Parameter Estimation via Conditional Expectation --- A Bayesian Inversion

When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp.\ functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by actual observations of the response of the real system. In a probabilistic setting, Bayes's theory is the proper mathematical background for this identification process. The possibility of being able to compute a conditional expectation turns out to be crucial for this purpose. We show how this theoretical background can be used in an actual numerical procedure, and shortly discuss various numerical approximations

Intra-uterine fetal demise caused by amniotic band syndrome after standard amniocentesis

The amniotic band syndrome represents a prime example of exogenous disruption of an otherwise normal feta I development. It may be a sequel of invasive diagnostic procedures such as amniocentesis or fetal blood sampling. A 38-year-old gravida II, para II delivered a morphologically normal male stillborn at term. The pregnancy history had been unremarkable but for an early 2nd-trimester amniocentesis. Cause of the intra-uterine fetal demise was noted to be an amniotic band constricting the umbilical cord, An amniotic band is a rare but potentially fatal condition which may be induced by, e.g., invasive prenatal procedures. Such bands are not usually diagnosed prenatally; however, selected patients with augmented risk may profit from intensive ultrasound evaluation including Doppler studies. Copyright (C) 2000 S. Karger AG, Basel

Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format

We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with significantly extended numerical experiments. Some unused material is remove

Aspects of the community structure and reproductive ecology of the common sea-urchin Parechinus angulosus Leske around the Cape Peninsula, South Africa

The geographic distributionof Parechinus angulosus according to present II records extend from Umhlali (Zululand) to Luderitz Bay (Day, 1974). The urchins are therefore adapted to a wide range of temperatures. Maximum fluctuations probably occur along the stretches of coastline affected by upwelling, the South-West, and west coasts, with differentials of twelve degrees Centigrade and more. Furthermore, these fluctuations may take place within a matter of a few hours or even less, i.e., almost capriciously Like temperature changes encountered on land but, unlike, the latter, by virtue of movement of different water masses, rather than cooling rate. The importance of temperature as a regulator of reproductive cycles in marine organisms, specifica'ly invertebrates, is discussed in Giese and Pearse's (1974) detailed review on the subject. A local study by Newman (1969) considers the effect of temperature on the spawning behaviour of abalone (Haliotis midae). This study was commenced in the light of this information and subjective preliminary observations underwater, which revealed striking density and relative size differences between urchins on the Atlantic side of the Peninsula and in the warmer False Bay. A detailed in situ record of sea temperatures at two study sites, Robben Island and Millers Point was obtained by installing temperature measuring devices. This monitoring covered a period of one year, while coupled with monthly collections of urchins in the mature size range. The variations of male and female gonad indices were found to be essentially in agreement, though lagging behind sea temperature changes. The spawning periods of urchins at both study sites was similar, occurring in spring and autumn, despite the significantly different water temperature averages. A clear pattern was observed at Millers Point, while being poorly defined at Robben Island. This is in agreement with the findings of Newman (1967, 1969), whose observations included data from St. Helena Bay to beyond Port Elizabeth. Newman found higher spawning intensities at localities having a clear seasonal pattern of sea temperature variations, compared to those of low seasonality, in other words, better synchrony of gamete release. A possible mechanism was mentioned by Giese & Pearse (1974), suggesting that the ripest individuals may be induced to spawn by a sharp temperature change, thereby stimulating other less mature individuals to follow suit. The degree of perturbation necessary to initiate such a sequence can vary widely. Many species are known to have breeding seasons in areas where temperature fluctuations are slight, such as parts of the polar, tropical, and abyssal ocean. My observations indicate that a confused temperature regime, involving rapid changes, such as occur during periodic upwelling cycles, results in repeated, incomplete spawn-outs. Consequently, gonad indices are always lower at Robben Island tha·n at Millers Point. Urchins at both localities show a low reproductive status during winter. Stained histological thin sections were prepared of gonad portions of selected urchins of both sexes covering all _maturity. (Gonad index) stages and compared with lipid values obtained for the same respective animals. It was found that during maturation, there is a gradual build-up of gametes passing through the well-known developmental stages, rather than a snap conversion of accumulated storage material (I only tested for lipid for practical reasons) into sperms or eggs. Even relatively regressed {low gonad index) urchins could be induced to release fertilizable spawn, by injection of potassium chloride solution. Lipid values form a surprisingly small portion of the total dry mass of the urchin, when it is realised that it is the gonads which function as the major storage organ in addition to the . intestine (Boolotian, 1966). Except for a few high, possibly spurious, values, male 1 ipid levels were on the whole lower than those found for females. It would have been interesting to have values for glycogen as well for comparison, particularly since calorific values obtained for representative male and female Parechinus by micro-bomb calorimetry were found to be in very close agreement, in fact practically identical. The significant mean size differences between urchins of the 2 study sites could be shown to be very 1 ikely due to the availability of food. This was shown by a comparison of gut states of urchins collected during the period of study. Although it is true that suspended food material is more plentiful in an upwelling locality, like Robben Island and unlike Millers Point, this situation is only important to the planktonic and recently settled juvenile stages of Parechinus. Once the animal has reached a test diameter of 10 mm (Greenwood, 1980), macroscopic and attached food sources A~~ required. In this respect Robben Island is clearly limiting. Hence urchins of a smaller mean size relative to Millers Point enter the reproductive life eyelet, since only that fractfon over and above the animal 1 s metabolic requirements is available for growth and reproduction. The significantly lower (one quarter) gamete output of urchins from Robben Island is evidence of this fact. Recruitment of Parechinus into the benthonic stage appears to be strongly dependent on substrate, in that the eroded, slaty structure of the reefs at Robben Island afford many suitable microhabitats to juveniles at this particularly vulnerable stage, while the rounded granite boulder terrain at Millers Point does not offer a comparable number and choice of such sites. Tegner and Dayton (1977) showed that substrate type may play an important role in the recruitmaitof a Californian species, Strongylocentrotus purpura~us, which like Parechinus is flexible in its recruitment behaviour, which means that the urchins select suitable sites from a range of choices, while - S. franciscanus juveniles shelter under the spine canopy of adults. The size class study sites must be seen in this light. In effect then we have a situation, where substrate type and availability jointly select against the smaller size classes at Millers Point. The strongly contagious distribution of Pa:l'echinus is very likely a response to the proximity of food. Since debris tends to accumulate on horizontal surfaces, urchins can be expected to be more numerous there than on inclined substrate. This was found to be true in a study by Vel imi rov et al. (1977), who made the dividing line between horizontal and vertical at 45° inclination. In addition, minor adjustments according to sea strate take place. It was found that under turbulent conditions urchins shelter on rockfaces away from I the sea and under overhangs to avoid strong swell action and contact with loose material such as torn off kelp plants. Predation was not studied and therefore only tentative impressions gained per chance in the field are at hand. Robbeneiland lying in a security area is largely undisturbed and harbours many rock lobsters (Jasus lalandii), which prey on Pa:l'echinus in addition to their main food, the ribbed mussel (Aulacomya ater). Indeed, lobsters were seen to feed on urchins on a number of occasions, apparentl_y selecting the smaller individuals. rare for many years through overexploitation. At Millers Point Jasus has been There appears to be no significant departure from the 1 :1 sex·ratio at either site according to Chi-square analysis. However, a more powerful analysis of variance and co-variance indicated a higher number of males at Robbeneiland. The discrepancy may be due to the fact that while Chi-square analysis was done on small (30 batches of urchins collected on a monthly basis and comprising mainly sexually matureindividuals at various stages of gonadal development, the Anova involved material obtained during seasonal transects including a wide range of size classes, particularly at Robbeneiland . Indeterminate urchins were excluded from the analysis, but nevertheless, misidentifications favouring males did occur. It would therefore appear that there is little reason to assume that the sex ratio of Parechinus is significantly unsymmetrical

To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case

In parametric equations - stochastic equations are a special case - one may want to approximate the solution such that it is easy to evaluate its dependence of the parameters. Interpolation in the parameters is an obvious possibility, in this context often labeled as a collocation method. In the frequent situation where one has a "solver" for the equation for a given parameter value - this may be a software component or a program - it is evident that this can independently solve for the parameter values to be interpolated. Such uncoupled methods which allow the use of the original solver are classed as "non-intrusive". By extension, all other methods which produce some kind of coupled system are often - in our view prematurely - classed as "intrusive". We show for simple Galerkin formulations of the parametric problem - which generally produce coupled systems - how one may compute the approximation in a non-intusive way