Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by A. Neubauer in 1994. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error. Finally, two converse results are proved and the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.Comment: 29 page

Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error. In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathe and Pereverzev in 2003, mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathe and Perevezev are shown. In particular, spectral regularization methods having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur

Dynamics and pattern of a managed coniferous forest landscape in Oregon

We examined the process of fragmentation in a managed forest landscape by comparing rates and patterns of disturbance (primarily clear-cutting) and regrowth between 1972 and 1988 using Landsat imagery. A 2589-km(exp 2) managed forest landscape in western Oregon was classified into two forest types, closed-canopy conifer forest (CF) (typically, greater than 60% conifer cover) and other forest and nonforest types (OT) (typically, less than 40 yr old or deciduous forest). The percentage of CF declined from 71 to 58% between 1972 and 1988. Declines were greatest on private land, least in wilderness, and intermediate in public nonwilderness. High elevations (greater than 914 m) maintained a greater percentage of CF than lower elevations (less than 914 m). The percentage of the area at the edge of the two cover types increased on all ownerships and in both elevational zones, whereas the amount of interior habitat (defined as CF at least 100 m from OT) decreased on all ownerships and elevational zones. By 1988 public lands contained approximately 45% interior habitat while private lands had 12% interior habitat. Mean interior patch area declined from 160 to 62 ha. The annual rate of disturbance (primarily clear-cutting) for the entire area including the wilderness was 1.19%, which corresponds to a cutting rotation of 84 yr. The forest landscape was not in a steady state or regulated condition which is not projected to occur for at least 40 yr under current forest plans. Variability in cutting rates within ownerships was higher on private land than on nonreserve public land. However, despite the use of dispersed cutting patterns on public land, spatial patterns of cutting and remnant forest patches were nonuniform across the entire public ownership. Large remaining patches (less than 5000 ha) of contiguous interior forest were restricted to public lands designated for uses other than timber production such as wilderness areas and research natural areas

Measuring forest landscape patterns in the Cascade Range of Oregon, USA

This paper describes the use of a set of spatial statistics to quantify the landscape pattern caused by the patchwork of clearcuts made over a 15-year period in the western Cascades of Oregon. Fifteen areas were selected at random to represent a diversity of landscape fragmentation patterns. Managed forest stands (patches) were digitized and analyzed to produce both tabular and mapped information describing patch size, shape, abundance and spacing, and matrix characteristics of a given area. In addition, a GIS fragmentation index was developed which was found to be sensitive to patch abundance and to the spatial distribution of patches. Use of the GIS-derived index provides an automated method of determining the level of forest fragmentation and can be used to facilitate spatial analysis of the landscape for later coordination with field and remotely sensed data. A comparison of the spatial statistics calculated for the two years indicates an increase in forest fragmentation as characterized by an increase in mean patch abundance and a decrease in interpatch distance, amount of interior natural forest habitat, and the GIS fragmentation index. Such statistics capable of quantifying patch shape and spatial distribution may prove important in the evaluation of the changing character of interior and edge habitats for wildlife

Regularization Methods for Ill-Posed Problems in Multiple Hilbert Scales

Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, regularization methods on these scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases convergence is proved and orders and optimal orders of convergence are shown.Comment: 32 pages, 2 figure

Associations of the metabolic syndrome and its components with cognitive impairment in older adults

BACKGROUND: The metabolic syndrome (MetS) is an established cardiovascular risk factor. Here, we investigated its role in cognitive impairment. METHODS: Baseline data from 202 participants (aged 65 to 87 years) of the BioCog study were used. All were free of clinical dementia (MMSE≥24/30). Cognitive impairment was defined as the lowest tertile of a cognitive summary score. Multiple logistic regression analyses examined associations of body mass index (BMI), triglycerides (TG), high-density lipoprotein (HDL-C), glucose and glycated hemoglobin A1c (HbA1c) levels with the odds of cognitive impairment. MetS was defined as ≥3 of its 5 components obesity (BMI ≥ 30 kg/m2), elevated TG (TG ≥1.7 mmol/L), reduced HDL-C (males:  0.05). Results for HDL-C were similar when HDL-C, glucose, BMI and TG were entered into a single model (OR 2.56 per 1 mmol/L reduction, 95% CI 1.09, 5.88, p = 0.031) and when cerebrovascular disease and coronary heart disease were additionally controlled for (OR 2.56 per 1 mmol/L reduction, 95% CI 1.06, 6.25, p = 0.036). Among the 5 MetS components, participants with elevated TG were at 2-fold increased odds of impairment (OR 2.09, 95% CI 1.08, 4.05, p = 0.028) including when the remaining 4 MetS components were entered (OR 2.23, 95% CI 1.07, 4.65, p = 0.033), but the finding was no longer statistically significant when cerebrovascular disease and coronary heart disease were additionally controlled for (p = 0.11). Presence of MetS and of obesity, reduced HDL-C, elevated glucose or elevated blood pressure were not significantly associated with impairment (all p > 0.05). CONCLUSION: Our findings support low HDL-C as an independent risk marker of cognitive impairment in older age. The need for research into mediatory and confounding factors, and re-evaluation of traditional cut-off points is highlighted. TRIAL REGISTRATION: The study was registered on 15th October 2014 at clinicaltrials.gov (NCT02265263)

Maternal deaths in Bloemfontein, South Africa -1986 - 1992

Objective. Determination of the maternal mortality ratio and the main causes of maternal death.Setting. Pelonomi Hospital, a tertiary care and referral hospital in Bloemfontein.Methods. Review of prospectively completed structured questionnaires on all maternal deaths from 1986 to 1992.Results. The maternal mortality ratio at our institution was 171 per 100000 live births. Haemorrhage (25%), infection (24%) and hypertensive disease (18%) were the most important causes of death. Seventy-one per cent were direct obstetric deaths and 23% indirect; in the remaining 6%, the cause was uncertain. Of all deaths, 35% were considered preventable.Conclusions. The maternal mortality ratio has decreased since our previous report for the period 1980 - 1985, and haemorrhage has replaced infection as the leading cause of death

A note on the differences of computably enumerable reals

We show that given any non-computable left-c.e. real α there exists a left-c.e. real β such that α≠β+γ for all left-c.e. reals and all right-c.e. reals γ. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Loef random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact

Stability of neuropsychological test performance in older adults serving as normative controls for a study on postoperative cognitive dysfunction

OBJECTIVE: Studies of postoperative cognitive dysfunction (POCD) rely on repeat neuropsychological testing. The stability of the applied instruments, which are affected by natural variability in performance and measurement imprecision, is often unclear. We determined the stability of a neuropsychological test battery using a sample of older adults from the general population. Forty-five participants aged 65 to 89 years performed six computerized and non-computerized neuropsychological tests at baseline and again at 7 day and 3 months follow-up sessions. Mean scores on each test were compared across time points using repeated measures analyses of variance (ANOVA) with pairwise comparison. Two-way mixed effects, absolute agreement analyses of variance intra-class correlation coefficients (ICC) determined test-retest reliability. RESULTS: All tests had moderate to excellent test-retest reliability during 7-day (ICC range 0.63 to 0.94; all p < 0.01) and 3-month intervals (ICC range 0.60 to 0.92; all p < 0.01) though confidence intervals of ICC estimates were large throughout. Practice effects apparent at 7 days eased off by 3 months. No substantial differences between computerized and non-computerized tests were observed. We conclude that the present six-test neuropsychological test battery is appropriate for use in POCD research though small sample size of our study needs to be recognized as a limitation. Trial registration ClinicalTrials.gov Identifier NCT02265263 (15th October 2014)