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.
157
questions
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1answer
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What is the relationship between shape functions, interpolation functions, and degrees of freedom?
I am a newbie in FEM. I would like to get clarity regarding a few questions on shape functions in this post (please use as simple language as possible).
What is the relation between Shape function ...
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0answers
66 views
How to approach geographic data interpolation by distance?
let's say I have a set of geographic locations (lat, lng) resulting from a query. Those locations have some kind of internal ranking, my set is sorted by this number in a descending order.
Now I'm ...
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1answer
182 views
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 ...
1
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1answer
50 views
Monotonicity preserving interpolant in 1D
I have a dataset $\{x_i, y_i\}_{i=0}^{n-1}$ where $x_0 < x_1 < \cdots x_{n-1}$ (not uniformly spaced), and, in addition $y_0 < y_1 < \cdots y_{n-1}$. So it feels natural to assume that $...
3
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1answer
195 views
Get the roots of a Hermite interpolating polynomial
I am using Python 3.7 to write a program that requires me to calculate the root of the Hermite interpolating polynomial, given two points $\epsilon_0$, $\epsilon_1$, the function ($d(\epsilon_0)$,$d(\...
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3answers
150 views
Bilinear interpolation for large grids
I need to make a bilinear interpolation of a function $Y(i,j)$ tabulated on a $100\times 100$ grid. I tried to do it with the Fortran polin2.f and ...
3
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2answers
99 views
How to reconstruct a 2D field from its integral?
General question
I work on the plane where I have a two-dimensional shape $V$ that is cut in a collection of parts $\{V_i\}$ that do not overlap
$
V_i ~~\text{s.t.}~~ \bigcup_i \overline{V}_i = \...
2
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2answers
259 views
Interpolation vs. Neural network
I am seeking knowledge from the community. I am solving a transport PDE (conservation of solute mass) using COMSOL. At each Newton-Raphson iteration, I need to update a constant called $Kd$ for some ...
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0answers
136 views
Derivative of Whittaker-Shannon interpolant
Last time we looked at how to improve the accuracy of Whittaker-Shannon interpolation, where user njuffa demonstrated that judicious use of sin_pi could greatly ...
1
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1answer
53 views
Interpolating the gradient of a cylindrically symmetric potential field that's 'supposed to' obey the Laplace equation?
The script below tries to implement a Jacobi iterative relaxation of a potential field for an electrostatic lens.
It's hot-off-the-press and I've just started to debug and look for things to test it ...
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1answer
426 views
Gnuplot: How can I fit a range of points (out of the entire data) to a function?
I have a set of data obtained for the I-V characteristics of an LED.
...
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0answers
38 views
extrapolation/interpolation in fmincg.m
Can you tell me these equations come from where in MATLAB fmincg.m?
...
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1answer
81 views
Interpolation of function onto mesh gives different results, depending on mesh density
I wanted to test the numerical accuracy of my program. For that I wanted to interpolate the function $$f=I_0\exp\left(-100x^2\right)\exp(-100y^2)$$ onto a grid, defined on $$\Omega=[0,1]^2$$ by using ...
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1answer
69 views
Creating an Interpolation of a w = f(x,y,z) function
I am trying to finish a series of interpolation functions.
The problem is more related with organizing the data than how to do the interpolations.
Using the RegularGridInterpolator, I created this ...
0
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1answer
391 views
Problems with python's interp 2D
I am writing some functions to interpolate data. While using interp2D, somehow, a sample matrix works but when I change the size of the matrix, it returns an error.
...
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0answers
86 views
Grid Data Interpolation
What are the most sophisticated methods for interpolating a scalar field say Electric or Magnetic Field on a 3-D grid?
I have scalar data on a meshgrid with equal spacing. I would like to use an ...
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0answers
30 views
Cardinal B-Splines with derivative information
Have Schoenberg's cardinal B-splines been extended to accept derivative information at each knot, similar to how Lagrange interpolation can be improved by Hermite interpolation?
3
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0answers
99 views
WENO5 scheme in a staggered grid
I'm trying to use the finite-difference WENO scheme to solve the 2D density conservation law with axial symmetry (coordinates $r,z$):
$\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v}) = \...
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6answers
2k views
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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0answers
381 views
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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1answer
441 views
Python - Fitting a function to data without using Scipy
I'm trying to write a program in python which doesn't need to use extra packages like numpy and scipy. In one part of the project, if I can interpolate a function to a set of data, I can save ...
5
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2answers
162 views
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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1answer
166 views
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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1answer
55 views
Wanted: smoothing time domain transform
Let $A$ be a finite (and small-ish) set of positive real numbers and 0. Let $B$ be a subset of $\mathbb N^0$, up to some (small-ish) bound.
I have a function $f(t)$, $A \rightarrow B$ that is ...
4
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0answers
67 views
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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3answers
143 views
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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0answers
32 views
Are planar 2nd order NURBS curve segments exactly the segments of conic sections?
I think it is well known that 2nd order NURBS curve can describe any segment of a conic section. Such a segment has 7 degrees of freedom, take an ellipse arc for example: the ellipse itself has 5 ...
2
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1answer
1k views
Three dimensional irregular grid data interpolation to regular grid
I have three-dimensional radar reflectivity data obtained as voxels (scans, rays, altitudes). The data has been sampled at irregular spacings and I want to convert this into a regular grid. In ...
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0answers
65 views
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 ...
4
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2answers
125 views
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 ...
3
votes
0answers
45 views
1D irregular interpolation with $O(1)$ or $O(\log(N))$ evaluation
In Floater's paper on barycentric rational interpolation, he shows that a stable interpolant using irregularly space points can be evaluated in $O(N)$ operations.
For equally spaces samples, cubic b-...
4
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1answer
97 views
Interpolation estimates for $H^1$ into $P_1$
As far as I can tell, nodal interpolation estimates proven with Bramble / Hilbert require higher regularity of the functions being interpolated. E.g. with linear elements in 2 dimensions, one needs $v ...
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1answer
1k views
Numpy.polyfit with regularization
I am trying to use the numpy polyfit method to add regularization to my solution.
My non-regularized solution is
...
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2answers
822 views
Polynomial Interpolation with Matlab polyfit
Given $N$ data points, does polyfit of degree $N-1$ produces the unique interpolating polynomial?
For concreteness, here is a code example:
...
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2answers
225 views
How to represent CFD result when I use grid-centered FVM?
My variables are stored at the center of the cells. How can I transfer these values to grid points? If I calculate the algebraic average value there may be a shock.
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1answer
84 views
Approximate function with minimum number of evaluations
I'm trying to approximate an unknown, non-linear function $f: A \to \mathbb{R}$ (in my case: $A = B \times B$ where $B$ is an infinite but compact subset of $\mathbb{R}$). That is, evaluating $f$ is ...
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1answer
185 views
Barycentric interpolation equivalent for irregular hexahedra
I have a mesh with irregular hexaedra and I need a fast way to interpolate values at points inside these cells. I know that trilinear interpolation does not work well for large skews. Barycentric ...
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0answers
68 views
1-D local interpolation of sampled function that is polynomial in 1/x
I have a number of evenly-spaced samples of functions that are polynomial in $1/x$ where $x$ is a continuous variable on $x\in\left[a,b\right]$, i.e. $$f(x_i)=\sum_{n=0}^{N} f_n x_i^{-n}.$$ I want to ...
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0answers
75 views
Find B-spline coefficients from values on collocation points
The context of my question is how to compute high order derivates on direct numerical simulation of turbulent channel flow. It is of particular interest for fluid dynamics and turbulence research.
I ...
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1answer
156 views
Find hidden sequence $a_n = a_{n-1} + T $ , with period $T$, given some “random” numbers
I have this data plotted on a graph in which all points have the same value on the y-axis, e.g a constant integer "c", while the x-axis is the time in seconds.
So, for a c = 25 on the y-axis, there ...
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0answers
218 views
Numerical integral of oscillating function with known zeros
I have a function that I need to numerically integrate from $0$ to $+\infty$, given by:
$$I = \int_0^{+\infty} \mathrm{d}x\,x\,T^2(x)f(x)$$
where $T^2$ is an interpolated function that goes to $1$ ...
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0answers
437 views
Polynomial approximation - Vandermonde matrix creation - precision
I am trying to fit a polynomial through 340 points in a 3D space, i.e.
$$f(x,y,z) = k$$
I asked previously about the theory behind polynomial interpolation here ->
Polynomial interpolation
Few ...
0
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1answer
167 views
Underdetermined/overdetermined polynomial interpolation
I am trying to apply a polynomial interpolation to 340 points in a 4D space, i.e.,
$$f(x,y,z)=k\, .$$
What I would like to understand is this: if I use a 6th order polynomial I will end up with 343 ...
1
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2answers
417 views
Vandermonde matrix DG Hestaven
I am trying to understand the nodal and modal basis formulation from the book of Hesthaven (Nodal Discontinuous Galerkin Methods, Hesthaven, Jan S., Warburton, Tim). For $N=2$, I get the Vandermonde ...
4
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1answer
299 views
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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2answers
297 views
What would be a good, fast, way to interpolate a point in 3D space
So I have a 3D mesh made of elements with eight nodes, 12 sides per cell, and in the course of my simulations, I would have to interpolate data from those nodes onto a point inside the cell with a ...
1
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1answer
102 views
Fast table interpolation on regular time data for ODE
I am using scipy.integrate.odeint to simulate the reaction of a system with known input signals via integration. The simplified code below illustrates what I'm ...
3
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2answers
433 views
Parametrized spline - oscilating second derivative
I am using a parametrized natural spline to interpolate 2D curves. For example something like this:
Where I parametrize both the x and the y coordinate with a parameter that increases by one between ...
3
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2answers
68 views
Bounded approximation to a bounded function
I have a non-negative function $f(x) \ge 0$ defined on the interval $[a,b]$. I would like to have a finite-dimensional approximation to this function that is guaranteed to be non-negative on $[a,b]$. ...
2
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1answer
128 views
Higher order interpolation in DWR method
Based on page $35$ of the book:
(W. Bangerth and R. Rannacher, "Adaptive Finite Element Methods for Solving Differential Equations", Birkhäuser, 2003,)
for computing the error in dual weighted ...
