Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

Downscaling from GCM precipitation: a benchmark for dynamical and statistical downscaling methods

A precipitation downscaling method is presented using precipitation from a general circulation model (GCM) as predictor. The method extends a previous method from monthly to daily temporal resolution. The simplest form of the method corrects for biases in wet-day frequency and intensity. A more sophisticated variant also takes account of flow-dependent biases in the GCM. The method is flexible and simple to implement. It is proposed here as a correction of GCM output for applications where sophisticated methods are not available, or as a benchmark for the evaluation of other downscaling methods. Applied to output from reanalyses (ECMWF, NCEP) in the region of the European Alps, the method is capable of reducing large biases in the precipitation frequency distribution, even for high quantiles. The two variants exhibit similar performances, but the ideal choice of method can depend on the GCM/reanalysis and it is recommended to test the methods in each case. Limitations of the method are found in small areas with unresolved topographic detail that influence higher-order statistics (e.g. high quantiles). When used as benchmark for three regional climate models (RCMs), the corrected reanalysis and the RCMs perform similarly in many regions, but the added value of the latter is evident for high quantiles in some small regions.

Short term hydroelectric scheduling combining network flow and interior point approaches

In this paper, short term hydroelectric scheduling is formulated as a network flow optimization model and solved by interior point methods. The primal-dual and predictor-corrector versions of such interior point methods are developed and the resulting matrix structure is explored. This structure leads to very fast iterations since it avoids computation and factorization of impedance matrices. For each time interval, the linear algebra reduces to the solution of two linear systems, either to the number of buses or to the number of independent loops. Either matrix is invariant and can be factored off-line. As a consequence of such matrix manipulations, a linear system which changes at each iteration has to be solved, although its size is reduced to the number of generating units and is not a function of time intervals. These methods were applied to IEEE and Brazilian power systems, and numerical results were obtained using a MATLAB implementation. Both interior point methods proved to be robust and achieved fast convergence for all instances tested. (C) 2004 Elsevier Ltd. All rights reserved.

Kinetic energy conserving integrators for Gaussian thermostatted SLLOD

A new integration scheme is developed for nonequilibrium molecular dynamics simulations where the temperature is constrained by a Gaussian thermostat. The utility of the scheme is demonstrated by its application to the SLLOD algorithm which is the standard nonequilibrium molecular dynamics algorithm for studying shear flow. Unlike conventional integrators, the new integrators are constructed using operator-splitting techniques to ensure stability and that little or no drift in the kinetic energy occurs. Moreover, they require minimum computer memory and are straightforward to program. Numerical experiments show that the efficiency and stability of the new integrators compare favorably with conventional integrators such as the Runge-Kutta and Gear predictor-corrector methods. (C) 1999 American Institute of Physics. [S0021-9606(99)50125-6].

Toward bacterial protein sub-cellular location prediction:single-class discrimminant models for all gram- and gram+ compartments

Based on Bayesian Networks, methods were created that address protein sequence-based bacterial subcellular location prediction. Distinct predictive algorithms for the eight bacterial subcellular locations were created. Several variant methods were explored. These variations included differences in the number of residues considered within the query sequence - which ranged from the N-terminal 10 residues to the whole sequence - and residue representation - which took the form of amino acid composition, percentage amino acid composition, or normalised amino acid composition. The accuracies of the best performing networks were then compared to PSORTB. All individual location methods outperform PSORTB except for the Gram+ cytoplasmic protein predictor, for which accuracies were essentially equal, and for outer membrane protein prediction, where PSORTB outperforms the binary predictor. The method described here is an important new approach to method development for subcellular location prediction. It is also a new, potentially valuable tool for candidate subunit vaccine selection.

INVESTIGATIONS INTO MECHANISMS UNDERLYING EXTREME WAVE FORMATIONS AND COMPUTATIONALLY INTENSIVE SIMULATIONS

Various mechanisms have been proposed to explain extreme waves or rogue waves in an oceanic environment including directional focusing, dispersive focusing, wave-current interaction, and nonlinear modulational instability. The Benjamin-Feir instability (nonlinear modulational instability), however, is considered to be one of the primary mechanisms for rogue-wave occurrence. The nonlinear Schrodinger equation is a well-established approximate model based on the same assumptions as required for the derivation of the Benjamin-Feir theory. Solutions of the nonlinear Schrodinger equation, including new rogue-wave type solutions are presented in the author's dissertation work. The solutions are obtained by using a predictive eigenvalue map based predictor-corrector procedure developed by the author. Features of the predictive map are explored and the influences of certain parameter variations are investigated. The solutions are rescaled to match the length scales of waves generated in a wave tank. Based on the information provided by the map and the details of physical scaling, a framework is developed that can serve as a basis for experimental investigations into a variety of extreme waves as well localizations in wave fields. To derive further fundamental insights into the complexity of extreme wave conditions, Smoothed Particle Hydrodynamics (SPH) simulations are carried out on an advanced Graphic Processing Unit (GPU) based parallel computational platform. Free surface gravity wave simulations have successfully characterized water-wave dispersion in the SPH model while demonstrating extreme energy focusing and wave growth in both linear and nonlinear regimes. A virtual wave tank is simulated wherein wave motions can be excited from either side. Focusing of several wave trains and isolated waves has been simulated. With properly chosen parameters, dispersion effects are observed causing a chirped wave train to focus and exhibit growth. By using the insights derived from the study of the nonlinear Schrodinger equation, modulational instability or self-focusing has been induced in a numerical wave tank and studied through several numerical simulations. Due to the inherent dissipative nature of SPH models, simulating persistent progressive waves can be problematic. This issue has been addressed and an observation-based solution has been provided. The efficacy of SPH in modeling wave focusing can be critical to further our understanding and predicting extreme wave phenomena through simulations. A deeper understanding of the mechanisms underlying extreme energy localization phenomena can help facilitate energy harnessing and serve as a basis to predict and mitigate the impact of energy focusing.

Fractional control with a smith predictor

The advantageous use of fractional calculus (FC) in the modeling and control of many dynamical systems has been recognized. In this paper, we study the control of a heat diffusion system based on the application of the FC concepts. Several algorithms are investigated and compared, when integrated within a Smith predictor control structure. Simulations are presented assessing the performance of the proposed fractional algorithms.

Solving Fractional Diffusion-Wave Equations Using a New Iterative Method

Mathematics Subject Classification: 26A33, 31B10

Integral Transforms Method to Solve a Time-Space Fractional Diffusion Equation

Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.

Numerical Solution of Fractional Diffusion-Wave Equation with two Space Variables by Matrix Method

Mathematics Subject Classi¯cation 2010: 26A33, 65D25, 65M06, 65Z05.

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method.

ANNA: A new prediction method for bioassessment programs

1. Cluster analysis of reference sites with similar biota is the initial step in creating River Invertebrate Prediction and Classification System (RIVPACS) and similar river bioassessment models such as Australian River Assessment System (AUSRIVAS). This paper describes and tests an alternative prediction method, Assessment by Nearest Neighbour Analysis (ANNA), based on the same philosophy as RIVPACS and AUSRIVAS but without the grouping step that some people view as artificial. 2. The steps in creating ANNA models are: (i) weighting the predictor variables using a multivariate approach analogous to principal axis correlations, (ii) calculating the weighted Euclidian distance from a test site to the reference sites based on the environmental predictors, (iii) predicting the faunal composition based on the nearest reference sites and (iv) calculating an observed/expected (O/E) analogous to RIVPACS/AUSRIVAS. 3. The paper compares AUSRIVAS and ANNA models on 17 datasets representing a variety of habitats and seasons. First, it examines each model's regressions for Observed versus Expected number of taxa, including the r(2), intercept and slope. Second, the two models' assessments of 79 test sites in New Zealand are compared. Third, the models are compared on test and presumed reference sites along a known trace metal gradient. Fourth, ANNA models are evaluated for western Australia, a geographically distinct region of Australia. The comparisons demonstrate that ANNA and AUSRIVAS are generally equivalent in performance, although ANNA turns out to be potentially more robust for the O versus E regressions and is potentially more accurate on the trace metal gradient sites. 4. The ANNA method is recommended for use in bioassessment of rivers, at least for corroborating the results of the well established AUSRIVAS- and RIVPACS-type models, if not to replace them.

FGF-23 as a Predictor of Renal Outcome in Diabetic Nephropathy

Background and objectives Fibroblast growth factor 23 (FGF-23) has emerged as a new factor in mineral metabolism in chronic kidney disease (CKD). An important regulator of phosphorus homeostasis, FGF-23 has been shown to independently predict CKD progression in nondiabetic renal disease. We analyzed the relation between FGF-23 and renal outcome in diabetic nephropathy (DN). Design, setting, participants, & measurements DN patients participating in a clinical trial (enalapril+placebo versus enalapril+losartan) had baseline data collected and were followed until June 2009 or until the primary outcome was reached. Four patients were lost to follow-up. The composite primary outcome was defined as death, doubling of serum creatinine, and/or dialysis need. Results At baseline, serum FGF-23 showed a significant association with serum creatinine, intact parathyroid hormone, proteirturia, urinary fractional excretion of phosphate, male sex, and race. Interestingly, FGF-23 was not related to calcium, phosphorus, 25OH-vitamin D, or 24-hour urinary phosphorus. Mean follow-up time was 30.7 +/- 10 months. Cox regression showed that FGF-23 was an independent predictor of the primary outcome, even after adjustment for creatinine clearance and intact parathyroid hormone (10 pg/ml FGF-23 increase = hazard ratio, 1.09; 95% CI, 1.01 to 1.16, P = 0.02). Finally, Kaplan-Meier analysis showed a significantly higher risk of the primary outcome in patients with FGF-23 values of >70 pg/ml. Conclusions FGF-23 is a significant independent predictor of renal outcome in patients with macroalbuminuric DN. Further studies should clarify whether this relation is causal and whether FGF-23 should be a new therapeutic target for CKD prevention. Clin J Am Soc Nephrol 6: 241-247, 2011. doi: 10.2215/CJN.04250510

Increased Immunoglobulin G Anti-Paracoccidioides brasiliensis Serum Antibody Avidity as a Predictor of Favorable Posttherapeutic Evolution in Paracoccidioidomycosis

Paracoccidioidomycosis is endemic in Latin America, and ca. 80% of all cases occur in Brazil. Little is known about antibody avidity or the evolution of such avidity in the posttherapeutic period for the different clinical presentations of the disease. In the present study, we evaluated 53 patients with paracoccidioidomycosis and calculated the avidity index. Medium-and high-avidity antibodies were found in 79.5% of patients with chronic presentation (n = 39). Among patients with the acute form (n = 14), 57.1% of the antibodies presented low avidity. In the posttherapeutic period, there was a significant increase in antibody avidity in patients presenting with the chronic multifocal form. In our preliminary study, which needs to be confirmed using a larger number of samples, the optimized method for studying antibody avidity detected differences among the clinical presentations of the mycosis and indicated the value of the avidity index as a marker of posttherapeutic evolution of patients with a multifocal chronic form of the disease.