This thesis presents a systematic computational investigation of loss functions in solving inverse problems of partial differential equations. The primary efforts are spent on understanding optimization-based computational inversion with loss functions defined with the Wasserstein metrics and with deep learning models. The scientific contributions of the thesis can be summarized in two directions. In the first part of this thesis, we investigate the general impacts of different Wasserstein metrics and the properties of the approximate solutions to inverse problems obtained by minimizing loss functions based on such metrics. We contrast the results to those of classical computational inversion with loss functions based on the 𝐿² and 𝐻⁻ metric. We identify critical parameters, both in the metrics and the inverse problems to be solved, that control the performance of the reconstruction algorithms. We highlight the frequency disparity in the reconstructions with the Wasserstein metrics as well as its consequences, for instance, the pre-conditioning effect, the robustness against high-frequency noise, and the loss of resolution when data used contain random noise. We examine the impact of mass unbalance and conduct a comparative study on the differences and important factors of various unbalanced Wasserstein metrics. In the second part of the thesis, we propose loss functions formed on a novel offline-online computational strategy for coupling classical least-square computational inversion with modern deep learning approaches for full waveform inversion (FWI) to achieve advantages that can not be achieved with only one component. In a nutshell, we develop an offline learning strategy to construct a robust approximation to the inverse operator and utilize it to produce a viable initial guess and design a new loss function for the online inversion with a new dataset. We demonstrate through both theoretical analysis and numerical simulations that our neural network induced loss functions developed by the coupling strategy improve the loss landscape as well as computational efficiency of FWI with reliable offline training on moderate computational resources in terms of both the size of the training dataset and the computational cost needed.

Interval analysis is an essential tool in the construction of validated numerical solutions of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential Equations. A validated solution typically consists of guaranteed lower and upper bounds for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an interval function (enclosure) containing all solutions of the problem. IVP for ODE: The central point of discussion is the wrapping effect. A new concept of wrapping function is introduced and applied in studying this effect. It is proved that the wrapping function is the limit of the enclosures produced by any method of certain type (propagate and wrap type). Then, the wrapping effect can be quantified as the difference between the wrapping function and the optimal interval enclosure of the solution set (or some norm of it). The problems with no wrapping effect are characterized as problems for which the wrapping function equals the optimal interval enclosure. A sufficient condition for no wrapping effect is that there exist a linear transformation, preserving the intervals, which reduces the right-hand side of the system of ODE to a quasi-isotone function. This condition is also necessary for linear problems and "near" necessary in the general case. Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for the wave equation is considered. It is proved that under certain conditions the problem is an operator equation with an operator of monotone type. Using the established monotone properties, an interval (validated) method for numerical solution of the problem is proposed. The solution is obtained step by step in the time dimension as a Fourier series of the space variable and a polynomial of the time variable. The numerical implementation involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We propose the combined use of periodic splines and Fourier series for representing discontinuous functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid. / Mathematical Sciences / D. Phil. (Mathematics)

First we consider implicit finite difference schemes on uniform grids in time and space for second order linear stochastic partial differential equations of parabolic type. Under sufficient regularity conditions, we prove the existence of an appropriate asymptotic expansion in powers of the the spatial mesh and hence we apply Richardson's method to accelerate the convergence with respect to the spatial approximation to an arbitrarily high order. Then we extend these results to equations where the parabolicity condition is allowed to degenerate. Finally, we consider implicit finite difference approximations for deterministic linear second order partial differential equations of parabolic type and give sufficient conditions under which the approximations in space and time can be simultaneously accelerated to an arbitrarily high order.

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function <strong>n</strong> ε W^1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state <strong>n</strong> to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV² (Ω,S<sup>2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.

Made available in DSpace on 2015-03-04T18:57:49Z (GMT). No. of bitstreams: 1 danielmoita_tesefinal.pdf: 1307742 bytes, checksum: 19f5820728d061b2b71cf62a0ffd0433 (MD5) Previous issue date: 2011-04-27 / The interaction between large-scale ux fields and precipitation fields in the tropical atmosphere can produce sharp boundaries between dry and humid regions (limited ahead by the rainfall front). As the speed of propagation of disturbances in these regions are di_erent, they form a discontinuity (Days and Pauluis, 2009). This phenomenon can be represented by a system of equations formed by the shallow water equations and the conservation equation of water vapor coupled by a nonlinear source term (Frierson et al., 2004). The aim of this study is to compare the consistency of results and performance of numerical simulation models that use ten di_erent numerical methods for solving these equations, including the finite di_erence methods: Leapfrog, Lax-Wendro_ and Leapfrog with _lters; the upwind method developed by Walcek (2000) and used in Freitas et al. (2011); and the finite volume methods: Godunov, Lax-Wendro_, Minmod, MC, Superbee and Bean-Warming. These models are tested with nonlinear conditions and, at the end, the results show that, albeit more complex and with a higher computational cost in identical simulations, the finite volume method is more appropriate to simulate such a phenomenon because it provides a more accurate solution when dealing with these discontinuities, thereby producing more realistic results than in the other cases. These results may have impact on the design of new models for the operational centers of weather and climate. / A interação entre campos de fluxos de larga escala e campos de precipitação na atmosfera tropical pode apresentar fronteiras abruptas entre regiões secas e regiões úmidas (limitadas pela frente de precipitação). Como a velocidade de propagação de distúrbios nessas regiões é diferente, forma-se uma descontinuidade (Dias e Pauluis, 2009). Tal fenômeno pode ser representado por um sistema de equações formado pelas equações da água rasa e pela equação de conservação do vapor d'água acopladas por um termo fonte não linear (Frierson et al., 2004) . O objetivo do presente trabalho é comparar a consistência dos resultados e o desempenho de modelos de simulação numérica que utilizam dez métodos numéricos diferentes para resolver essas equações, dentre eles os métodos das diferenças finitas: Leapfrog, Lax-Wendroff e Leapfrog com filtros; Upwind desenvolvido por Walcek (2000) e utilizado em Freitas et al. (2011); e os métodos de volumes finitos: Godunov, Lax-Wendroff, Minmod, MC, Superbee e Bean-Warming. Estes modelos são testados com condições não lineares e, ao final, os resultados mostram que o método dos volumes finitos, mesmo sendo mais complexo e ter um custo computacional maior em simulações idênticas, é mais adequado para simular tal fenômeno pois fornece uma solução mais precisa ao lidar com as descontinuidades, gerando, assim, resultados mais realísticos que nos outros casos. Esses resultados podem ter impacto no desenho dos novos modelos para os centros operacionais de previsão de tempo e clima.

Made available in DSpace on 2015-03-04T18:57:54Z (GMT). No. of bitstreams: 1 dilberto.pdf: 3192288 bytes, checksum: c781d5e4d13d5a0028c8c410967fe213 (MD5) Previous issue date: 2009-08-14 / Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior / In this thesis we study models of plane beams governed by Timoshenko s hypothesis and models of curved beams governed by Bresse s hypothesis in the presence of dissipative mechanism, which act partially on the rotation function in the transverse section or on the transverse displacement ones. We realize an analytic study of these models and we show they are exponentially stable, if and only if, the velocities of wave propagations are equal. Such result is more interesting on the point of mathematical view whereas in the practice the velocities of wave propagations are never equal. We study in the general case the polynomial stability property and we show the dissipative systems are stable and, in these situations, the decay rate can be improved according to the regularity of the initial data. In the specific cases of the models of curved beams, the differential factor is in the mathematical techniques we use, which they are much more sophisticated. Finally we realize a numerical study of the dissipative models using semi-discrete and totally discrete models in finite differences, purposing to avoid the problem of shear locking and to we confirm the theoretical results developed here. / Neste trabalho estudamos modelos de vigas planas governados pelas hipóteses de Timoshenko e modelos de vigas curvas governados pelas hipóteses de Bresse, na presença de mecanismos dissipativos atuando parcialmente, quer sobre a função de rotação na seção transversal ou sobre a função de deslocamento transversal. Desenvolvemos um estudo analítico desses modelos e mostramos que eles são exponencialmente estáveis se, e somente se, as velocidades de propagações de ondas são iguais. Este resultado é interessante do ponto de vista matemático, visto que na prática as velocidades de propagações de ondas nunca são iguais. No caso geral, estudamos a propriedade de estabilidade polinomial e mostramos que os sistemas dissipativos são polinomialmente estáveis, com taxas de decaimento que podem ser melhoradas de acordo com a regularidade dos dados iniciais. Nos casos específficos dos modelos de vigas curvas, o fator diferencial reside nas técnicas matemáticas que aplicamos, as quais são muito mais sofisticadas. Finalmente realizamos um estudo numérico dos modelos dissipativos usando modelos semidiscretos e totalmente discretos em diferenças finitas, com a preocupação de se evitar o problema de trancamento no cortante e para comprovarmos os resultados teóricos desenvolvidos nesta tese.

In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.

University of Macau / Faculty of Science and Technology / Department of Mathematics

Studying the zeros of a parameter dependent operator F defined on a Hilbert space H is a fundamental problem in mathematics. When the Hilbert space is finite dimensional, continuation provides, via predictor-corrector algorithms, efficient techniques to numerically follow the zeros of F as we move the parameter. In the case of infinite dimensional Hilbert spaces, this procedure must be applied to some finite dimensional approximation which of course raises the question of validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced zero for the finite dimensional system can be used to explicitly define a set which contains a unique zero for the infinite dimensional problem F: HxR->Im(F). We use this new validated continuation to study equilibrium solutions of partial differential equations, to prove the existence of chaos in ordinary differential equations and to follow branches of periodic solutions of delay differential equations. In the context of partial differential equations, we show that the cost of validated continuation is less than twice the cost of the standard continuation method alone.

Within the last decade, a new type of signal acquisition has emerged called Compressive Sensing that has proven especially useful in providing a recoverable representation of sparse signals. This thesis presents similar results for Compressive Parametric Estimation. Here, signals known to lie on some unknown parameterized subspace may be recovered via randomized compressive measurements, provided the number of compressive measurements is a small factor above the product of the parametric dimension with the subspace dimension with an additional logarithmic term. In addition to potential applications that simplify the acquisition hardware, there is also the potential to reduce the computational burden in other applications, and we explore one such application in depth in this thesis. Source localization by matched-field processing (MFP) generally involves solving a number of computationally intensive partial differential equations. We introduce a technique that mitigates this computational workload by ``compressing'' these computations. Drawing on key concepts from the recently developed field of compressed sensing, we show how a low-dimensional proxy for the Green's function can be constructed by backpropagating a small set of random receiver vectors. Then, the source can be located by performing a number of ``short'' correlations between this proxy and the projection of the recorded acoustic data in the compressed space. Numerical experiments in a Pekeris ocean waveguide are presented which demonstrate that this compressed version of MFP is as effective as traditional MFP even when the compression is significant. The results are particularly promising in the broadband regime where using as few as two random backpropagations per frequency performs almost as well as the traditional broadband MFP, but with the added benefit of generic applicability. That is, the computationally intensive backpropagations may be computed offline independently from the received signals, and may be reused to locate any source within the search grid area. This thesis also introduces a round-robin approach for multi-source localization based on Matched-Field Processing. Each new source location is estimated from the ambiguity function after nulling from the data vector the current source location estimates using a robust projection matrix. This projection matrix effectively minimizes mean-square energy near current source location estimates subject to a rank constraint that prevents excessive interference with sources outside of these neighborhoods. Numerical simulations are presented for multiple sources transmitting through a generic Pekeris ocean waveguide that illustrate the performance of the proposed approach which compares favorably against other previously published approaches. Furthermore, the efficacy with which randomized back-propagations may also be incorporated for computational advantage (as in the case of compressive parametric estimation) is also presented.