Evaluation of flood risk in response to climate variability

Standard procedures for forecasting flood risk (Bulletin 17B) assume annual maximum flood (AMF) series are stationary, meaning the distribution of flood flows is not significantly affected by climatic trends/cycles, or anthropogenic activities within the watershed. Historical flood events are therefore considered representative of future flood occurrences, and the risk associated with a given flood magnitude is modeled as constant over time. However, in light of increasing evidence to the contrary, this assumption should be reconsidered, especially as the existence of nonstationarity in AMF series can have significant impacts on planning and management of water resources and relevant infrastructure. Research presented in this thesis quantifies the degree of nonstationarity evident in AMF series for unimpaired watersheds throughout the contiguous U.S., identifies meteorological, climatic, and anthropogenic causes of this nonstationarity, and proposes an extension of the Bulletin 17B methodology which yields forecasts of flood risk that reflect climatic influences on flood magnitude. To appropriately forecast flood risk, it is necessary to consider the driving causes of nonstationarity in AMF series. Herein, large-scale climate patterns—including El Niño-Southern Oscillation (ENSO), Pacific Decadal Oscillation (PDO), North Atlantic Oscillation (NAO), and Atlantic Multidecadal Oscillation (AMO)—are identified as influencing factors on flood magnitude at numerous stations across the U.S. Strong relationships between flood magnitude and associated precipitation series were also observed for the majority of sites analyzed in the Upper Midwest and Northeastern regions of the U.S. Although relationships between flood magnitude and associated temperature series are not apparent, results do indicate that temperature is highly correlated with the timing of flood peaks. Despite consideration of watersheds classified as unimpaired, analyses also suggest that identified change-points in AMF series are due to dam construction, and other types of regulation and diversion. Although not explored herein, trends in AMF series are also likely to be partially explained by changes in land use and land cover over time. Results obtained herein suggest that improved forecasts of flood risk may be obtained using a simple modification of the Bulletin 17B framework, wherein the mean and standard deviation of the log-transformed flows are modeled as functions of climate indices associated with oceanic-atmospheric patterns (e.g. AMO, ENSO, NAO, and PDO) with lead times between 3 and 9 months. Herein, one-year ahead forecasts of the mean and standard deviation, and subsequently flood risk, are obtained by applying site specific multivariate regression models, which reflect the phase and intensity of a given climate pattern, as well as possible impacts of coupling of the climate cycles. These forecasts of flood risk are compared with forecasts derived using the existing Bulletin 17B model; large differences in the one-year ahead forecasts are observed in some locations. The increased knowledge of the inherent structure of AMF series and an improved understanding of physical and/or climatic causes of nonstationarity gained from this research should serve as insight for the formulation of a physical-casual based statistical model, incorporating both climatic variations and human impacts, for flood risk over longer planning horizons (e.g., 10-, 50, 100-years) necessary for water resources design, planning, and management.

Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

Variations of the FEAST Eigenvalue Algorithm

FEAST is a recently developed eigenvalue algorithm which computes selected interior eigenvalues of real symmetric matrices. It uses contour integral resolvent based projections. A weakness is that the existing algorithm relies on accurate reasoned estimates of the number of eigenvalues within the contour. Examining the singular values of the projections on moderately-sized, randomly-generated test problems motivates orthogonalization-based improvements to the algorithm. The singular value distributions provide experimentally robust estimates of the number of eigenvalues within the contour. The algorithm is modified to handle both Hermitian and general complex matrices. The original algorithm (based on circular contours and Gauss-Legendre quadrature) is extended to contours and quadrature schemes that are recursively subdividable. A general complex recursive algorithm is implemented on rectangular and diamond contours. The accuracy of different quadrature schemes for various contours is investigated.