Questions tagged [error-estimation]
The error-estimation tag has no usage guidance.
114
questions
0
votes
1answer
39 views
The proper way to assess the error of Jacobi iteration (for 2D Poisson equation)?
Motivation: I'm using 2D regular grid (it's actually a quadtree but I can still treat it as a finite difference thing if I weight-average the solution over smaller scale cells for the purpose of ...
0
votes
0answers
39 views
Cubature rule in unit Sphere in $\mathbb{R}^{3}$
I need to find the cubature rule for the following integration
$$\int_{S^{2}} f(s,\tilde{s})d\tilde{s} ds,$$
where $S^2$ is the unit sphere in $\mathbb{R}^{3}$.
1
vote
0answers
63 views
Determine truncation error of PDE discretization
The equation is $$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$
I'm discretizing this PDE using FVM as follows:
$0=x_0=x_{1/2}<x_1<x_{...
0
votes
1answer
81 views
Actual global error vs theoretical global error: How to combine theory with practice
I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation:
$y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
1
vote
1answer
62 views
Correct weighting in least squares fitting
I am trying to fit some data points $d_i$ to a non-linear model function $m_i$, which depends on a number of fit parameters $f_k$ (I want to determine these) and also on some known, constant values $...
2
votes
1answer
72 views
Step size updating scheme adaptive embedded RK methods
If I have a RK method $y$ of order $p$ and a RK method $z$ of order $p-1$ I have read I can estimate the local error as $r_{n+1} = y_{n+1} - z_{n+1}$. First of all I don't see how this estimates the ...
3
votes
2answers
158 views
Asymptotic error of forward Euler
I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....
1
vote
1answer
147 views
Lower bound for bilinear form in FEM
I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries.
I did some research and found:
$$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
3
votes
0answers
103 views
$L^2$ norm error estimates of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
-2
votes
1answer
50 views
Calculate accuracy order/rate
I was doing error analysis of numerical scheme and I get $L_1$ error for each grid size with $N$ element. I was searching reference to compute accuracy order/rate from that error data but doesn't find ...
2
votes
1answer
36 views
Log-transformation of decision variables in parameter estimation
I am trying to find the diffusion coefficient ($D$) and the partition coefficient ($KLP$) using experimental data of desorption of a pollutant from a film into a liquid. This process can be modelled, ...
5
votes
2answers
160 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 ...
3
votes
2answers
92 views
Error control and sequence acceleration at the same time
In a posteriori error control for solving ODEs, one typically computes two different approximate solutions, one of which being "more accurate" and one of which being "less accurate". If $y_q^{n+1}$ is ...
1
vote
0answers
51 views
Decrease in slope during convergence analysis
I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides.
I used 5 refinements:
$dx = dy = dz = ...
4
votes
1answer
80 views
Error on a integral quantity with noise
First of all sorry if this is the wrong place to ask this question, I went to a few stack sites and thought here it would be more suitable.
My problem:
I have a physical quantity $F$ that depends on ...
1
vote
0answers
93 views
Global truncation error behavior at fixed time step
I am trying to solve the following diffusion equation problem:
$\frac{\partial f}{\partial t}=\frac{\partial (D\frac{\partial f}{\partial x})}{\partial x}+S$
$D=1+x^{2}+\sin(x)$
$f(x,0)=1 , f(0,t)...
1
vote
0answers
55 views
Numerical analysis: Chebyshev coefficient representation error
If $x_k$ are the Chebyshev nodes, that is for $n \in \mathbb{N}^*$, we have $x_k = \cos(\pi \frac{2k + 1}{n})$. Now suppose you have approximated values of $x_k$ and $x_{k'}$ for $k,k' \leq n$. In the ...
1
vote
3answers
91 views
Discretization Error amplification instead of stagnation to machine precision
I wrote a code on Python 2.7.5 to solve numerically the following differential equation.
$\frac{\partial^2f}{\partial x^2}=-S$
$S=\pi^{2}\sin(\pi x)$
S is chosen that way in order to have $f= \sin(\...
1
vote
0answers
16 views
Procedure to identify characteristic properties of unknown functions in a DAE model
I have a system of 1st order odes given by
$$
\dot{x_1}(t) = \alpha_1 f_1(x_1,t) + \beta_1 u(t) \\
\dot{x_2}(t) = \alpha_2 f_2(x_2,t) + \beta_2 u(t)
$$
They are constrained by an algebraic equation
...
3
votes
0answers
84 views
Stabilizing online average calculation
In Knuth, the following method for computing an average is presented:
\begin{align*}
M_{n} = M_{n-1} + (x_{n} - M_{n-1})/n
\end{align*}
(See here, if you don't have TAOCP.)
Assuming the samples all ...
2
votes
1answer
88 views
Taylor expansion of error - Finite elements approximation
In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the ...
4
votes
1answer
93 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 ...
4
votes
2answers
241 views
Error propagation through an FFT
If I take the Fourier transform of data $x \pm \sigma$, is there a standard approach to what the error in the outputs will be? Would the best way be a direct evaluation of the upper and lower bounds?
3
votes
1answer
78 views
FEM problem: how to get a feeling for size of problem
The following problem is given:
$$ - \bigtriangleup u = f, \quad \partial_n u = g \quad \text{on } \Gamma_N, \quad u = 0 \quad \text{on } \Gamma_D $$
with $\Gamma_N$ or $\Gamma_D$ denoting the ...
-3
votes
1answer
59 views
Modulus (absolute) of a function, its quadrature, and relevance of zeros
Modulus of a (discretized) function, $|f_h(x)|$, where $h$ refers to the mesh spacing, would, in general, have zeros, and those zeros would not necessarily lie exactly at the mesh points.
A naive <...
1
vote
0answers
88 views
Mesh error quantification using error norm for a sharp logistic function [closed]
Consider the following logistic function in 1D for the domain $0$ to $1$.
$$ f(x) = \frac{1}{1 + \exp(-50(x-0.5))} $$
Let's say I resolve this on two meshes:
first with 11 equispaced points (0.0, 0....
0
votes
1answer
30 views
Propagation of error in fitting two sets of data to each other
I have two sets of experimental data: $\phi(t)$ and $I(t)$. In theory they are related to each other as: $\phi(t) = nI(t) $. By fitting these curves together I can find the value of $n$ (which is a ...
0
votes
1answer
127 views
Error measure for a simple finite difference scheme
I am responsible for assisting with certain error measurements for an FE program. The idea is to set up some benchmark comparisons which we can use for future development of the program.
To simplify ...
1
vote
0answers
33 views
Round-off errors in ratio of Gaussian functions
I would like to understand how division of round-off errors can lead to large errors???
I am using a method to measure the frequency dependent dissipation in simulations performed by Discontinuous ...
4
votes
2answers
773 views
What does optimal convergence rate mean? How to prove whether a numerical method, e.g. FEM, has an optimal convergence rate or not?
I searched online about the way optimal is defined mathematically,but without any information acquired? How to prove whether a numerical method, e.g. FEM, has an optimal convergence rate or not? ...
2
votes
2answers
237 views
error estimator VS error indicator in the context of FEM error estimation
Is an error indicator an index for size measurement of something? And how is the relationship between the error estimator and error indicator? Is the error indicator just used for the (e.g. derivative)...
1
vote
0answers
33 views
Deterministic method to compute “Process noise covariance matrix, Q” for a Kalman filter when parameter variations of the model is known apriori
I am implementing a Kalman filter (for a linear ODE system for now).
My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
2
votes
0answers
89 views
Precision not improving by decreasing step-size in nonlinear Schrödinger
I tried to simulate soliton propagation by solving the nonlinear Schrödinger equation using the split-step Fourier method. The following is an example of the Matlab code copied from a textbook.
...
1
vote
0answers
26 views
How group similar data in a single classes and reduce the error
I have vehicles gps information in my real time traffic application. Averaging that information I know the speed of every road at any time. The problem is that is too much data to send back to the ...
2
votes
0answers
45 views
Quick evaluation of floating point Absolute Error
I need to to find a quick and dirty way to estimate the absolute error introduced by a series of agebraic operations of IEEE single precision floating point numbers, a pessimistic result is ok.
The ...
2
votes
1answer
66 views
Dirichlet term in error estimations
I am working on a method based on moving least square approximation where shape functions do not satisfy Kroneker Delta property. So Dirichlet boundary condition should be enforced.
I usually used ...
1
vote
0answers
150 views
Estimate $L_2$ norm of a elliptic problem with unknown exact solution on finite element method
I have the elliptic problem $$-\Delta u = 1,\,\,\Omega\subset\mathbb{R}^2$$
with $u=0$ on $\partial\Omega,$ with $\Omega=[-1,1]^2\backslash([0,1]\times[-1,0])$ and I want to estimate the $L_2$ error ...
1
vote
2answers
94 views
Relationship between global and local error?
In some cases I have seen that if the local error is:
$Err = O (\Delta t^{p+1})$
where the global error is p. So if local error is 3, global will be 2.
Does somebody know where it comes? For ...
7
votes
2answers
275 views
How can I derive the a priori error estimate for a symmetric bilinear form using lagrange finite elements?
Suppose that I'm solving the poisson equation by the finite element method by lagrange elements. I know that the error can be measured in a variety of ways, depending on which norm you choose. For a ...
1
vote
1answer
339 views
matlab lsqcurvefit parameter estimation journey
The lsqcurvefit solution in matlab converges at different solutions depending upon the initial guess:
Surface represents the error (SSE) between model and data at various combinations of parameters ...
4
votes
1answer
214 views
$L^2$-error in FEM: how to compute integral over reference element?
I have the following problem. The domain is $(0,1)$ and we consider a uniform triangulation on $\hat{\Omega}$ with elements $K_i = [i/N,(i+1)/N]$ and $X_h^1$ the linear finite element space. I wrote ...
5
votes
0answers
150 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 ...
1
vote
1answer
4k views
What is “tolerance” in ODE45 in Matlab?
I have used ode45 in Matlab. And, it is my understanding that the 4 and the 5 are for the order of the global and local error, respectively. So, the global error ...
2
votes
1answer
177 views
What is the error associated with Fornberg's algorithm?
Bengt Fornberg derived a general way to compute the weights for arbitrary finite difference schemes in two papers: his 1988 paper and (better) his 1998 paper.
What are the numerical errors ...
1
vote
0answers
233 views
Limit to precision of step-size
When solving an equation of the following form:
$$
\begin{aligned}
\frac {\partial A}{\partial t} &= EB - A
\\
\frac {\partial B}{\partial t} &= EA - B
\\
\frac {\partial E}{\partial t} &...
0
votes
2answers
82 views
how to estimate convergence orders? what about this formula?
I have solved a PDE with an analytical equation. Through operator splitting I divided the PDE into one PDE and one ODE, using a sequential approach.
Finally for different $dt$, I got euclidian norm ...
1
vote
1answer
49 views
How can I compute the difference between shape function and dual solution in dwr?
I am trying to find error estimation based on weighted residual technique:
$$Q(e_h)=\sum_{k \in {\cal T}}\eta_k, $$
where
$$\eta_k=\int_kR(z-v_h)d\Omega+\int_{\partial k}J(z-v_h)ds,$$
or in the weak ...
4
votes
0answers
106 views
Stable computation of $\log\sum x_i$ from $\log x_i$, with many terms
Kahan's summation algorithm is a method to compute sums:
$$\sum x_i$$
with many terms, without significant error.
I want to do this with very large numbers, and instead of the numbers themselves, I ...
3
votes
2answers
134 views
Adaptive tolerance for nested quadrature
I am doing a nested integral via quadrature. To give a definite example, lets say:
$$
\int_0^2dx \left[x + \int_0^x dy \, 2y\right]
$$
So effectively I'm integrating $x + x^2$ from 0 to 2 (although ...
1
vote
0answers
50 views
Square error estimate for adaptive mesh refinement
In a particular implementation(Finite volume advection using upwind) of adaptive mesh refinement the error square estimate for a cell C is given as
$$
\sum_{i = x,y,z} vol * \frac{1}{12} * h^{2} * (\...
