# 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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### Storing large amounts of interpolation data

Overview and Prior Research I am looking for a way store a (in principle arbitrarily) large "3D-table" for interpolation/ lookup in combination with python. I have considered CSV files, but,...
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### QUICK scheme derivation

I am reading about QUICK scheme for calculating the value of unknown variable $\phi$ in finite volume method. Given a locally one dimensional flow, we assume the value of $\phi$ is computed as a 2nd ...
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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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### 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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### How to interpolate stress at unknown points from the stress values available based on geometrical position for constant load?

I am working on a combined contact, bending, and torsion problem. I have data on geometrical points and their instantaneous stress components. However, based on the available data, I have to ...
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### What are the benefits of using machine learning for interpolation over traditional interpolation methods?

I am trying to get a better understanding of the application of function approximation with machine learning. My question is simple, how does function approximation with ML compare to traditional ...
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### Shape function for quarter point element from degenerate quadrilateral

I want an element as the one shown following. Nodes 6 and 8 are in the quarter position. Whether eight node quadrilateral element or six node triangle function can be used directly？ If not, how to ...
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### Projecting nodal solution to gauss points with certain accuracy

I am having a problem that was also mentioned at the accepted answer to this question by Wolfgang Bangerth. I need to calculate, as it was specified at the question at the link, F1 integral and for ...
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### What are the best ways to interpolate a vector field inside (convex) polygons?

I want to interpolate a vector field inside convex polygons in a polygonal mesh. For triangular meshes the scheme uses a piecewise constant interpolation in the triangle, discretized at the center of ...
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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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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-...