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Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.


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Journal ArticleDOI
TL;DR: In this paper, two kinds of typical on-off switches represented in idealized simple models are investigated: the zero-order discontinuous switch (Type I) and the first-order switching (Type II) in both time-continuous and discrete circumstances.
Abstract: Since the accuracy of the tangent linear approximation of moist physics in a mesoscale model is case dependent, the problem related to the variational data assimilation with physical “on–off” processes is studied further in both time-continuous and discrete circumstances. Two kinds of typical on–off switches represented in idealized simple models are investigated: the zero-order discontinuous switch (Type I) and the first-order discontinuous switch (Type II). The main results are as follows. For Type I: 1) For the case in which the model is time continuous, the gradient of the cost function with respect to the initial condition exists except for the threshold. 2) In the time-discrete case, there are zigzag discontinuities in the cost function, and the method that keeps the switches in the tangent linear model the same as in the forward model (called Zou's method) is able to compute the correct gradient where it exists. An optimization with this gradient might yield a local minimum rather than the...

24 citations

Journal ArticleDOI
TL;DR: The proposed model is intended to extend the range of applicability of perturbation models when applied to inclusions that are non-point-like, and gives a better prediction for absolute values of the scattering and absorption coefficients of inclusions, when the inclusion optical properties are higher than the surrounding background.
Abstract: We report what to our knowledge is a novel perturbation approach for time-resolved transmittance imaging in diffusive media, based on the diffusion approximation with extrapolated boundary conditions. The model relies on the method of Pade approximants and consists of a nonlinear approximation of time-resolved transmittance curves in the presence of an inclusion. The proposed model is intended to extend the range of applicability of perturbation models when applied to inclusions that are non-point-like. We test the model on different tissue phantoms with scattering only, absorbing only, and both scattering and absorbing inclusions. Maps of the optical properties are displayed, and the results are compared with those obtained by means of the usual linear approximation of time-resolved transmittance curves. We found that the nonlinear approach gives a better prediction for absolute values of the scattering and absorption coefficients of inclusions, when the inclusion optical properties are higher than the surrounding background. Furthermore, better-resolved spots and a reduced cross talk between the two parameters are found in the reconstructed maps. Because the range of the optical properties spanned by the considered phantoms covers the values expected for optical mammography, the application of the reported reconstruction method to in vivo images of a breast appears promising from a diagnostic viewpoint.

24 citations

Journal ArticleDOI
TL;DR: In this paper, the model reduced 4D-Var (Vermeulen and Heemink, 2006 ) is investigated to test its feasibility in ecosystem application, and two experiments are conducted in a 1D ecological model.

24 citations

Posted Content
TL;DR: In this article, the authors give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev-Korobov-type smoothness and mixed SSA-analytic type smoothness in the infinite-dimensional case.
Abstract: We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev-Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case where specific summability properties of the smoothness indices are fulfilled. These estimates are then applied to the linear approximation of functions from the associated spaces in terms of the $\varepsilon$-dimension of their unit balls. Here, the approximation is based on linear information. Such function spaces appear for example for the solution of parametric and stochastic PDEs. The obtained upper and lower bounds of the approximation error as well as of the associated $\varepsilon$-complexities are completely independent of any dimension. Moreover, the rates are independent of the parameters which define the smoothness properties of the infinite-variate parametric or stochastic part of the solution. These parameters are only contained in the order constants. This way, linear approximation theory becomes possible in the infinite-dimensional case and corresponding infinite-dimensional problems get tractable.

24 citations

Book ChapterDOI
TL;DR: A more accurate and valid procedure for characterizing the behavior of estimates of parameters in nonlinear models is discussed, illustrating the approach and the insights it gives using a model for frontal elution affinity chromatography and a compartment model.
Abstract: Publisher Summary This chapter focuses on parameter estimates from nonlinear models. The methods require some extra computing after the model has been fitted to a data set but the computing is efficient and easily accomplished. Fitting nonlinear models to data relies heavily on procedures used to fit linear models. The chapter presents a review of fitting linear models, including how to assess the quality of parameter estimates for such fits and discusses fitting nonlinear models and application of linear model methods for assessing the quality of parameter estimates for nonlinear models. The chapter also discusses a more accurate and valid procedure for characterizing the behavior of estimates of parameters in nonlinear models, illustrating the approach and the insights it gives using a model for frontal elution affinity chromatography and a compartment model. Profile plots can be extremely useful in nonlinear model building because they remove the gross dangers involved when using linear approximation standard errors and confidence regions.

24 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202229
202197
2020134
2019124
2018147