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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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
25 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?
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0answers
83 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
149 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
191 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 ...
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2answers
156 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
154 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
51 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 ...
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0answers
65 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
130 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
30 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 ...
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1answer
525 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
63 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 ...
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2answers
103 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 ...
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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-...
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1answer
86 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
650 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
492 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
220 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
81 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
124 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
56 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
64 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
155 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
116 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
332 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 ...
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1answer
110 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 ...
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2answers
298 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
190 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
219 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 ...
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1answer
81 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 ...
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2answers
281 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 ...
4
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2answers
67 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 $[a,b]$. ...
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1answer
96 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 ...
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0answers
225 views
How to perform Taylor series expansion consisting of cell averaged derivatives in a computational element?
I have encountered the term cell "Cell averaged derivatives" and "Taylor series expansion using cell averaged derivatives about centroid" in the context of polynomial representation in any cell (...
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2answers
219 views
Given x,y,z data of a periodic object, calculate the period of the object (if possible)
So the problem I am working on is as such. Given the x,y,z data of a periodic object over time (from the origin in 3d space) (need not be uniform), calculate the period of the object (if the data ...
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0answers
138 views
Integral image resizing
I'm trying to find an approach for integral image resizing.
I found out that I can do it with a bilinear interpolation method, but with this approach I can only resizing by the factor which is a power ...
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2answers
113 views
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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2answers
126 views
citations for numerical lookup table interpolation of P/ODE(s) RHS
I'm not sure that this *overflow is right place to ask.... Sorry if it is off topic.
Does anyone know a citation (scientific article or book) for a numerical trick (method), when we tabulate a right-...
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1answer
427 views
Chebyshev approximation by projection vs interpolation
Suppose we want to approximate a function $f: [a, b] \rightarrow \Re$ with a Chebyshev series:
$$ f(x) \approx \sum_{k=0}^n c_k \, T_k\left( \frac{2x-b-a}{b-a} \right) $$
where $T_k(x) = \cos(k\, \...
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1answer
162 views
Line integral along the edge of an isoparametrically mapped quadrilateral
I need to integrate a function along the edge of a quadrilateral (boundary integral). For example, the function is $f(x,y)=x^3+y^3$, the quadrilateral coordinates are $(0,0),(2,-1),(3,2),(1,3)$ and ...
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0answers
116 views
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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0answers
100 views
computational tool for higher order Lagrangian interpolation for finite element
In finite element, I can calculate the Lagrangian interpolation shape functions for each degree of freedom in an element, from the the number of nodal degrees of freedom and the number of nodes ...
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1answer
177 views
Methods for interpolating from points that are not on a regular grid?
I'm working on a project where I'll need to be able to interpolate scalar potential values at arbitrary points in a 3-D box from a large (potentially millions to billions of points) collection of ...
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3answers
99 views
Polynomial reconstruction on unstructured grids
For a 1D grid I can calculate a Lagrange polynomial through an arbitrary set of points for the reconstruction of a polynomial function.
In 2D I have an unstructured grid and want to interpolate the ...
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1answer
84 views
Using two reference values for a scalar variable: What's the name of this type of problem?
I don't really know where to ask this one...
In fact, I am not sure I can define it properly.
Here goes...
Let's say I take measurements.
In order to "normalize" these measurements, I divide their ...
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2answers
138 views
Integrate result of finite element calculation in MATLAB
I have a PDE that I solve in MATLAB over a 2D domain using the "recommended workflow", i.e. the functions descg, ...
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0answers
59 views
Sensitivity Analysis — Total Variation for a function with categorical arguments?
I have an application in sensitivity analysis of complex system models with moderately nonlinear interactions between arguments
Arguments potentially dozens or hundreds in number
Arguments mostly ...
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1answer
424 views
Interpolation of velocities on staggered grid (in PIC)
Edit: (copying from my comment)
Let's consider the inverse problem when I need to transfer velocities from particles to the grid (inverse bilinear interpolation). How'd I transfer a particle's x-...
