La magnetoencephalographie (meg) et l'electroencephalographie (eeg) possèdent une resolution temporelle exceptionnelle qui les destine naturellement a l'observation et au suivi des processus electrophysiologiques sous-jacents. Cependant, il n'existe pas a ce jour de méthode d'exploitation des signaux meg et eeg qui puisse les faire prétendre au statut de veritables méthodes d'imageries. En effet, la modélisation de la production des champs magnétiques et des différences de potentiels électriques recueillis sur le champ necessite a priori la prise en compte de la géométrie complexe de la tete et des proprietes de conductivite des tissus. Enfin, l'estimation des générateurs est un probleme qui fondamentalement ne possede pas de solution unique. La motivation initiale de notre travail a concerne le developpement d'approches permettant d'obtenir une tomographie corticale de l'electrophysiologie. Nous avons alors mis en uvre une modelisation markovienne du champ d'intensite des sources en proposant des modeles spatio-temporels adaptes a la meeg, et notamment aux variations morphologiques locales des structures anatomiques corticales. De plus, nous avons exploite cette notion d'ajustement local afin de proposer une nouvelle méthode de fusion de données meg et eeg au sein d'un seul et unique problème inverse. Nous avons également accorde une importance particulière a l'évaluation des méthodes proposées. Ainsi, et pour aller au-dela des simulations numériques souvent trop limitatives, nous avons mis au point un fantôme physique adapte a la meg et a l'eeg qui nous a permis d'etudier les performances des estimateurs en association avec des modeles de tete a divers degres de realisme. Enfin, nous proposons une première application a ces méthodes dans le cadre de l'identification de réseaux épileptiques chez les patients souffrant d'épilepsie partielle.

On étudie, dans cette thèse, la diffusion d'ondes dans différents milieux nonlinéaires possédant une bande interdite naturelle. On montre, en particulier, l'existence d'un comportement de bistabilité dans les milieux régis, soit par l'équation de sine-Gordon (chaîne de pendules courte, réseaux de jonctions Josephson, double couches à effet Hall quantique), soit par l'équation de Schrödinger nonlinéaire (milieu Kerr et milieu de Bragg), dans les cas discrets et continus. Ces différents milieux sont soumis à des conditions aux bords périodiques, dont la fréquence est prise dans la bande interdite et avec une amplitude déterminant l'état de stabilité du système. En effet, pour une amplitude suffisante (supratransmission), le milieu n'est plus réfléchissant et absorbe de l'énergie, faisant passer le signal de sortie d'un état d'amplitude évanescente vers un état de très grande amplitude. On donne, par ailleurs, une description analytique complète de la bistabilité qui permet de comprendre les différents états stationnaires observés dans ces milieux et de prédire le passage d'un état à un autre.

More than 16 years post-apartheid, South Africa is still regarded as the most unequal society in the world. The government is facing various obstacles and challenges in improving the standard of living and quality of life for all its citizens, for example in facilitating the access to clean drinking water and sanitation, building houses and providing basic education. In addition, the country is facing the world’s largest HIV/AIDS epidemic with a national prevalence rate of 18.1 %, equalling approximately 5.7 million people who are currently infected. (Pressly, 2009; UNAIDS, 2008c) Against this background, the aim of this thesis was to study the Madwaleni community, situated in a deeply rural area of the former apartheid homeland Transkei. Applying the Community Oriented Primary Care approach, a strategy of ‘community assessment and diagnosis’ was used to obtain a holistic community profile and to determine the perceptions of its community members regarding their health and social problems and needs, intending to make recommendations to health care providers working at Madwaleni Hospital regarding future health education and disease prevention programmes. (Brown and Fee, 2002) This research used a cross-sectional design. In a preliminary survey, qualitative data was collected in short interviews with health care providers working at Madwaleni Hospital (N=46). The information served as a basis to develop and design parts of the Madwaleni community survey questionnaire. The questionnaire consisted of 36 questions, complying with the aim and objectives of this thesis. It was used for the structured interviews with the main study population, all of whom were members of the Madwaleni community (N=200), whereas half of the main study population were men and half were women, then again, half were unaware of their HIV status and half were HIV+ and had joined the Madwaleni HIV/AIDS programme. Key findings 1) Madwaleni community profile and characteristics Thoughtful sexual behaviour: Particularly interesting in light of the HIV/AIDS epidemic, more than 90 % of the sexually active community members were monogamous at the time of the survey. While only 36.4 % of the men and women unaware of their HIV status used condoms, 76.5 % of the HIV+ community members claimed to do so, indicating that the Madwaleni HIV wellness programme and especially its counselling and health education components are adequate and valuable in serving their purpose. High rates of illiteracy and insufficient education: Only 56.5 % of the interviewed community members were ‘functionally literate’ at the time of the survey. Of those, only 8 % had received a matriculation and not one of the community members had received any higher degree. In addition, 19.5 % of the sampled men and women were not able to read at all. High rates of unemployment, poverty and dependency on welfare grants: Only 20 % of the Madwaleni community members were employed at the time of the survey. Taking the daily income per capita as a reference, one third of the community members suffered from ‘moderate poverty’, defined as an income of 1 to 2 US $ per day, while the other two thirds suffered from ‘extreme poverty’, defined as an income of less than 1 US $ per day, although more than 90 % of the corresponding households received at least one type of welfare grant already. Large household sizes and predominance of traditional dwellings: In the Madwaleni community, an average of eight people lived together per household at the time of the survey, whereas 95 % of the community members lived in traditional dwellings, constructed from freely occurring natural resources. In need of safe drinking water, sanitary systems and access to electricity: More than 80 % of the Madwaleni community members obtained their drinking water from rivers or stagnant dams, while only 6.5 % used rain water and 9.5 % had access to piped water. In addition, almost 70 % of the community members had no access to any sanitary systems, using nearby bushes instead. Furthermore, more than 90 % had no access to electricity. The majority used paraffin for cooking, candles for lighting and wood for heating their homes. Small-scale cultivation to provide an extra source of food: In the Madwaleni area, 90 % of the families owned a small garden patch attached to their houses, used for small-scale cultivation. In addition, almost 90 % owned livestock, mainly poultry, cattle and goats. Crops and animals were used to provide an extra source of food; however, not one of the households could solely live on subsistence farming. Difficulties in accessing health care facilities: On average, each of the community members needed three-quarters of an hour to access their closest clinic and almost one and a half hours to reach Madwaleni Hospital, with 40 % and 60 % respectively depending on public taxi transport to get there. No substantial improvement of the living circumstances since apartheid: Comparing the Madwaleni community characteristics with corresponding data from apartheid-times, no substantial improvement of the living circumstances and conditions could be noticed, proving that governmental and non-governmental actions, programmes and services have not yet reached all remote communities. Similar community characteristics in the neighbouring communities: Comparing these characteristics with corresponding features of communities in the immediate or surrounding areas, namely Cwebe, Ntubeni, Mboya, Shixini and Zithulele, various similarities could be detected, indicating that the living circumstances and conditions might be generalisable to a certain degree, at least to deeply rural communities in the former Transkei area. More disadvantaged than the general South African population: The Madwaleni community differed significantly from the general South African population in 75 % of the compared characteristics. For example, amongst the community members the illiteracy rate (21.7 % vs. 13.6 %, p = 0.002) and unemployment rate (80.5 % vs. 25.5 %, p < 0.001) were significantly higher. In addition, the ‘poverty headcount ratio of 2 US $ per day’ showed that significantly more people were suffering from poverty in the Madwaleni area (92.2 % vs. 34 %, p < 0.001). The Madwaleni community members were less likely to have access to clean drinking water, along with significantly higher proportions of them using river water as their main source of drinking water (75.5 % vs. 5.1 %, p < 0.001). Also, they were less likely to have access to any sanitation or toilet facilities (31.3 % vs. 91.8 %, p < 0.001) or to electricity (8.5 % vs. 80.2 %, p < 0.001). 2) Weightiest health and social problems as experienced by the Madwaleni community In the Madwaleni area, the three health problems with the highest impact on the community were TB, HIV/AIDS and hypertension. On the basis of the applied 3-to-0-point rating matrix, they were rated by more than 95 % of the community members as being relevant problems, with mean values of 2.33, 2.30 and 2.14 respectively. Interestingly, women rated HIV/AIDS higher than men. Musculoskeletal problems and headache were additional health problems with relevant impact on the Madwaleni community, rated by more than 90 %, with mean values above 1.80. While pain and discomfort experienced by PLWHA have been recognised and researched before, there are no corresponding studies on rural communities and further research is necessary to identify the contributing factors. Additional relevant health problems: Interestingly, six health problems were rated higher by HIV untested than by HIV+ community members, namely bilharzia/ schistosomiasis, epilepsy, Herpes Zoster, HIV/AIDS, lung infections and stroke. Since the HIV+ men and women were educated about and screened for all of those diseases within the Madwaleni HIV/AIDS programme, this might explain the deviating rating patterns between the different sub-samples. Moreover, these results demonstrate that health education and disease prevention programmes are able to reduce the perceived burden of health problems and might therefore serve as a substantial argument in their favour. Interestingly, for the Madwaleni community, social matters had a higher impact on their lives than health problems, whereas the three social problems with the highest impact on the community were alcohol abuse, dependency on social grants and smoking. They were rated by more than 98 % of the community members as being relevant problems, with mean values of 2.75, 2.73 and 2.72 respectively. In accordance with these findings, employment & lack of work opportunities, education & illiteracy, food supply and poverty were additional social problems with relevant impact in the Madwaleni area, rated by more than 90 %, with mean values above 2.00. 3) Recommendations for future health education and disease prevention programmes At the time of the survey, the three most relevant health education and disease prevention topics for the Madwaleni community were HIV/AIDS, TB and healthy nutrition. They were rated by more than 95 % of the community members as being relevant health education problems, with mean values of 2.65, 2.51 and 2.36 respectively. In addition, STIs, alcohol & drug-related problems, water & sanitation and body & muscle pain were rated as the subsequent issues of relevance, with mean values above 2.00, supporting the identified community characteristics as well as the listing of the weightiest health and social problems. In addition, valuable new insight could be gained. For instance, HIV untested men rated the topic HIV/AIDS lower than all other community members, which is particularly interesting since men only constitute a minority of 20 % of the people testing for HIV in the Madwaleni area. Besides, topics not previously considered, such as injury prevention and basic first aid, were in-fact relevant for more than 85 % of the community members and require further attention. Furthermore, deviating rating patterns between men and women and the corresponding need for gender-specific educational workshops became evident, for example, for men about prostate & testicular cancer check-up or erectile dysfunction and for women about breast & cervical cancer check-up & papsmears or nutrition & growth. In addition, HIV+ community members rated depression & stress and psychiatric diseases higher than HIV untested men and women, with further studies required to identify the underlying reasons for these deviating rating patterns. Taking all findings from this Madwaleni community survey into consideration, health care providers working at the hospital and its peripheral clinics should first and foremost concentrate their efforts on maintaining the existing programmes, particularly, the Madwaleni HIV/ARV programme and the workshops on hypertension and diabetes mellitus. In addition, if qualified and motivated personnel can be recruited and the necessary funding can be raised, future health education and disease prevention programmes should focus on TB, alcohol & substance abuse-related problems as well as water & sanitation.

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Tikhonov-type regularization of linear and nonlinear ill-posed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the l1-norm is used as penalty term, the l1-regularization was studied by numerous authors although the non-reflexivity of the Banach space l1 and the fact that such penalty functional is not strictly convex lead to serious difficulties. We consider the case that the sparsity assumption is narrowly missed. This means that the solutions may have an infinite number of nonzero but fast decaying components. For that case we formulate and prove convergence rates results for the l1-regularization of nonlinear operator equations. In this context, we outline the situations of Hölder rates and of an exponential decay of the solution components.

We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.

The focus of this paper is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.