Linear Approximation (linear + approximation)

Distribution by Scientific Domains

Kinds of Linear Approximation

  • piecewise linear approximation


  • Selected Abstracts


    Pyrene and chrysene fate in surface soil and sand microcosms

    ENVIRONMENTAL TOXICOLOGY & CHEMISTRY, Issue 2 2001
    J. Chadwick Roper
    Abstract Polycyclic aromatic hydrocarbons (PAHs) are major components of wastes from municipal gas plants and many wood preservatives. Soil contaminated with these wastes is a potential threat to human health because of the carcinogenicity of many PAHs. This study follows the fate of two four-ring PAHs, pyrene and chrysene, in three matrices: an adapted soil (obtained from a site contaminated with PAHs for more than 75 years), an uncontaminated soil (with and without an inoculum of adapted soil), and sand mixed with an inoculum of adapted soil. Radiolabeled pyrene, chrysene, and salicylic acid (a metabolite of PAH biodegradation) were used to trace the mineralization, transformation, extractability, and formation of an unextractable residual over time. Linear approximations of the rates of these processes were made. High-performance liquid chromatography (HPLC) analysis of extracts from inoculated soil showed the transient formation of two known metabolites: 1-hydroxypyrene (from pyrene) and 1-hydroxy-2-naphthoic acid (from chrysene). The amount of extractable label diminished steadily over the course of the study in systems that were not inhibited with sodium azide, whereas the amount of extractable label remained relatively constant in inhibited systems. Correspondingly, the amount of nonextractable residual label generally increased during each incubation in uninhibited systems, whereas the amount of this residual label remained relatively constant in inhibited systems. In contrast, the rate and extent of mineralization varied widely across matrix types. This suggests that alterations of the PAH that impact extractability and residual formation are common, in contrast to mineralization, which was apparently limited to adapted communities. [source]


    Dynamic systems with high damping rubber: Nonlinear behaviour and linear approximation

    EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 13 2008
    Andrea Dall'Asta
    Abstract High damping rubber (HDR) shows a quite complex constitutive behaviour, which is nonlinear with respect to strain and is dependent on the strain rate. In addition, it exhibits a transient response during which the material properties change (scragging or more generally the Mullins effect). A number of recent works were dedicated to analysing and modelling material behaviour. This paper studies the nonlinear dynamics of systems with restoring force produced by HDR-based devices in order to propose a procedure to define equivalent linear models considering both transient and stationary behaviours. The reliability of these linear models is tested by evaluating the upper and lower bounds of the seismic response of a structural system equipped with HDR-based devices (structural system with dissipative bracings and isolated systems). Copyright © 2008 John Wiley & Sons, Ltd. [source]


    Effects of toroidal HVDC ground electrode on earth-return circuits

    EUROPEAN TRANSACTIONS ON ELECTRICAL POWER, Issue 5 2002
    W. Machczy
    This paper presents a method of evaluation of currents and potentials, excited by conductive effects of high-voltage direct-current (HVDC) transmission system, along two parallel earth-return circuits such as pipelines and cables buried in the vicinity of a toroidal ground electrode. It is assumed that the system considered is linear and that the earth is an isotropic, homogeneous medium of finite conductivity. Conductive coupling between earth-return circuits is taken into account, whereas the reaction of the conductors' currents on the electrode current is disregarded. The transmission line model of the conductor with earth-return, a segmental linear approximation of the curve of the primary earth potential distribution along the conductor and the concept of superposition have been used in the method. It should be pointed out, that the method does not require the time consuming numerical integration. The technical applications of the method are illustrated by examples. [source]


    A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2004
    T. Chen
    Abstract This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second-order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least-squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure-stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure-computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free-surface problems. Copyright © 2004 John Wiley & Sons, Ltd. [source]


    An approach to the linguistic summarization of time series using a fuzzy quantifier driven aggregation

    INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 5 2010
    Janusz Kacprzyk
    We extend our previous work on the linguistic summarization of time series data meant as the linguistic summarization of trends, i.e. consecutive parts of the time series, which may be viewed as exhibiting a uniform behavior under an assumed (degree of) granulation, and identified with straight line segments of a piecewise linear approximation of the time series. We characterize the trends by the dynamics of change, duration, and variability. A linguistic summary of a time series is then viewed to be related to a linguistic quantifier driven aggregation of trends. We primarily employ for this purpose the classic Zadeh's calculus of linguistically quantified propositions, which is presumably the most straightforward and intuitively appealing, using the classic minimum operation and mentioning other t -norms. We also outline the use of the Sugeno and Choquet integrals proposed in our previous papers. We show an application to the absolute performance type analysis of time series data on daily quotations of an investment fund over an 8-year period, by presenting first an analysis of characteristic features of quotations, under various (degrees of) granulations assumed, and then by listing some more interesting and useful summaries obtained. We propose a convenient presentation of linguistic summaries focused on some characteristic feature exemplified by what happens "almost always," "very often," "quite often," "almost never," etc. All these analyses are meant to provide means to support a human user to make decisions. © 2010 Wiley Periodicals, Inc. [source]


    Neutron imaging with bent perfect crystals.

    JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 3 2001

    Neutron imaging with bent crystals is considered in the linear approximation of neutron optics. A matrix formalism analogous to that of conventional lens optics is developed. Various imaging conditions are discussed in relation to the crystal deformation type and to possible applications in neutron scattering. All focusing conditions known previously are included as particular cases. Arrangements combining high resolution in imaging with high resolution in scattering are examined. The concept of Bragg mirrors as devices for non-dispersive imaging is introduced. The most powerful application appears to be that of Bragg mirrors combined with the time-of-flight method. Imaging with a thick packet of silicon wafers at the spatial resolution of a single thin wafer is demonstrated. [source]


    Integrated multi-echelon supply chain design with inventories under uncertainty: MINLP models, computational strategies

    AICHE JOURNAL, Issue 2 2010
    Fengqi You
    Abstract We address in this article a problem that is of significance to the chemical industry, namely, the optimal design of a multi-echelon supply chain and the associated inventory systems in the presence of uncertain customer demands. By using the guaranteed service approach to model the multi-echelon stochastic inventory system, we develop an optimization model to simultaneously determine the transportation, inventory, and network structure of a multi-echelon supply chain. The model is an MINLP with a nonconvex objective function including bilinear, trilinear, and square root terms. By exploiting the properties of the basic model, we reformulate this problem as a separable concave minimization program. A spatial decomposition algorithm based on the integration of Lagrangean relaxation and piecewise linear approximation is proposed to obtain near global optimal solutions with reasonable computational expense. Examples for specialty chemicals and industrial gas supply chains with up to 15 plants, 100 potential distribution centers, and 200 markets are presented. © 2009 American Institute of Chemical Engineers AIChE J, 2010 [source]


    Consideration of the linear concentration increase of the unbound drug fraction in plasma

    JOURNAL OF PHARMACEUTICAL SCIENCES, Issue 2 2009
    Leonid M. Berezhkovskiy
    Abstract The concentration of drug in plasma may not necessarily be much less than that of the drug binding proteins, as often considered. Thus the unbound drug fraction could be concentration dependent, which should be taken into account in the interpretation of drug pharmacokinetics and modeling. It is shown that the increase of the unbound drug fraction, fu, can be very accurately considered proportional to the drug plasma concentration for a relatively wide range of concentrations. Equations for the calculation of fu in this linear range are obtained, as well the limiting drug concentration when the linear approximation of fu is applicable. The suggested approach greatly simplifies the calculation of fu and can be ready used in pharmacokinetic calculations and PK-PD models, as well as for the prediction of the change of fu due to the variation of protein concentrations in plasma. Naproxen protein binding in human plasma is considered as an illustration of the method. © 2008 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 98:383,393, 2009 [source]


    Amplitude,shape approximation as an extension of separation of variables

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2008
    N. Parumasur
    Abstract Separation of variables is a well-known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude,shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well-established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi-implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude,shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Limitations of a linear model for the hurricane boundary layer

    THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 641 2009
    Stefanie Vogl
    Abstract The linear model for the steady boundary layer of a rapidly rotating axisymmetric vortex is derived from a detailed scale analysis of the full equations of motion. The previously known analytic solution is re-appraised for vortices of hurricane scale and strength. The internal consistency of the linear approximation is investigated for such a vortex by calculating from the solution the magnitude of the nonlinear terms that are neglected in the approximation compared with the terms retained. It is shown that the nonlinear terms are not negligibly small in a large region of the vortex, a feature that is consistent with the scale analysis. We argue that the boundary-layer problem is well-posed only at outer radii where there is subsidence into the layer. At inner radii, where there is ascent, only the radial pressure gradient may be prescribed and not the wind components at the top of the boundary layer, but the linear problem cannot be solved in these circumstances. We examine the radius at which the vertical flow at the top of the boundary layer changes sign for different tangential wind profiles relevant to hurricanes and show that this is several hundred kilometres from the vortex centre. This feature represents a further limitation of the linear model applied to hurricanes. While the present analysis assumes axial symmetry, the same limitations presumably apply to non-axisymmetric extensions to the linear model. Copyright © 2009 Royal Meteorological Society [source]


    State waypoint approach to continuous-time nonlinear optimal control problems

    ASIAN JOURNAL OF CONTROL, Issue 6 2009
    Mohamadhadi Honarvarmahjoobin
    Abstract In this paper, we propose an optimal control technique for a class of continuous-time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit-form suboptimal solution, by means of linear approximation. Thus the original continuous-time nonlinear control problem reduces to a finite-dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior-reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Discriminant Analysis for Longitudinal Data with Multiple Continuous Responses and Possibly Missing Data

    BIOMETRICS, Issue 1 2009
    Guillermo Marshall
    Summary Multiple outcomes are often used to properly characterize an effect of interest. This article discusses model-based statistical methods for the classification of units into one of two or more groups where, for each unit, repeated measurements over time are obtained on each outcome. We relate the observed outcomes using multivariate nonlinear mixed-effects models to describe evolutions in different groups. Due to its flexibility, the random-effects approach for the joint modeling of multiple outcomes can be used to estimate population parameters for a discriminant model that classifies units into distinct predefined groups or populations. Parameter estimation is done via the expectation-maximization algorithm with a linear approximation step. We conduct a simulation study that sheds light on the effect that the linear approximation has on classification results. We present an example using data from a study in 161 pregnant women in Santiago, Chile, where the main interest is to predict normal versus abnormal pregnancy outcomes. [source]


    Normal form representation of control systems

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 5 2002
    Daizhan Cheng
    Abstract This paper is to investigate the normal form representation of control systems. First, as numerical tools we develop an algorithm for normal form expression and the matrix representation of the Lie derivative of a linear vector field over homogeneous vector fields. The concept of normal form is modified. Necessary and sufficient conditions for a linear transformation to maintain the Brunowsky canonical form are obtained. It is then shown that the shift term can always be linearized up to any degree. Based on this fact, linearization procedure is proposed and the related algorithms are presented. Least square linear approximations are proposed for non-linearizable systems. Finally, the method is applied to the ball and beam example. The efforts are focused on the numerical and computer realization of linearization process. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Life table response experiment analysis of the stochastic growth rate

    JOURNAL OF ECOLOGY, Issue 2 2010
    Hal Caswell
    Summary 1.,Life table response experiment (LTRE) analyses decompose treatment effects on a dependent variable (usually, but not necessarily, population growth rate) into contributions from differences in the parameters that determine that variable. 2.,Fixed, random and regression LTRE designs have been applied to plant populations in many contexts. These designs all make use of the derivative of the dependent variable with respect to the parameters, and describe differences as sums of linear approximations. 3.,Here, I extend LTRE methods to analyse treatment effects on the stochastic growth rate log ,s. The problem is challenging because a stochastic model contains two layers of dynamics: the stochastic dynamics of the environment and the response of the vital rates to the state of the environment. I consider the widely used case where the environment is described by a Markov chain. 4.,As the parameters describing the environmental Markov chain do not appear explicitly in the calculation of log ,s, derivatives cannot be calculated. The solution presented here combines derivatives for the vital rates with an alternative (and older) approach, due to Kitagawa and Keyfitz, that calculates contributions in a way analogous to the calculation of main effects in statistical models. 5.,The resulting LTRE analysis decomposes log ,s into contributions from differences in: (i) the stationary distribution of environmental states, (ii) the autocorrelation pattern of the environment, and (iii) the stage-specific vital rate responses within each environmental state. 6.,As an example, the methods are applied to a stage-classified model of the prairie plant Lomatium bradshawii in a stochastic fire environment. 7.,Synthesis. The stochastic growth rate is an important parameter describing the effects of environmental fluctuations on population viability. Like any growth rate, it responds to differences in environmental factors. Without a decomposition analysis there is no way to attribute differences in the stochastic growth rate to particular parts of the life cycle or particular aspects of the stochastic environment. The methods presented here provide such an analysis, extending the LTRE analyses already available for deterministic environments. [source]


    Efficient MILP formulations for the simultaneous optimal peptide tag design and downstream processing synthesis

    AICHE JOURNAL, Issue 9 2009
    João M. Natali
    Abstract Novel and efficient linear formulations are developed for the problem of simultaneously performing an optimal synthesis of chromatographic protein purification processes, and the concomitant selection of peptide purification tags, that result in a maximal process improvement. To this end, two formulations are developed for the solution of this problem: (1) a model that minimizes both the number of chromatographic steps in the final purification process flow sheet and the composition of the tag, by use of weighted objectives, while satisfying minimal purity requirements for the final product; and (2) a model that attempts to find the maximal attainable purity under constraints on the maximum number of separation techniques and tag size. Both models are linearized using a previously developed strategy for obtaining optimal piecewise linear approximations of nonlinear functions. Proposed are models to two case studies based on protein mixtures with different numbers of proteins. Results show that the models are capable of solving to optimality all the implemented cases with computational time requirements of under 1 s, on average. The results obtained are further compared with previous nonlinear and linear models attempting to solve the same problem, and, thus, show that the approach represents significant gains in robustness and efficiency. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source]


    First-Order Schemes in the Numerical Quantization Method

    MATHEMATICAL FINANCE, Issue 1 2003
    V. Bally
    The numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first-order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are proposed. [source]


    Dynamic ,back-off' analysis: use of piecewise linear approximations

    OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 2 2003
    José L. Figueroa
    Abstract The operating point of a process is usually computed by optimizing an objective function, e.g. the profit, subject to some plant characteristics. Typically, the resulting point lies on the boundary of the operating region. At this point, the presence of disturbances can easily cause constraint violations and make the process move to the unfeasible region. Then, it is necessary to move the operating point away into the feasible region by considering the effect that the expected disturbances will have on the operation of the plant. The purpose of this paper is to present an efficient algorithm to modify the operating point in order to keep feasibility (both in steady-state and along transitory) in the process operation against the disturbances. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Reduced modeling and state observation of an activated sludge process

    BIOTECHNOLOGY PROGRESS, Issue 3 2009
    Isabelle Queinnec
    Abstract This article first proposes a reduction strategy of the activated sludge process model with alternated aeration. Initiated with the standard activated sludge model (ASM1), the reduction is based on some biochemical considerations followed by linear approximations of nonlinear terms. Two submodels are then obtained, one for the aerobic phase and one for the anoxic phase, using four state variables related to the organic substrate concentration, the ammonium and nitrate-nitrite nitrogen, and the oxygen concentration. Then, a two-step robust estimation strategy is used to estimate both the unmeasured state variables and the unknown inflow ammonium nitrogen concentration. Parameter uncertainty is considered in the dynamics and input matrices of the system. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009 [source]