Questions tagged [interpolation]
Interpolation is the process of estimating the values of a function, when the function's values are known only at a particular set of points. Questions on interpolation in one or more dimensions, as well as algorithms for doing so, should have this tag.
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What is the preferred and efficient approach for interpolating multidimensional data?
What is the preferred and efficient approach for interpolating multidimensional data?
Things I'm worried about:
performance and memory for construction, single/batch evaluation
handling dimensions ...
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Why do equi-spaced points behave badly?
Experiment description:
In Lagrange interpolation, the exact equation is sampled at $N$ points (polynomial order $N - 1$) and it is interpolated at 101 points. Here $N$ is varied from 2 to 64. Each ...
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What is the best way to find discontinuities of a black-box function?
It was suggested that this might be a better place for this question than Mathematics Stack Exchange where I asked it before.
Suppose one has a black-box function which can be evaluated anywhere (...
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Example of a continuous function that is difficult to approximate with polynomials
For teaching purposes I'd need a continuous function of a single variable that is "difficult" to approximate with polynomials, i.e. one would need very high powers in a power series to "fit" this ...
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Efficient interpolation method for unstructured grids?
I would like to know a good method for interpolating data between two unstructured grids, where one grid is a coarser version of the other.
Efficiency is very important to me since I'm solving a ...
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How to interpolate multipoint data to the cell centres of an unstructured mesh?
I have sets of multipoint field data, each point data set relates to a single cell of an unstructured mesh. The goal is to interpolate the data to the cell centre, directly or indirectly, in the most ...
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Implications of thermodynamic inconsistency in CFD calculations
During my PhD work, I had to use tabulated values of thermodynamic properties of gases in some Computational Fluid Dynamics (CFD in short) simulations.
My tables are discretized in temperature and ...
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Do RBF kernel matrices tend to be ill-conditioned?
I use RBF kernel function to implement one kernel based machine learning algorithm(KLPP),
the resulting kernel matrix $K$ $$K(i,j)= \exp\left({\frac{-(x_{i}-x_{j})^2}{ \sigma_{m}^2}}\right)$$
is ...
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When we use Bernstein polynomials in application
When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...
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Fast (approximate) evaluation of Chebyshev polynomial
Is there a preferred way how to implement a fast (approximate) evaluation of the Chebyshev interpolation polynomial on uniform grid (given the function values at the Chebyshev nodes)? My problem is ...
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Help deciding between cubic and quadratic interpolation in line search
I'm performing a line search as part of a quasi-Newton BFGS algorithm. In one step of the line search I use a cubic interpolation to move closer to the local minimizer.
Let $f : R \rightarrow R, f \...
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What is the most accurate interpolation method for a 3D-flowfield on a structured grid?
I solve multi-species, compressible Navier-Stokes equations on a 3D structured grid. I have obtained a solution on a given grid (let's say a relatively coarse one). I want now to refine my grid and ...
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High Order derivatives of splines using SciPy
I have created a spline to fit my data in python using:
spline=scipy.interpolate.UnivariateSpline(energy, fpp, k=4)
The equation I want to use involves a ...
Karim Sutton
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Polynomial interpolation on a regular hexagon
Background
We know a function $f$ on the vertices of a regular hexagon, as follows
$$\left( 1, \ 0, \ f_{0}\right), \ \left( \frac{1}{2}, \ \frac{\sqrt{3}}{2}, \ f_{1}\right), \ \left( - \frac{1}...
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Generating harmonic polynomials in cartesian coordinates
TLDR: Are these polynomials really harmonic polynomials, and how can I generate them?
Long version:
I want to describe an electrostatic potential $\Phi(x,y,z)$ over a source-free volume, by using a ...
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What are these oscillations?
I have a function $g(x)$ defined numerically that is sort of in between a Gaussian and a Lorentzian. It decays much slower than a Gaussian, but still faster than a simple inverse power.
I need to ...
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Interpolate 2D data
I generated a cartesian grid in Python using NumPy's linspace and meshgrid, and I obtained some data over this 2D grid from an ...
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Intermediate values (interpolation) after Runge-Kutta calculation
I have a numerical ODE simulation that I computed at fixed time step $h$ using a 4-th order Runge-Kutta method (RK4), producing a series of results $(x_1,y_1), (x_2,...
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Linear interpolation in Fortran
Is there a Fortran subroutine which performs linear interpolation in one-dimenional data?
I need something similar to MATLAB function interp1.
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Evaluating an integral numerically at many points
Given a real function $f$, how can one efficiently evaluate $\int_0^{a_i}f(x)dx$ for millions of different $a_i$?
Applying a standard quadrature method (such as Simpson's rule or Gaussian quadrature) ...
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Piecewise polynomial interpolation: Hermite vs Lagrange
I am a bit confused of the qualitative behavior of the two methods. Consider quadratic case, start by having points $x_i$, where I know the value and points $y_j$, where the values to be found. If I ...
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Automatic differentiation of barycentric rational functions
By a barycentric rational interpolant we understand a function of the form
\begin{align*}
r(t) := \frac{\sum_{i=0}^{n-1} \frac{w_i y_i}{t-t_i} }{ \sum_{i=0}^{n-1} \frac{w_i}{t-t_i}}
\end{align*}
In ...
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Full Multigrid Prolongation Operator
I am looking into full multigrid, FMG, and several sources, including these slides, that a lot of people are referring to, state that the prolongation operator used in FMG the first time you visit a ...
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Interpolation by Solving a Minimization Problem (Optimization)
I will try to give the motivation behind this problem and later the math formality.
Given a grayscale image (1 Channel - $M \times N$ Matrix).
Someone marks some pixels as anchors.
Now, you need to ...
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restriction and interpolation in multigrid method
I need detailed explanation of the formula below
A2=I1*A1*I2
I suppose this formula computes matrix A2 on a coarse grid and here A1 is original matrix on fine ...
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Is there a Gauss-Laguerre integration routine in Python?
I am reading the book "Numerical Recipes in Fortran 77: The Art of Scientific Computing" (Second Edition) and I came across some methods for numerical integration of 1D functions. More specifically ...
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interpolation combined with methods of characteristics can cause oscillations for the transport equation?
I would like to know about the effect of using a higher order interpolator for the methods of characteristics. I am solving $$u_t+a(x,t)u_x=0$$ with some nonsmooth initial data $u_0(x)$ by the method ...
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Continuation procedure to solve for a 2D curve that satisfies f(x,y) = 0
I have some function of $R^2$, that must be numerically computed. For instance, I might be interested in a real-valued contour integral that begins from (x,y) = 0.
$$
f(x,y) = \Re\left[\int_0^{x + iy}...
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What spline functions are used in Section 13.9 of "Numerical Recipes in C"?
I asked a similar question on MathSE but with more added fluff, but didn't really get any straight answers, so I figured I'd ask here. Computing Fourier coefficients of a function using the FFT is ...
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Grid mapping from an unstructured triangular mesh to a regular rectangular mesh
I am modeling fracture propagation using a 2-D dynamic unstructured grid. As the fracture propagates over time, the elements move accordingly. For a given time step, I would like to interpolate the ...
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Fortran, making a more efficient bilinear interpolation
I'm trying to write an efficient bilinear (2D)-interpolation, after reading some recipes, as a fortran-mex for Matlab that is used extensively throughout a long algorithm of solar image processing, ...
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What is the most efficient approach to interpolate values between two FEM meshes in 2D?
I am looking for efficient algorithm to interpolate values from one unstructured 2D mesh grid to another. Both grids are constructed using the same type of elements (triangles or quadrilaterals). Both ...
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Cressman interpolation and objective analysis
I have read this question and answer – Interpolation of scattered data to a regular grid in python and I am doing something similar as I have temperature values of the atmosphere at different heights ...
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Integrating highly oscillatory functions
I have a logarithmic grid, upon which i have two functions that are similar to this one (this is only the last 100 points):
These are essentially very similar to a Sin function at this point. I need ...
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Approximating a step function with polynomials
The Weierstrass approximation theorem says any continuous function $f(x): [0,1] \to \mathbb{R}$ can be approximated uniformly by polynomials. Given any $\epsilon$, we can find $p(x) = x^n + \dots $ ...
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Interpolation schemes to move data between cells and nodes
I work on non-graded quadtree grids where the entire grid is a hierarchy of cells specified using a quadtree data structure, where, in general, there is no constraint regarding the relative size of ...
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SciPy interpolation with Univariate Splines
I have coded a routine for interpolation with B-splines, only to discover later that this functionality is already included in Python's SciPy.
However, I do not understand one parameter in the SciPy ...
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Which way is the right way to compute the integrals in finite element methods?
Finite element methods involve integrals of functions that are not polynomials, and these integrals must be computed numerically.
For example, suppose that $f$ is the right-hand side of a Poisson ...
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Automatic finite differences
Given numbers $x, y \in \mathbb{R}$ where
$$\frac{|y-x|}{|x|}$$
is small, and code that implements the function $f$ with a sequence of arithmetic operations, I would like to compute to high accuracy ...
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Minimal surface solution in Python
Note: this question was also posted in StackOverflow and math.stackexchange.
I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane ...
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Polynomial approximation spaces
I often see people using products of 1-D polynomials to do interpolation or projection of smooth multivariate functions over grids or cells because it is intuitive and simple to implement. What are ...
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Computing inverse functions of functions of two variables
There are several functions of two or three variables that I am working with. For this question I have made a small set showing the resistivity, $\rho$, in n$\Omega$m, of copper as a function of its ...
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Adaptive Table Lookup for Expensive Function Evaluation
I have a function that is expensive to evaluate whose inputs are n-dimensional (n is the order of a dozen or two). I need the output of this function at each node and each time step for a PDE ...
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Padua-type pointset for functions singular on line $x=y$
The Padua points $\mathrm{Pad}_{n} \subset [-1,1]^{2}$ are a unisolvent pointset with optimal growth of Lebesgue constant, described in detail here. With some work they can be used to generate a ...
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How to optimally choose points for multivariable Hermite interpolation?
I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input.
I would like to create interpolation polynomial for it.
In one-dimensional case ...
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How do I perform chebyshev interpolation from a to b with custom angle range?
Typically Chebyshev interpolation from $-1$ to $1$ with angle from $0$ to $\pi$:
$\xi_j=\cos \left ({\pi j \over N}\right )$
$x_j=(1+\xi_j) * {L \over 2}$
$w$:
$w_0=\pi/(2N)$
$w_{1,...,N-1}=\pi/(N)$...
user5273
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What is the difference between $u_h$ and $I_h(u)$ in finite element literature?
In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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Interpolate the orbital coordinates of an object using coordinates and velocities vector
I have a table of the orbital coordinates and velocities of an object with time steps of 1 minute.
Now I would like to interpolate this to a finer time increments with time steps of the order of 1 ...
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Interpolating a mathematical function using a Hermite Cubic Finite Element Space
I have a Hermite Cubic Finite Element Space on a computer in the form of Matlab m-files. More specifically, I can evaluate four "shape functions" $N_1, N_2, N_3,$ and $N_4$, for which the following ...
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Interpolation with the roots of orthogonal polynomials & Spectral expansion
I'm a bit confused about the relationships between these two approximation methods mentioned in the title.
Does this kind of interpolation also belongs to the field of spectral methods?
Are the ...
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