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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
135 views

Nodal functionals in finite element analysis

I have a quintic Hermite basis functions [-1,1] for FEM applications, I wanted to check if it's nodal functionals are proper. Could someone explain nodal functionals in details and give an example of ...
0 votes
1 answer
136 views

Quintic Hermite shape functions

I am trying to use quintic Hermite basis functions for FEM applications, could someone please direct me to the general formula that would help me generate quintic Hermite shape functions? In natural ...
1 vote
0 answers
66 views

shape functions of interpolating a piecewise polynomial with continuous 0-th and 1-st derivatives

shape functions are the basis functions that interpolate a function in a subdomain using polynomials. linear interpolation is probably the most convenient approach which results in so-called "...
6 votes
2 answers
2k views

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 ...
1 vote
2 answers
224 views

Finding weighted average of curves

This is related to my previous post here I have a dataset with values of multiple curves. An example plot is shown below. I want to scale the curves (move up/down) so that all curves overlap. The ...
7 votes
1 answer
168 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 ...
2 votes
1 answer
4k 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 ...
3 votes
0 answers
93 views

Why weighted harmonic mean for pchip slopes leads to monotone interpolator

In Fritsch and Carlson's paper on monotone interpolation, they identify numerous conditions under which a cubic Hermite interpolator will be monotone. For example: On the subinterval $[t_i, t_{i+1}]$ ...
0 votes
2 answers
132 views

Conservative interpolation from a 1D grid to another 1D grid

I am given with a function $f(x)$ on a grid $X_{old}\in \{x_{min},x_{max}\}$ with a non uniform spacing. I need to interpolate that function on a new log-spaced grid $X_{new}\in\{x^{\prime}_{min},x^{\...
2 votes
1 answer
277 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 ...
4 votes
2 answers
355 views

Chebyshev/Lagrange polynomials in spectral methods

I am currently trying to familiarise myself with (Pseudo-)Spectral Methods for solving differential equations. Now, I am struggling to understand some obviously crucial concept of this approach. The ...
3 votes
1 answer
120 views

Interpolating 2D data on a hemisphere in order to have $C^2$ function but no overshoot

I am interpolating a 2D dataset on a hemisphere, and I am currently using scipy.Rbf that I like for its simplicity. I am defining the norm of the interpolator with ...
1 vote
3 answers
445 views

If FEM is exact at the nodes, why do first and second-order elements give very different results?

I'm looking at the solution to a structural mechanics problem that is modeled with first-order elements and then as a comparison with second-order elements. It is clear that the first-order elements ...
0 votes
1 answer
210 views

Is 'natural neighbor' interpolation better than linear for unstructured function interpolation?

Natural neighbor interpolation is defined here, it is an intriguing method that uses voronoi diagrams. Notably it is smooth almost everywhere whereas linear interpolation is only piecewise linear. I ...
0 votes
1 answer
1k views

Fitting a monotonically increasing spline function

I want to fit a monotonically increasing smooth spline function for a dataset Code: ...
2 votes
0 answers
89 views

Modal representations of nodal tensor product Galerkin elements

Nodal discontinuous Galerkin methods on simplices, like those described in Hesthaven and Warburton, have the nice property that the number of nodes is equal to the minimum number needed to represent a ...
1 vote
0 answers
41 views

Integral from function approximations

I have some data which I cannot manage to model and fit with a known function, so let’s say that they are a sample from the unknow function $f(x)$, which look a sort of skewed bell-shaped distribution....
2 votes
1 answer
92 views

Interpolation of 1D solution from an original grid to a new grid

I have a solution of a 1D wave on a grid (tangent hyperbolic variation) and now I want to interpolate the obtained solution to a new grid with the same number of points as the previous grid but the ...
5 votes
2 answers
207 views

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 ...
0 votes
0 answers
98 views

Advance a Interpolation

Note; No special knowledge of Pykrige is needed to answer the question, as I already mention examples in the question! Hi I would like to use Universal Kriging in my code. For this I have data that ...
2 votes
0 answers
433 views

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 ...
1 vote
0 answers
75 views

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 $...
2 votes
1 answer
294 views

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} & \...
2 votes
1 answer
174 views

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,...
0 votes
0 answers
176 views

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}...
4 votes
2 answers
387 views

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,...
1 vote
1 answer
291 views

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 ...
1 vote
2 answers
177 views

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}...
1 vote
1 answer
110 views

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+...
4 votes
2 answers
191 views

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 ...
5 votes
3 answers
238 views

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 ...
0 votes
0 answers
233 views

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 ...
0 votes
1 answer
227 views

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 ...
0 votes
2 answers
560 views

How to interpolate a set of points with a continuous closed B-spline curve?

I have been learining the NURBS theory by the classical textbook "The NURBS Book" this year. In the chapter 9, the author introduced the method of non-rational B-spline curve interpolation ...
1 vote
1 answer
251 views

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 ...
8 votes
5 answers
541 views

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}...
0 votes
1 answer
57 views

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 ...
1 vote
0 answers
130 views

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 ($C_0$ ...
3 votes
1 answer
139 views

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 ...
0 votes
1 answer
259 views

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 ...
1 vote
0 answers
48 views

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 ...
2 votes
1 answer
725 views

Interpolation using compactly supported radial basis function

I have been struggling for two days with the following problem. I would like to do a $d$-dimensional interpolation over some data. I tried first to use polyharmonic splines, but when the size of data ...
1 vote
0 answers
169 views

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 ...
4 votes
4 answers
161 views

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 ...
1 vote
1 answer
466 views

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} + ...
2 votes
2 answers
665 views

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 ...
1 vote
1 answer
2k views

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 ...
5 votes
2 answers
2k views

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 ...
11 votes
1 answer
218 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 vote
0 answers
78 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 ...