Topic
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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TL;DR: In this paper, the effect of nonlinearities on these nonlinear optimal perturbations [herein, nonlinear singular vectors (NLSVs)] is examined in terms of structure and dynamics.
Abstract: Singular vector (SV) analysis has proved to be helpful in understanding the linear instability properties of various types of flows. SVs are the perturbations with the largest amplification rate over a given time interval when linearizing the equations of a model along a particular solution. However, the linear approximation necessary to derive SVs has strong limitations and does not take into account several mechanisms present during the nonlinear development (such as wave–mean flow interactions). A new technique has been recently proposed that allows the generalization of SVs in terms of optimal perturbations with the largest amplification rate in the fully nonlinear regime. In the context of a two-layer quasigeostrophic model of baroclinic instability, the effect of nonlinearities on these nonlinear optimal perturbations [herein, nonlinear singular vectors (NLSVs)] is examined in terms of structure and dynamics. NLSVs essentially differ from SVs in the presence of a positive zonal-mean shear a...
21 citations
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TL;DR: It is shown that the method using quadratic approximation based on Taylor series about means after selection is the most efficient.
Abstract: Two methods of deriving linear selection indices for non-linear profit functions have been proposed. One is by linear approximation of profit, and another is the graphical method of Moav and Hill (1966). When profit is defined as the function of population means, the graphical method is optimal. In this paper, profit is defined as the function of the phenotypic values of individual animals; it is then shown that the graphical method is not generally optimal. We propose new methods for constructing selection indices. First, a numerical method equivalent to the graphical method is proposed. Furthermore, we propose two other methods using quadratic approximation of profit: one is based on Taylor series about means before selection, and the other is based on Tayler series about means after selection. Among these different methods, it is shown that the method using quadratic approximation based on Taylor series about means after selection is the most efficient.
21 citations
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TL;DR: In this paper, the motion of a phase boundary in the Earth caused by temperature and pressure excitations at the Earth's surface is determined under a linear approximation using a sum of convolutions of pressure and temperature Green's functions with the corresponding excitations.
Abstract: Summary. The motion of a phase boundary in the Earth caused by temperature and pressure excitations at the Earth’s surface is determined under a linear approximation. The solution is found as a sum of convolutions of pressure and temperature Green’s functions with the corresponding excitations. The Green’s functions are given under the form of Laplace transforms that can be inverted either by numerical evaluation of a branch cut integral or by inversion of a series expansion. This solution is a generalization of a solution previously derived by Gjevik. This latter solution is the first term in the series expansion. The relaxation times associated with the phase boundary motion are of the order of 105-107yr for the olivine-spinel phase transition and of 106-107yr for the basalt-eclogite transition. The linear approximation remains valid for long times only if the phase boundary moves slowly.
21 citations
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TL;DR: In this paper, an adaptive space-time registration-based data compression procedure is proposed to align local features in a fixed reference domain, followed by a space time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and a hyper-reduction procedure to speed up online computations.
Abstract: We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.
21 citations
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TL;DR: In this paper, the existence and stability of solutions to the initial-value problem associated with two-way fully dispersive wave models has been investigated using state-of-the-art numerical tools.
21 citations