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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Complex interpolation for isotopic (precipitation) data

Is there a package that interpolates precipitation data taking into account mountains and oceans? I have so far used Numpy and Basemap but as you can see in the code, the data from Europe affect the ...
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How can I reduce the artifact in "Thin Plate Spline" interpolation?

At the Top "right", there is the 2D-density plot of the recorded data (actual), fewer in number. Recorded data has been sampled a on the 8 arms of a regular octagon. These 8 arms are placed ...
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Generate polynomial basis through a sequence of SVD

I need help to understand how to use the result given by an algorithm for constructing an orthonormal polynomial basis over $L^{2}(X)$, where $X\subset\mathbb{R}^2$, with respect to the inner product $...
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Geometric multigrid transfer operators constraints

I have been reading up on constraints used to construct prolongation and restriction operators for geometric multigrid. One such constraint is that the prolongation matrix is the transpose of the ...
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Prolongation and restriction operators in multigrid for high order PDEs

If I have the Poisson equation $\Delta u = f$ a standard transfer operator (for a regular grid) is the full weighting/bilinear interpolation scheme: $$K = \frac{1}{4}\begin{bmatrix}\frac{1}{4} & \...
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Interpolation question

I have a set of data $(x_i,y_i,y'_i)$, $i=1,\dots,N$ and I want to fit an interpolating curve $f(x)$ which matches both the data $y_i$ and the first derivatives $y'_i$ at the nodes $x_i$, \begin{align}...
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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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Classical global estimate for $H^1$ error

I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$. $$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
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Sum of norms over elements is not equal to norm over the whole $\Omega$

In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$: $$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
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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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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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Spline interpolation for vector-valued data in 3D space

I have output from a 3D linear elasticity finite element simulation which uses linear tetrahedral elements, such that the displacement is continuous over the nodes but the gradient is not ($C_0$ ...
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At which stage to interpolate?

Assume I have a multidimensional grid $G$. I consider two functions $f(x)$ and $g(x)$. I have solved the values for the functions over all grid points $x \in G$. Let me now be interested in some third ...
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Finite element interpolation

I have a finite element solver that I implemented in MATLAB. I am calculating a specific potential function that is constant in each tetrahedral element. The question is, I want to be able to ...
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Interpolate location based on triangular sensor array

There are 3 sensors (A, B, C) on a plane, located in the corners of a (known) equilateral triangle. I want to calculate the (2D) location of an object (X) inside that triangle. One sensor returns one ...
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Interpolation of a Concave Mesh

I would like to have an algorithm that interpolates the values attached to nodes in a concave mesh mesh. To be me precise, assume we have a point cloud P (e.g. in 3 dimensions) and a list of edges E ...
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Getting torsion and curvature out of ODE solution skeleton

Suppose I have solved an ODE $v'(t) = f(t,x)$ via some adaptive stepper, such as RK4 or Dormand-Prince, generating a list of points $\{(t_i, v_i, v_i' = f(t_i, v_i))\}_{i=0}^{n-1}$. I wish to use this ...
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Factorization of cubic spline interpolation matrix

In cubic spline interpolation, we use the set of knots and function values $(x_i,y_i),i=1,...,n$ to construct a (tridiagonal) system of equations for the unknowns $\sigma_i$: $$ h_{i-1}\sigma_{i-1} + ...
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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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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 $...
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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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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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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 = \...
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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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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 ...
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Interpolating the gradient of a cylindrically symmetric potential field expected obey the Laplace equation, especially near/across r=0

The script below tries to implement a Jacobi iterative relaxation of a potential field for an electrostatic lens. In order to plot electric field lines and calculate trajectories for charged particles,...
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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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extrapolation/interpolation in fmincg.m

Can you tell me these equations come from where in MATLAB fmincg.m? ...
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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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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 ...
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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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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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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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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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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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5 votes
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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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