Abstract

Coronary heart disease is the most common cause of death in the world and non-invasive medical imaging techniques like positron emission tomography (PET) are important to help diagnose a patient suffering from atherosclerosis as early as possible. Unfortunately, dynamic PET measurements using radioactive water to examine blood flow create challenging image reconstruction and parameter estimation problems.Non-negative matrix factorization (NMF) has been successfully used as a data analysis tool for many different applications. In this thesis, we will motivate the use of NMF through model-based approaches to dynamic PET reconstruction and examine the results and performance of different NMF algorithms when applied to dynamic PET measurements with very poor statistics.

Abstract

The identification of production functions from data is an important task in the modelling of economic growth. In this paper, we consider a non-parametric approach to this identification problem in the context of the spatial Solow model which allows for rather general production functions, in particular convex–concave ones that have recently been proposed as reasonable shapes. We formulate the inverse problem and apply Tikhonov regularization. The inverse problem is discretized by finite elements and solved iteratively via a preconditioned gradient descent approach. Numerical results for the reconstruction of the production function are given and analysed at the end of this paper.

Abstract

The aim of this paper is to discuss potential advances in PET kinetic models anddirect reconstruction of kinetic parameters. As a prominent example we focus on atypical task in perfusion imaging and derive a system of transport-reaction-diffusionequations, which is able to include macroscopic flow properties in addition to theusual exchange between arteries, veins, and tissues.

For this system we propose an inverse problem of estimating all relevant parame-ters from PET data. We interpret the parameter identification as a nonlinear inverseproblem, for which we formulate and analyze variational regularization approaches.For the numerical solution we employ gradient-based methods and appropriate split-ting methods, which are used to investigate some test cases.

Abstract

Positron-Emission Tomography (PET) is an imaging technique in nuclear medicine used to image physiological processes. A major obstacle is the need for dynamic image reconstruction from low quality PET-data, which applies in particular for tracers (radioactive water) with fast decay like H215O when looking for improved spatial resolution.

Here we present a model-based approach to overcome those difficulties. We derive a set of differential equations able to represent the kinetic behavior of H215O PET tracers during cardiac perfusion. In this model one takes into account the exchange of materials between artery, tissue and vein, which predicts the tracer activity if the reaction rates, velocities, and diffusion coefficients are known. One then interprets the computation of these distributed parameters (spatially dependent only) as a nonlinear inverse problem, which we solve using variational regularization approaches. For the minimization we use the gradient-based methods and Forward-Backward Splitting. The main advantage is the reduction of the degrees of freedom, which makes the problem overdetermined and thus allows to proceed to low quality data.

Abstract

In this paper the classical Solow model is extended, by consideringspatial dependence of the physical capital and population dynamics, andby introducing a nonconcave production function. The physical capitaland population evolution equations are governed by semilinear parabolicdifferential equations which describe their evolution over time and space.

The convergence to a steady state according to different hypotheses onthe production function is discussed. The analysis is focused on an S-shaped production function, which allows the existence of saddle pointsand poverty traps. The evolution of this system over time, and its con-vergence to the steady state is described mainly through numerical simulations.

Abstract

In this paper the classical Solow model is extended, by considering spatial dependence of the physical capital and technological progress, and by introducing a nonconcave production function. The physical capital and technological progress accumulation equations are governed by semilinear parabolic differential equations which describe their evolution over time and space. The convergence to a steady state according to different hypotheses on the production function is discussed. The analysis is focused on an S-shaped production function, which allows the existence of saddle points and poverty traps. The evolution of this system over time, and its convergence to the steady state is described mainly through numerical simulations.

Abstract

The Solow model is an economic model to describe economic growth. In this thesis we will provide an extension to this model by introducing a space dimension into the model and a production function that is non-concave. Furthermore, we will consider different kinds of technological progress.

The resulting model is governed by a semilinear parabolic partial differential equation which describes the evolution of capital in time. We will consider the solution of the partial differential equation as our direct problem and show its well-posedness. Furthermore, we will present the numerical solution of this problem and examine the influence of the respective parameters on the model.

We will then try to identify the production function in a situation where noisy data of the capital distribution over time is attainable. We will formulate the corresponding inverse problem, introduce a method to solve this inverse problem numerically and present some numerical results.