optical fiber in DT. Fig. 3. Cross-sectional plot through the center of the reference and reconstructed p r images. Fig. 4. Cross-sectional plot through the center of the reference and reconstructed n r distributions. In Chapter 5, the SS scheme is applied to our main application, viz. photoacoustic tomography (PAT) for the recovery of the absorbed energy map, the optical absorption coefficient and the chromophore concentrations in soft tissues. Nevertheless, the main contribution of this chapter is to provide a single-step method for the recovery of the optical absorption field from both simulated and experimental time-domain PAT data. A single-step direct recovery is shown to yield better reconstruction than the generally adopted two-step method for quantitative PAT. Such a quantitative reconstruction maybe converted to a functional image through a linear map. Alternatively, one could also perform a one-step recovery of the chromophore concentrations from the boundary pressure, as shown using simulated data in this chapter. Being a Monte Carlo scheme, the SS scheme is highly parallelizable and the availability of such a machine-ready inversion scheme should finally enable PAT to emerge as a clinical tool in medical diagnostics. Given below in Fig. 5 is a comparison of the optical absorption map of the Shepp-Logan phantom with the reconstruction obtained as a result of a direct (1-step) recovery. Fig. 5. The (a) exact and (b) reconstructed optical absorption maps of the Shepp-Logan phantom. The x- and y-axes are in m and the colormap is in mm-1. Chapter 6 concludes the work with a brief summary of the results obtained and suggestions for future exploration of some of the schemes and applications described in this thesis.