No / Three-dimensional (3D) representations of com- plex geometric shapes, especially when they are recon- structed from magnetic resonance imaging (MRI) and com- puted tomography (CT) data, often result in large polygon meshes which require substantial storage for their handling, and normally have only one fixed level of detail (LOD). This can often be an obstacle for efficient data exchange and interactive work with such objects. We propose to re- place such large polygon meshes with a relatively small set of coefficients of the patchwise partial differential equation (PDE) function representation. With this model, the approx- imations of the original shapes can be rendered with any desired resolution at interactive rates. Our approach can di- rectly work with any common 3D reconstruction pipeline, which we demonstrate by applying it to a large reconstructed medical data set with irregular geometry.

Background Oedema, commonly known as tissue swelling, occurs mainly on the leg and the arm. The condition may be associated with a range of causes such as venous diseases, trauma, infection, joint disease and orthopaedic surgery. Oedema is caused by both lymphatic and chronic venous insufficiency, which leads to pooling of blood and fluid in the extremities. This results in swelling, mild redness and scaling of the skin, all of which can culminate in ulceration. Methods We present a method to model a wide variety of geometries of limbs affected by oedema and venous ulcers. The shape modelling is based on the PDE method where a set of boundary curves are extracted from 3D scan data and are utilised as boundary conditions to solve a PDE, which provides the geometry of an affected limb. For this work we utilise a mixture of fourth order and sixth order PDEs, the solutions of which enable us to obtain a good representative shape of the limb and associated ulcers in question. Results A series of examples are discussed demonstrating the capability of the method to produce good representative shapes of limbs by utilising a series of curves extracted from the scan data. In particular we show how the method could be used to model the shape of an arm and a leg with an associated ulcer. Conclusion We show how PDE based shape modelling techniques can be utilised to generate a variety of limb shapes and associated ulcers by means of a series of curves extracted from scan data. We also discuss how the method could be used to manipulate a generic shape of a limb and an associated wound so that the model could be fine-tuned for a particular patient.

We present an efficient method for reconstructing complex geometry using an elliptic Partial Differential Equation (PDE) formulation. The integral part of this work is the use of three-dimensional curves within the physical space which act as boundary conditions to solve the PDE. The chosen PDE is solved explicitly for a given general set of curves representing the original shape and thus making the method very efficient. In order to improve the quality of results for shape representation we utilize an automatic parameterization scheme on the chosen curves. With this formulation we discuss our methodology for shape representation using a series of practical examples.

For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the non-normality of the linearized operator. These new "mostly linear" theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored. The main goal of this work is to begin to study the role of nonlinearity in transition. We use model problems to illustrate that small unmodeled disturbances can cause transition through movement or bifurcation of equilibria. We also demonstrate that small wall roughness can lead to transition by causing the linearized system to become unstable. Sensitivity methods are used to obtain important information about the disturbed problem and to illustrate that it is possible to have a precursor to predict transition. Finally, we apply linear feedback control to the model problems to illustrate the power of feedback to delay transition and even relaminarize fully developed chaotic flows. / Ph. D.

Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases. In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method. An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T. In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique.

<div>In this work, we introduce a framework for constructing light-weight neural network approximations to the solution operators for partial differential equations (PDEs). Using a data-driven offline training procedure, the resulting operator network models are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a neural network. Importantly, the network models can be calibrated to specific distributions of input data in order to reflect properties of real-world data encountered in practice. The networks thus provide specialized solvers tailored to specific use-cases, and while being more restrictive in scope when compared to more generally-applicable numerical methods (e.g. procedures valid for entire function spaces), the operator networks are capable of producing approximations significantly faster as a result of their specialization.</div><div><br></div><div>In addition, the network architectures are designed to place pointwise posterior distributions over the observed solutions; this setup facilitates simultaneous training and uncertainty quantification for the network solutions, allowing the models to provide pointwise uncertainties along with their predictions. An analysis of the predictive uncertainties is presented with experimental evidence establishing the validity of the uncertainty quantification schema for a collection of linear and nonlinear PDE systems. The reliability of the uncertainty estimates is also validated in the context of both in-distribution and out-of-distribution test data.</div><div><br></div><div>The proposed neural network training procedure is assessed using a novel convolutional encoder-decoder model, ConvPDE-UQ, in addition to an existing fully-connected approach, DeepONet. The convolutional framework is shown to provide accurate approximations to PDE solutions on varying domains, but is restricted by assumptions of uniform observation data and homogeneous boundary conditions. The fully-connected DeepONet framework provides a method for handling unstructured observation data and is also shown to provide accurate approximations for PDE systems with inhomogeneous boundary conditions; however, the resulting networks are constrained to a fixed domain due to the unstructured nature of the observation data which they accommodate. These two approaches thus provide complementary frameworks for constructing PDE-based operator networks which facilitate the real-time approximation of solutions to PDE systems for a broad range of target applications.</div>

The purpose of this thesis is to develop a vertex model framework that can be used to perform computational experiments related to the dynamics of epithelial tissues in developmental systems. We focus on three example systems: the Drosophila wing imaginal disc, the Drosophila epidermis and the visceral endoderm of the mouse embryo. Within these systems, key questions pertaining to size-control mechanisms and coordination of cell migration remain unanswered and are amenable to computational testing. The vertex model presented here builds upon existing frameworks in three key ways. Firstly, we include novel force terms, representing, for example, the reaction of a cell to being compressed and its shape becoming distorted during a highly dynamic process such as cell migration. Secondly, we incorporate a model of diffusing morphogenetic growth factors within the vertex framework, using an arbitrary Lagrangian-Eulerian formulation of the diffusion equation and solving with the finite-element method (FEM). Finally, we implement the vertex model on the surface of an ellipsoid, in order to simulate cell migration in the mouse embryo. Throughout this thesis, we validate our model by running simple simulations. We demonstrate convergence properties of the FEM scheme and discuss how the time taken to solve the system scales with tissue size. The model is applied to biological systems and its utility demonstrated in several contexts. We show that when growth is dependent on morphogen concentration in the Drosophila wing disc, proliferation occurs preferentially in regions of high concentration. In the Drosophila epidermis, we show that a recently proposed mechanism of compartment size-control, in which a growth-factor is released in limited amounts, is viable. Finally, we examine the phenomenon of rosettes in the mouse embryo, which occur when five or more cells meet at a common vertex. We show, by running simulations both with and without rosettes, that they are crucial facilitators of ordered migration, and are thus critical in the patterning of the early embryo.

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.

Neste trabalho, tivemos como objetivo estudar a existência de soluções fracas não triviais para o problema elíptico com não linearidade crítica { - &Delta;u = &lambda;u + u2* - 1+ + g(x, u+) + f(x); em &Omega; u = 0; sobre &part; &Omega; , (P) onde &Omega; é um domínio limitado com fronteira suave em &#8477;N, com N &ge; 3, 2* = 2N / (N - 2) é o expoente crítico de Sobolev, u+ = max(u; 0), g &isin; C(&Omega;&#772; x &#8477;, &#8477;+), &lambda; > &lambda;1, &lambda; &notin; &sigma; (- &Delta;) e f &isin; Lr> (&Omega;), com r > N. Com o intuito de observar as mudanças que ocorrem do caso subcrítico para o crítico e as diferentes técnicas variacionais para a resolução de problemas elípticos, estudamos, inicialmente, um problema um pouco mais antigo que (P), que, por sua vez, motivou seu estudo. Tal problema é { - &Delta;u = &lambda; u + up+ +f; em &Omega; u = 0; sobre &part; &Omega;(P\') onde consideramos o caso subcrítico, ou seja, quando p &isin; (1; 2* - 1). Com o auxílio do TEOREMA DE ENLACE verificamos que tanto (P) quanto (P\') têm pelo menos duas soluções fracas não triviais. / In this work, we aimed to study the existence of nontrivial weak solutions for the elliptic problem with critical non-linearity { - &Delta;u = &lambda;u + u2* - 1+ + g(x, u+) + f(x); in &Omega; u = 0; on &part; &Omega; , (P) where &Omega; is a bounded domain with smooth boundary in &#8477;N, with N &ge; 3, 2* = 2N / N -2 is the critical Sobolev exponent, u+ = max(u; 0), g &isin; C(&Omega;&#772; x &#8477;, &#8477;+), &lambda; > &lambda;1, &lambda; &notin; &sigma; (- &Delta;) and f &isin; Lr (&Omega;), with r > N. In order to observe different variational techniques for solving elliptic problems, we studied initially a problem a little older than (P), which, in turn, led to its study. This problem is { - &Delta;u = &lambda; u + up+ +f; in&Omega; u = 0; on &part; &Omega;(P\') where we consider the subcritical case, that is, when p &isin; (1, 2* - 1). With the aid of the LINKING THEOREM we see that both (P) and (P\') have at least two nontrivial weak solutions.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we present the proof of a result due to Lars Hörmander which establishes a necessary condition for a linear operator with variable coefficients is globally resolvable. / Nesse trabalho apresentaremos a demonstração de um resultado devido à Lars Hörmander, que estabelece uma condição necessária para que um operador linear com coeficientes variáveis seja globalmente resolúvel.