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.
174
questions
1
vote
1answer
85 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 $r$.
$$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
1
vote
2answers
163 views
Sum of norms over elements is not equal to norm over the whole $\Omega$
In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$:
$$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
4
votes
2answers
157 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
3answers
161 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
0answers
98 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
1answer
129 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 ...
1
vote
1answer
88 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 ...
0
votes
1answer
40 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 ...
8
votes
5answers
261 views
Polynomial interpolation on a regular hexagon
Background
We know a function $f$ on the vertices of a regular hexagon, as follows
$$\left( 1, \ 0, \ f_{0}\right), \ \left( \frac{1}{2}, \ \frac{\sqrt{3}}{2}, \ f_{1}\right), \ \left( - \frac{1}...
1
vote
0answers
43 views
Spline interpolation for vector-valued data in 3D space
I have output from a 3D linear elasticity finite element simulation which uses linear tetrahedral elements, such that the displacement is continuous over the nodes but the gradient is not ($C_0$ ...
3
votes
1answer
122 views
At which stage to interpolate?
Assume I have a multidimensional grid $G$. I consider two functions $f(x)$ and $g(x)$. I have solved the values for the functions over all grid points $x \in G$. Let me now be interested in some third ...
0
votes
1answer
102 views
Finite element interpolation
I have a finite element solver that I implemented in MATLAB. I am calculating a specific potential function that is constant in each tetrahedral element. The question is, I want to be able to ...
1
vote
0answers
38 views
Interpolate location based on triangular sensor array
There are 3 sensors (A, B, C) on a plane, located in the corners of a (known) equilateral triangle.
I want to calculate the (2D) location of an object (X) inside that triangle.
One sensor returns one ...
0
votes
0answers
62 views
Interpolation of a Concave Mesh
I would like to have an algorithm that interpolates the values attached to nodes in a concave mesh mesh.
To be me precise, assume we have a point cloud P (e.g. in 3 dimensions) and a list of edges E ...
3
votes
4answers
138 views
Getting torsion and curvature out of ODE solution skeleton
Suppose I have solved an ODE $v'(t) = f(t,x)$ via some adaptive stepper, such as RK4 or Dormand-Prince, generating a list of points $\{(t_i, v_i, v_i' = f(t_i, v_i))\}_{i=0}^{n-1}$.
I wish to use this ...
1
vote
1answer
130 views
Factorization of cubic spline interpolation matrix
In cubic spline interpolation, we use the set of knots and function values $(x_i,y_i),i=1,...,n$ to construct a (tridiagonal) system of equations for the unknowns $\sigma_i$:
$$
h_{i-1}\sigma_{i-1} + ...
2
votes
2answers
206 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
1answer
660 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 ...
1
vote
0answers
71 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 ...
11
votes
1answer
198 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
1answer
57 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
votes
1answer
376 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(\...
1
vote
3answers
372 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
votes
2answers
111 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
votes
2answers
875 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 ...
3
votes
0answers
140 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
vote
1answer
121 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 ...
1
vote
1answer
3k 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.
...
0
votes
0answers
43 views
extrapolation/interpolation in fmincg.m
Can you tell me these equations come from where in MATLAB fmincg.m?
...
0
votes
1answer
86 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 ...
0
votes
1answer
157 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
votes
1answer
2k 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.
...
1
vote
0answers
90 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 ...
0
votes
0answers
39 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
votes
0answers
117 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}) = \...
16
votes
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 ...
5
votes
0answers
646 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 ...
0
votes
1answer
581 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
votes
2answers
176 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 ...
8
votes
1answer
189 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 ...
1
vote
1answer
63 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 ...
5
votes
0answers
74 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 ...
7
votes
3answers
165 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) ...
2
votes
0answers
33 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
votes
1answer
2k 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 ...
5
votes
0answers
67 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
votes
2answers
151 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
48 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
votes
1answer
109 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 ...
0
votes
1answer
2k 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
...
