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Concept# Logarithmic scale

Summary

A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way. As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount, on a logarithmic scale, every unit of length corresponds to multiplying the previous value by the same amount. Hence, such a scale is nonlinear: the numbers 1, 2, 3, 4, 5, and so on, are not equally spaced. Rather, the numbers 10, 100, 1000, 10000, and 100000 would be equally spaced. Likewise, the numbers 2, 4, 8, 16, 32, and so on, would be equally spaced. Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph.
Common uses
The markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.
The following are examples of commonly used logarithmic scales, where a larger quantity results in a

Official source

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Giovanni Migliorati, Fabio Nobile, Erik Gustaf Bogislaw Von Schwerin

We analyse the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is Uncertainty Quantification (UQ) for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the monovariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero, provided the number of samples scales quadratically with the dimension of the polynomial space. Several numerical tests are presented both in the monovariate and multivariate case, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function

2014, ,

We analyse the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is Uncertainty Quantification (UQ) for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the monovariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero, provided the number of samples scales quadratically with the dimension of the polynomial space. Several numerical tests are presented both in the monovariate and multivariate case, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function.

Giovanni Migliorati, Fabio Nobile, Erik Gustaf Bogislaw Von Schwerin

In this work we consider the random discrete $L^2$ projection on polynomial spaces (hereafter RDP) for the approximation of scalar quantities of interest (QOIs) related to the solution of a partial differential equation model with random input parameters. In the RDP technique the QOI is first computed for independent samples of the random input parameters, as in a standard Monte Carlo approach, and then the QOI is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach. We consider several examples including the Darcy equations with random permeability, the linear elasticity equations with random elastic coefficient, and the Navier--Stokes equations in random geometries and with random fluid viscosity. We show that the RDP technique is well suited to QOIs that depend smoothly on a moderate number of random parameters. Our numerical tests confirm the theoretical findings in [G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, Analysis of the Discrete $L^2$ Projection on Polynomial Spaces with Random Evaluations, MOX report 46-2011, Politecnico di Milano, Milano, Italy, submitted], which have shown that, in the case of a single uniformly distributed random parameter, the RDP technique is stable and optimally convergent if the number of sampling points is proportional to the square of the dimension of the polynomial space. Here optimality means that the weighted $L^2$ norm of the RDP error is bounded from above by the best $L^\infty$ error achievable in the given polynomial space, up to logarithmic factors. In the case of several random input parameters, the numerical evidence indicates that the condition on quadratic growth of the number of sampling points could be relaxed to a linear growth and still achieve stable and optimal convergence. This makes the RDP technique very promising for moderately high dimensional uncertainty quantification.