Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Ecologie benthique de la Lagune Ebrié. Théorie sur la méthode du DR' pour la mise en évidence de biocoenoses

The theoretical aspect of the DR' method was presented. It suggestes the association of some characteristic transects, the organisms of which will be used to describe the first biocoenoses of the Ebrié lagoon.

Biologie de Ethmalosa fimbriata (Bowdich) en Côte d'Ivoire. 2 - Étude de la croissance en lagune par la méthode de Petersen

Ethmalosa growth curves (calculated by the least squares method) were determined from weekly samplings in Ebrié Lagoon. In order to obtain more accurate results than with a modal decomposition, the author used directly the modal values of the samples. One-year-old ethmalosa is about 15 cm long (fork length). For older fish, growth data seem to be disturbed by migrations: fish measuring >25 cm do not appear in the lagoon. Ethmalosa would spend the first year of its life in the lagoon, where it hatches and reproduces, and would migrate to the sea during its second year.

Identification of Sex and Female’s Reproductive Stage in Commercial Fish Species through the Quantification of Ribosomal Transcripts in Gonads

The estimation of maturity and sex of fish stocks in European waters is a requirement of the EU Data Collection Framework as part of the policy to improve fisheries management. On the other hand, research on fish biology is increasingly focused in molecular approaches, researchers needing correct identification of fish sex and reproductive stage without necessarily having in house the histological know-how necessary for the task. Taking advantage of the differential gene transcription occurring during fish sex differentiation and gametogenesis, the utility of 5S ribosomal RNA (5S rRNA) and General transcription factor IIIA (gtf3a) in the molecular identification of sex and gametogenic stage was tested in different economically-relevant fish species from the Bay of Biscay. Gonads of 9 fish species (, Atlantic, Atlantic-chub and horse mackerel, blue whiting, bogue, European anchovy, hake and pilchard and megrim), collected from local commercial fishing vessels were histologically sexed and 5S and 18S rRNA concentrations were quantified by capillary electrophoresis to calculate a 5S/18S rRNA index. Degenerate primers permitted cloning and sequencing of gtf3a fragments in 7 of the studied species. 5S rRNA and gtf3a transcript levels, together with 5S/18S rRNA index, distinguished clearly ovaries from testis in all of the studied species. The values were always higher in females than in males. 5S/18S rRNA index values in females were always highest when fish were captured in early phases of ovary development whilst, in later vitellogenic stages, the values decreased significantly. In megrim and European anchovy, where gonads in different oogenesis stages were obtained, the 5S/18S rRNA index identified clearly gametogenic stage. This approach, to the sexing and the quantitative non-subjective identification of the maturity stage of female fish, could have multiple applications in the study of fish stock dynamics, fish reproduction and fecundity and fish biology in general.

Quantification of drag and lift imposed by pop-up satellite archival tags and estimation of the metabolic cost to cownose rays (Rhinoptera bonasus)

The recent development of the pop-up satellite archival tag (PSAT) has allowed the collection of information on a tagged animal, such as geolocation, pressure (depth), and ambient water temperature. The success of early studies, where PSATs were used on pelagic fishes, has spurred increasing interest in the use of these tags on a large variety of species and age groups. However, some species and age groups may not be suitable candidates for carrying a PSAT because of the relatively large size of the tag and the consequent energy cost to the study animal. We examined potential energetic costs to carrying a tag for the cownose ray (Rhinoptera bonasus). Two forces act on an animal tagged with a PSAT: lift from the PSATs buoyancy and drag as the tag is moved through the water column. In a freshwater flume, a spring scale measured the total force exerted by a PSAT at flume velocities from 0.00 to 0.60 m/s. By measuring the angle of deflection of the PSAT at each velocity, we separated total force into its constituent forces — lift and drag. The power required to carry a PSAT horizontally through the water was then calculated from the drag force and velocity. Using published metabolic rates, we calculated the power for a ray of a given size to swim at a specified velocity (i.e., its swimming power). For each velocity, the power required to carry a PSAT was compared to the swimming power expressed as a percentage, %TAX (Tag Altered eXertion). A %TAX greater than 5% was felt to be energetically significant. Our analysis indicated that a ray larger than 14.8 kg can carry a PSAT without exceeding this criterion. This method of estimating swimming power can be applied to other species and would allow a researcher to decide the suitability of a given study animal for tagging with a PSAT.

Quantification and reduction of unobserved mortality rates for snow, southern Tanner, and red king crabs (Chionoecetes opilio, C. bairdi, and Paralithodes camtschaticus) after encounters with trawls on the seafloor

Unobserved mortalities of nontarget species are among the most troubling and difficult issues associated with fishing, especially when those species are targeted by other fisheries. Of such concern are mortalities of crab species of the Bering Sea, which are exposed to bottom trawling from groundfish fisheries. Uncertainty in the management of these fisheries has been exacerbated by unknown mortality rates for crabs struck by trawls. In this study, the mortality rates for 3 species of commercially important crabs—red king crab, (Paralithodes camtschaticus), snow crab (Chionoecetes opilio) and southern Tanner crab (C. bairdi)—that encounter different components of bottom trawls were estimated through capture of crabs behind the bottom trawl and by evaluation of immediate and delayed mortalities. We used a reflex action mortality predictor to predict delayed mortalities. Estimated mortality rates varied by species and by the part of the trawl gear encountered. Red king crab were more vulnerable than snow or southern Tanner crabs. Crabs were more likely to die after encountering the footrope than the sweeps of the trawl, and higher death rates were noted for the side sections of the footrope than for the center footrope section. Mortality rates were ≤16%, except for red king crab that passed under the trawl wings (32%). Herding devices (sweeps) can expand greatly the area of seafloor from which flatfishes are captured, and they subject crabs in that additional area to lower (4–9%) mortality rates. Raising sweep cables off of the seafloor reduced red king crab mortality rates from 10% to 4%.