Rescaling Technique Research Articles

A non-linear mixed effect dynamic model using adherence to treatment to describe long-term therapy outcome in HIV-patients

Aim Poor drug-adherence is an important factor explaining the resurgence of HIV-1 virus. Complex non-linear models have been developed to describe the population dynamics of HIV virus, but they are not used in clinical trials due to their complexity. Linearized models have been applied to real data. However, they can only explain the decay of the virus following antiviral treatment. The objective of our project is to develop a population non-linear mixed effect model characterizing the long-term dynamics of viral load in clinical data, and to quantify the effect of adherence in the dynamic of HIV virus. Methods The basic model incorporates physiologically meaningful variables (free virus, total T-cells, and latent T-cells), uses standard rescaling techniques to guarantee identifiability of its parameters given measurements of the free virus (viral load), and takes into account intra-subject variability. Drug-adherence is incorporated on the basic reproductive ratio of the virus (RR) as follows: RR=λ+γ*Aj(t), where λ is the RR for non compliers (Aj(t)=0) and λ+γ, γ≤ 0, is the RR for perfect compliers (Aj(t)=1). The model is applied on real AIDS clinical data. Results We show that adherence affects the RR. Perfect adherence decreases RR by 3% resulting in an important reduction of RNA levels. Conclusions The model may be used to draw biologically relevant interpretations from repeated HIV-1 RNA measurements and quantify the relative effect of drug-adherence on viral response in HIV-infected patients. Clinical Pharmacology & Therapeutics (2005) 77, P90–P90; doi: 10.1016/j.clpt.2004.12.238

Expanding Lorenz Attractors through Resonant Double Homoclinic Loops

In this paper we study the existence of Lorenz attractors in the unfolding of resonant double homoclinic loops in dimension three. Our results generalize the ones obtained in [C. Robinson, SIAM J. Math. Anal., 32 (2000), pp. 119--141] in two ways. First, we obtain attractors instead of weak attractors obtained there. Second, we enlarge considerably the region in the parameter space corresponding to flows presenting expanding Lorenz attractors. The proof is based on rescaling techniques [J. Palis and F. Takens, Hyperbolicity and Sensitive Choatic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors, Cambridge University Press, Cambridge, UK, 1993] to obtain convergence to noncontinuous maps.

Mathematical modeling of the movement of suspended particles subjected to acoustic and flow fields

When particles are subjected to an acoustic field particle trajectories depend on the particle and fluid compressibility and density values. Hence a combination of acoustic and flow fields on particles can be used to deflect and trap, or to segregate and/or fractionate fine particles in fluid suspensions. Using particle physics in an acoustic field, a mathematical model was developed to calculate trajectories of deflected particles due to the application of acoustic standing waves. The resulting second order ordinary differential equation was quite stiff and hence difficult to solve numerically and did not have a closed form solution. The analysis of the above equation showed that the basic problems with numerical solutions could not be ameliorated through the use of standard rescaling techniques. A combination of phase space and asymptotic analysis turns out to be far more useful in obtaining approximate solutions. An approximate solution was derived which enabled the calculation of the particle trajectories and concentration at collection planes in the acoustic field. Analysis of the solution showed that all the particles move toward the pressure node to which the particles are supposed to move. Particles with 2 μm diameter took approximately 20 s to reach that node. Then at the bench scale, the above technology was implemented by building a flow chamber with two transducers at opposite ends to generate an acoustic standing wave. SiC particle trajectories were tracked using captured digital images from a high-resolution microscope. The displacements of SiC particles due to an acoustic force were compared with the mathematical model predictions. For input power levels between 3.0 and 5.0 W, the experimental data were comparable to mathematical model predictions. Hence it was concluded that the proposed approximate solution was both quantitatively and qualitatively closer to experimental results than the simplified form ignoring the second order term reported in the literature.

Robust stabilization of uncertain high order systems via smooth output feedback

We investigate the robust stabilization problem for a family of uncertain nonlinear systems with uncontrollable/unobservable linearization. To achieve global robust stabilization using a single output feedback controller, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system. Using this rescaling technique combined with the non-separation principle based design methods (Qian and Lin, 2002a; Yang and Lin, 2003), we develop a robust output feedback control scheme for uncertain nonlinear systems satisfying a homogeneous growth condition. Both a smooth state feedback controller and a homogeneous observer are designed for the rescaled system using only the knowledge of the bounding system rather than the uncertain system itself.

Assessment of inflow boundary conditions for compressible turbulent boundary layers

A description of different inflow methodologies for turbulent boundary layers, including validity and limitations, is presented. We show that the use of genuine periodic boundary conditions, in which no alteration of the governing equations is made, results in growing mean flow and decaying turbulence. Premises under which the usage is valid are presented and explained, and comparisons with the extended temporal approach [T. Maeder, N. A. Adams, and L. Kleiser, “Direct simulation of turbulent supersonic boundary layers by an extended temporal approach,” J. Fluid Mech. 429, 187 (2001)] are used to assess the validity. Extending the work by Lund et al. [J. Comput. Phys. 140, 233 (1998)], we propose an inflow generation method for spatial simulations of compressible turbulent boundary layers. The method generates inflow by reintroducing a rescaled downstream flow field to the inlet of a computational domain. The rescaling is based on Morkovin’s hypothesis [P. Bradshaw, “Compressible turbulent shear layers,” Annu. Rev. Fluid Mech. 9, 33 (1977)] and generalized temperature–velocity relationships. This method is different from other existing rescaling techniques [S. Stolz and N. A. Adams, “Large-eddy simulation of high-Reynolds-number supersonic boundary layers using the approximate deconvolution model and a rescaling and recycling technique,” Phys. Fluids 15, 2398 (2003); G. Urbin and D. Knight, “Large-eddy simulation of a supersonic boundary layer using an unstructured grid,” AIAA J. 39, 1288 (2001)], in that a more consistent rescaling is employed for the mean and fluctuating thermodynamic variables. The results are compared against the well established van Driest II theory and indicate that the method is efficient and accurate.

Scaling, asymmetry and a Fokker–Planck model of the fast and slow solar wind as seen by WIND

The solar wind plasma is a natural laboratory for studies of plasma turbulence. Long, evenly sampled satellite data sets are natural candidates for statistical studies and these are often performed in the context of magnetohydrodynamic (MHD) turbulence. In this paper, scaling properties of solar wind bulk plasma parameters are discussed. Low order probability density function (PDF) asymmetry analysis is applied to data that combines slow and fast solar wind. Results are compared to those obtained using a PDF rescaling technique. A break in scaling, identified by the rescaling method, is confirmed to occur at a temporal scale of ∼26 hours. Low asymmetry levels of the fluctuations PDF are identified for the quantities that also exhibit self-similar statistics. A generalized structure function analysis is then applied to the kinetic energy density obtained from slow and fast solar wind streams. The applicability of a Fokker–Planck model for slow and fast wind ρv2 fluctuations is investigated.

Large-eddy simulation of high-Reynolds-number supersonic boundary layers using the approximate deconvolution model and a rescaling and recycling technique

Spatially developing supersonic turbulent boundary layers at a Mach number of 2.5 and momentum-thickness Reynolds numbers at inflow of 4530 and 10 049, respectively, are computed with large-eddy simulation using the approximate deconvolution model [Stolz et al., Phys. Fluids 13, 2985 (2001)]. Turbulent inflow conditions are generated by rescaling the turbulent boundary layer at some distance downstream of inflow and reintroducing the rescaled mean profiles and fluctuation fields at inflow. This technique follows essentially that of Lund et al. [J. Comput. Phys. 140, 233 (1998)], albeit simplified and adapted for compressible flow. The simulations feature rather short spatial transient behavior and the results agree well with experimental data and theoretical correlations. The validity of assumptions of the strong Reynolds analogy is addressed.

Mode I weight functions for an orthotropic double cantilever beam

Approximate weight functions are proposed and validated numerically for an orthotropic double cantilever beam (DCB) loaded in mode I. They define the stress intensity factor at the crack tip due to a pair of point forces acting on the crack surfaces and have been deduced from the corresponding isotropic result using an orthotropy rescaling technique. The weight functions allow mode I large scale bridging problems in beams and plates to be formulated as integral equations, in terms of stress intensity factors at the crack tip, without the limitations imposed on accuracy by beam theory approximations. The proposed functions are applied to investigate the influence of the orthotropy of the material on the fracture behavior of DCBs in the presence of large scale bridging.

Large Eddy Simulation of a Compressible Boundary Layer

AbstractA large eddy simulation of a compressible boundary layer is performed. To generate an appropriate inflow distribution the rescaling technique for compressible flows is discribed. In this method Morkovin's hypothesis in which the total temperature fluctuations are neglected compared with the static temperature fluctuations is applied to rescale and generate the temperature profile at inlet. This new technique is used for various large eddy simulations of subsonic and supersonic three‐dimensional boundary layers of a flat plate. Simulation results for the time‐averaged mean flow and Reynolds stresses are compared with numerical and analytical data to demonstrate the high quality of the method.

Nonlinear rescaling vs. smoothing technique in convex optimization

We introduce an alternative to the smoothing technique approach for constrained optimization. As it turns out for any given smoothing function there exists a modification with particular properties. We use the modification for Nonlinear Rescaling (NR) the constraints of a given constrained optimization problem into an equivalent set of constraints. The constraints transformation is scaled by a vector of positive parameters. The Lagrangian for the equivalent problems is to the correspondent Smoothing Penalty functions as Augmented Lagrangian to the Classical Penalty function or MBFs to the Barrier Functions. Moreover the Lagrangians for the equivalent problems combine the best properties of Quadratic and Nonquadratic Augmented Lagrangians and at the same time are free from their main drawbacks. Sequential unconstrained minimization of the Lagrangian for the equivalent problem in primal space followed by both Lagrange multipliers and scaling parameters update leads to a new class of NR multipliers methods, which are equivalent to the Interior Quadratic Prox methods for the dual problem. We proved convergence and estimate the rate of convergence of the NR multipliers method under very mild assumptions on the input data. We also estimate the rate of convergence under various assumptions on the input data. In particular, under the standard second order optimality conditions the NR method converges with Q- linear rate without unbounded increase of the scaling parameters, which correspond to the active constraints. We also established global quadratic convergence of the NR methods for Linear Programming with unique dual solution. We provide numerical results, which strongly support the theory.

Vortex breakdown in turbulent pipe flows

In many technical applications turbulent flows with embedded slender vortices exist. Depending on the boundary conditions vortex breakdown can occur. The purpose of this work is to develop and implement a solution scheme for large-eddy simulations of vortex breakdown in turbulent pipe flows. One of the main problems in this simulation is the formulation of the inflow boundary condition for a fully developed turbulent flow with an embedded vortex. For that purpose a rescaling technique is developed in which a solution at a downstream location is inserted at the inflow boundary after an appropriate rescaling. To determine rescaling laws for pipe flows with an embedded vortex, analytical velocity profiles of swirling flows are first prescribed in a laminar flow. From the spatial development of the vortex a scaling law is deduced. In a next step this procedure is to be transferred to turbulent flows.

Uniqueness of the Infinite Open Cluster for High-density Percolation on Lattice Sierpinski Carpet

We prove the uniqueness of infinite open cluster for high-density bond percolation on lattice Sierpinski Carpet; forthermore, an alternative proof of the existence of phase transition of the model is given. A rescaling technique is developed and used as the main tool of our proofs.

Hölder continuity of local weak solutions for parabolic equations exhibiting two degeneracies

We consider equations of the form $${\partial}_{t} v - \mbox{div} ( \alpha (v) \nabla v) = 0 \ , $$ where $v \in [0,1]$ and $\alpha (v)$ degenerates for $v=0$ and $v=1$. We show that local weak solutions are locally Hölder continuous provided $\alpha$ behaves like a power near the two degeneracies. We adopt the technique of intrinsic rescaling developed by DiBenedetto.

Noise characterization of block-iterative reconstruction algorithms: I. Theory.

Researchers have shown increasing interest in block-iterative image reconstruction algorithms due to the computational and modeling advantages they provide. Although their convergence properties have been well documented, little is known about how they behave in the presence of noise. In this work, we fully characterize the ensemble statistical properties of the rescaled block-iterative expectation-maximization (RBI-EM) reconstruction algorithm and the rescaled block-iterative simultaneous multiplicative algebraic reconstruction technique (RBI-SMART). Also included in the analysis are the special cases of RBI-EM, maximum-likelihood EM (ML-EM) and ordered-subset EM (OS-EM), and the special case of RBI-SMART, SMART. A theoretical formulation strategy similar to that previously outlined for ML-EM is followed for the RBI methods. The theoretical formulations in this paper rely on one approximation, namely, that the noise in the reconstructed image is small compared to the mean image. In a second paper, the approximation will be justified through Monte Carlo simulations covering a range of noise levels, iteration points, and subset orderings. The ensemble statistical parameters could then be used to evaluate objective measures of image quality.

Homoclinic bifurcations of endomorphisms: The codimension one case

In this note, we consider the dynamic that appears when we unfold a quadratic degenerate homoclinic point of a generic one-parameter family of endomorphisms f μ . This is done through a rescaling technique. Among other facts, it follows from our theorem the abundance of strange expanding sets.

Multimagnon excitations in alternating spin/bond chains

The nature of multimagnon excitations in alternating spin/bond ferromagnetic chains are studied using a combination of scaling methods and the recursion method. Two-magnon excitations for a Heisenberg chain with either alternating spin magnitudes, $S$ and ${S}^{\ensuremath{'}},$ or alternating nearest-neighbor couplings, ${J}_{1}$ and ${J}_{2}$ can be studied using real-space rescaling techniques. These results are then used to investigate three-magnon excitations using the recursion method.

Analysis of hydraulic components using computational fluid dynamics models

This paper presents some results obtained during the computational fluid dynamics (CFD) analysis of internal flows inside a hydraulic component, using a scaling technique applied to numerical pre- and post-processing. The main aim of the work is to demonstrate the reduction of computational work needed for a complete analysis of component behaviour over a wide range of operating conditions. This result is achieved through the adoption of a methodology aimed at giving the highest level of generality to a non-dimensional solution, thereby overcoming the two major limitations encountered in the use of CFD in fluid power design: computer resources and time. In the case study, the technique was applied to a hydraulic distributor and computations were performed with a commercial computational fluid dynamics code. The key factor of this technique is the evaluation, for a given distributor opening, of the Reynolds number of the flow in the metering region. Provided that this number is high enough to ensure that the discharge coefficient has reached its asymptotic value, the characterization of the flow by a single non-dimensional numerical run can be shown. The theoretical contents of the analysis of the re-scaling technique, which focuses on the engineering information necessary in component design, are described in detail. The bases for its subsequent application to actual cases are then outlined. Finally, a fairly close correlation between numerical results and experimental data is presented.

Recovery of optical cross-section perturbations in dense-scattering media by transport-theory-based imaging operators and steady-state simulated data

We present a useful strategy for imaging perturbations of the macroscopic absorption cross section of dense-scattering media using steady-state light sources. A perturbation model based on transport theory is derived, and the inverse problem is simplified to a system of linear equations, WΔμ = ΔR, where W is the weight matrix, Δμ is a vector of the unknown perturbations, and ΔR is the vector of detector readings. Monte Carlo simulations compute the photon flux across the surfaces of phantoms containing simple or complex inhomogeneities. Calculation of the weight matrix is also based on the results of Monte Carlo simulations. Three reconstruction algorithms-conjugate gradient descent, projection onto convex sets, and the simultaneous algebraic reconstruction technique, with or without imposed positivity constraints-are used for image reconstruction. A rescaling technique that improves the conditioning of the weight matrix is also developed. Results show that the analysis of time-independent data by a perturbation model is capable of resolving the internal structure of a dense-scattering medium. Imposition of positivity constraints improves image quality at the cost of a reduced convergence rate. Use of the rescaling technique increases the initial rate of convergence, resulting in accurate images in a smaller number of iterations.

ASYMPTOTIC ANALYSIS OF A MODEL KINETIC EQUATION

For a simple model of a linear kinetic equation the exact solution is expanded in terms of a small parameter whose presence makes the equation, singularly perturbed. Various asymptotic expansion methods are analyzed and it is shown that the compressed method, which is related to the Chapman-Enskog asymptotic procedure, is the most accurate. This holds when the technique of time rescaling is applied to overcome the difficulties with the application of the standard asymptotic procedure.

Delamination of beams under transverse shear and bending

Two delamination problems are studied: a straight beam of rectilinearly orthotropic material under transverse loading and a curved beam (or C-specimen) with cylindrical material anisotropy under pure bending. Results are presented for each problem for the dependence on the crack length of both the energy release rate and a measure of the relative amount of mode II to mode I. The number of nondimensional parameters in the problems is fairly large, including two dimensionless elasticity parameters. Considerable simplification of the parametric dependence in the straight beam problem is achieved using an orthotropic rescaling technique due to Suo. Asymptotic solutions based on elastic beam theory are also presented. A by-product of the present analysis is the finding that frictionless contact of the crack faces near one of the crack tips has little effect on the other tip, even in the case when the size of the contact zone is large relative to the length of the crack.