Preconditioners for Generalized Saddle-Point Problems

Generalized saddle point problems arise in a number of applications, ranging from optimization and metal deformation to fluid flow and PDE-governed optimal control. We focus our discussion on the most general case, making no assumption of symmetry or definiteness in the matrix or its blocks. As these problems are often large and sparse, preconditioners play a critical role in speeding the convergence of Krylov methods for these problems. We first examine two types of preconditioners for these problems, one block-diagonal and one indefinite, and present analyses of the eigenvalue distributions of the preconditioned matrices. We also investigate the use of approximations for the Schur complement matrix in these preconditioners and develop eigenvalue analysis accordingly. Second, we examine new developments in probing methods, inspired by graph coloring methods for sparse Jacobians, for building approximations to Schur complement matrices. We then present an analysis of these techniques and their accuracy. In addition, we provide a mathematical justification for their use in approximating Schur complements and suggest the use of approximate factorization techniques to decrease the computational cost of applying the inverse of the probed matrix. Finally, we consider the effect of our preconditioners on four applications. Two of these applications come from the realm of fluid flow, one using a finite element discretization and the other using a spectral discretization. The third application involves the stress relaxation of aluminum strips at low stress levels. The final application involves mesh parameterization and flattening. For these applications, we present results illustrating the eigenvalue bounds on our preconditioners and demonstrating the theoretical justification of these methods. We also present convergence and timing results, showing the effectiveness of our methods in practice. Specifically the use of probing methods for approximating the Schur compliment matrices in our preconditioners is empirically justified. We also investigate the $h$-dependence of our preconditioners one model fluid problem, and demonstrate empirically that our methods do not suffer from a deterioration in convergence as the problem size increases

Stochastic method for in-situ damage analysis

Based on the physics of stochastic processes we present a new approach for structural health monitoring. We show that the new method allows for an in-situ analysis of the elastic features of a mechanical structure even for realistic excitations with correlated noise as it appears in real-world situations. In particular an experimental set-up of undamaged and damaged beam structures was exposed to a noisy excitation under turbulent wind conditions. The method of reconstructing stochastic equations from measured data has been extended to realistic noisy excitations like those given here. In our analysis the deterministic part is separated from the stochastic dynamics of the system and we show that the slope of the deterministic part, which is linked to mechanical features of the material, changes sensitively with increasing damage. The results are more significant than corresponding changes in eigenfrequencies, as commonly used for structural health monitoring.Comment: This paper is accepted by European Physical Journal B on November 2. 2012. 5 pages, 5 figures, 1 tabl

Chemical composition of aerosols collected over the tropical North Atlantic Ocean

Ambient aerosol samples were collected over the tropical northern Atlantic Ocean during the month of April 1996 onboard the R/V Seward Johnson. Dichotomous high-volume collector samples were analyzed for ferrous iron immediately after collection, while trace metals, anions, and cations were determined upon return to the laboratory. Data are analyzed with the aid of enrichment factor, principal component, and weighted multiple linear regression analyses. Average mineral aerosol concentrations amounted to 19.3±16.4 μg m^(−3) whereby the chemical characteristics and air mass back trajectories indicated the dust to be of a typical shale composition and Saharan origin. Calcite accounted for 3.0 and 7.9% of the mineral aerosol during the first and second halves of the cruise, respectively. Total iron concentrations (averaging 0.84±0.61 μg m^(−3)) are crustally derived, of which 0.51±0.56% is readily released as Fe(II). Eighty-six percent of this Fe(II) is present in the fine (<3 μm diameter) aerosol fraction and correlates with NSS-SO_4^(2−) and oxalate. Approximately 23% of the measured NSS-SO_4^(2−) in both size fractions appears to be biogenically derived, and the rest is of anthropogenic nature. Biogenic SO_4^(2−) /methanesulfonic acid (MSA) ratios could not be easily extracted by employing a multiple linear regression analysis analogous to that of Johansen et al. [1999], possibly due to the varying characteristics of the aerosol chemistry and air temperature during the cruise. Because of the presence of anthropogenic SO_4^(2−), the non-sea-salt (NSS)- SO_4^(2−)/MSA ratio, 37.4±6.4, is elevated over what would be expected if the NSS - SO_4^(2−) were purely biogenic. Cl^− depletion is seen in all samples and averages 18.3±9.1%. The release of Cl from the aerosol phase appears to occur through acid displacement reactions with primarily HNO_3 in the coarse and H_2SO_4 in the fine fraction

Chemical characterization of ambient aerosol collected during the southwest monsoon and intermonsoon seasons over the Arabian Sea: Anions and cations

Ambient aerosol samples were collected over the northern Indian Ocean during two 1 month-long research cruises (German R/V Meteor) that took place during the intermonsoon (May) and SW monsoon (July/August) of 1995. A high volume and two small volume collectors were used to collect samples, which were subsequently analyzed for ferrous iron, 32 elements, and anions and cations. The present paper focuses on the bulk aerosol material, the ions, while utilizing some of the trace metal data that were presented in more detail in our previous paper [Siefert et al., 1999]. Data are analyzed and interpreted with the aid of principal component and multiple linear regression analyses. Intermonsoon samples were strongly influenced by continental material, both of crustal and anthropogenic origin. The crustal component (24.5±13% of the total suspended particulate mass (TSP), 6.0±4.4 μg m^(−3)) contained 3.2% gypsum (CaSO_4). While more than half of the TSP (21.2±9.6 μg m^(−3)) during the SW monsoon was sea-salt-derived due to the strong winds prevailing during this season, only 1.7±1.1% (0.7±0.4 μg m^(−3)) was found to be of crustal origin. Sulfate (SO_4^(2−)) sources were determined and quantified with linear regression analyses utilizing specific tracers for the independent variables. Lead (Pb) was found to be a more reliable surrogate for anthropogenic SO_4^(2−) compared to nitrate (NO_3^−) during the relatively polluted intermonsoon. Soluble calcium (Ca^(2+)) served as the tracer for gypsum, and methane sulfonate (MSA) served as the tracer for biogenically derived SO_4^(2−) during both seasons. On the basis of this analysis, 75% of the non-sea-salt sulfate (NSS-SO_4^(2−)) (0.8±0.2 μg m^(−3), representing ∼2.4% of TSP) was found to be of biogenic origin during the SW monsoon with the remaining 25% of anthropogenic origin. During the intermonsoon, NSS-SO_4^(2−) accounted for 2.1±1.2 μg m^(−3) (∼9.2% of TSP) and had a composition that was 65% anthropogenic, 21% biogenic, and 14% gypsum-derived. Linear regression analyses revealed that the bio-SO_4^(2−)/MSA weight ratios appear to be consistent with the temperature dependence proposed by Hynes et al. [1986]. In this case the yield of SO_4^(2−) increased relative to MSA with an increase in temperature. Three samples during the SW monsoon, near the coast of Oman, showed lower temperatures, due to coastal upwelling, than the rest of the samples; at 24°C the bio-SO_4^(2−)/MSA weight ratio was 6.8±0.5. The remainder of the SW monsoon samples were collected at an average temperature of 27.2°C, for which the bio-SO_4^(2−)/MSA weight ratio was 13.5±4.4. At an average temperature of 28.9°C during the intermonsoon, sampling gave a ratio of 17.7±4.8. These observations indicate a temperature dependence factor between 24° and 29°C of 2.2 (i.e., a 2.2 increase in the ratio of bio-SO_4^(2−)/MSA with every degree temperature increase). Cl− deficits determined during both seasons appear to indicate that different mechanisms may govern the observed depletion of Cl− in each season

Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part II: application to partial differential equations

A template-based generic programming approach was presented in a previous paper that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded analysis algorithms. In this paper, we describe the implementation details for using the template-based generic programming approach for simulation and analysis of partial differential equations (PDEs). We detail several of the hurdles that we have encountered, and some of the software infrastructure developed to overcome them. We end with a demonstration where we present shape optimization and uncertainty quantification results for a 3D PDE application

Model-Assisted Pattern Search

Computer simulations of complex physical phenomena are used in many contexts, including that of engineering design. Increasingly scientists and engineers have also been trying to optimize problems defined by such simulations (e.g. to determine design parameters for a physical product). However, these problems often have several features that hinder the use of standard optimization techniques. The lack of derivative information and numerical error induced by the simulation can cause problems for derivative-based optimization methods. Likewise, extreme computational expense can make the use of direct search methods problematic. The Model-Assisted Pattern Search (MAPS) algorithm, which is the subject of this research, attempts to address the issue. While maintaining a pattern search framework, MAPS makes use of easily constructed surrogates to the objective function in order to speed the optimization process. Numerical results for MAPS and several other algorithms are presented here for a variety of different objective functions