Questions tagged [error-estimation]
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143
questions
1
vote
1answer
91 views
Question about step in the proof of standard discrete trace inequality
I'm studying from Guermond lecture notes available at https://www.math.tamu.edu/~guermond/M661_FALL_2019/chap12.pdf (see Lemma 12.8( Discrete trace inequality).)
Consider the simple case $p=r$, i.e. ...
4
votes
0answers
70 views
Global reconstruction defined elementwise in a-posteriori error estimator
This question is a follow-up of this previous one. In "Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems" by Georgoulis et al., an error estimator is ...
0
votes
1answer
85 views
How is the integral of a projection over an element $T$ computed in practice? (deal.II related)
I'm studying an error estimator for the equation $\nabla\cdot(\beta u) + cu = f$ and it contains the following term $$||f - cU_h - \Pi(f-c U_h) ||_T$$
where :
$\Pi$ is the local orthogonal $L^2$ ...
2
votes
1answer
736 views
Error in Simpson's 3/8 rule is higher than that of Simpson's 1/3 rule
For a given function $f(x)$, I have tried to find its numerical integral using Simpson's 1/3 and Simpson's 3/8 rules.
I then compare the solution from the numerical quadratures to the analytical ...
3
votes
0answers
55 views
Typo in a-priori error estimate in a Discontinuous Galerkin paper
I'm looking at this famous paper which is available in the link below:
Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
1
vote
1answer
90 views
Finding the source of numerical instability in a electrostatic problem solved by conformal mapping
I'm using conformal mapping to solve a 2D electrostatic problem (calculating the potential $u(x,y)$ in the plane). Let $C_1$ and $C_2$ be two circles at an electric potential $U_1$ and $U_2$, ...
3
votes
1answer
130 views
Proof of R. Verfürth paper on adaptive mesh and bubble functions
I'm studying adaptive meshes, and my professor wrote the following property for a bubble function ( see this scicomp post for the definition I'm using)$b_T$ defined on a triangle $T$.
$$||b_T \phi ||_{...
1
vote
1answer
88 views
Classical global estimate for $H^1$ error
I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$.
$$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
3
votes
1answer
101 views
How to find the optimum finite difference method for derivatives?
Related to: What are the negatives of using higher order finite diference schemes?
Problem: I have some discrete data of a trajectory $x_t$ with errors $\delta x_t$ of a physical system sampled at ...
3
votes
1answer
92 views
Computing the residual in a Dual Weighted Residual (DWR) method
I am in the process if computing the Dual-Weighted Residual (DWR) for a linear PDE with a linear functional but I am struggling with the residual part of the calculation.
For example suppose we want ...
1
vote
2answers
92 views
Stability condition FCTS method
The FTCS method comes from the discretization of a diffusion PDE like this:
$$
a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t}
$$
If I have the ...
0
votes
1answer
44 views
Round-off error step choice
In the Numerical Recipes in section 5.7.- Numerical derivatives the choice of the step size $h$ in the numerical derivative should lead to a difference between $x$ and $x+h$ representable by an exact ...
0
votes
1answer
52 views
Taylor expansion round-off error
In the Numerical Recipes in section 5.7.- Numerical derivatives it's introduced de roundoff error of:
$$
f^{\prime}(x) \approx \frac{f(x+h)-f(x)}{h}
$$
as (with $h$ an "exact" number):
$$
\...
2
votes
1answer
59 views
Initial condition precision
Is there a way to have an estimate of the error propagated on an ode numerical solution by the error of the initial conditions? I suppose this depend on the numerical method used and on the problem ...
1
vote
1answer
137 views
FEM solution for Poisson is not exact at nodes
Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$
I know that the FEM solution (...
0
votes
1answer
152 views
Perturbation problem using Runge-Kutta 4
I'm trying to evaluate the perturbations magnitude between 2 body orbiting a central one in three dimensions. In order to do this I need to have an estimate of the error, which I did using Richardson ...
2
votes
1answer
203 views
Comparing numerical solutions with very different time grids
I've read an article (Long-term integrations and stability of planetary orbits in our Solar system) in which the authors solved the problem of the absence of an analytical solution for the solar ...
0
votes
1answer
89 views
Error too large in leapfrog method for solving the wave equation of a vibrating string
I have been trying to figure out what I did wrong for the last two days. I do not know if I actually did something wrong or if the error is supposed to be this large in usual leapfrog problems. I ...
0
votes
1answer
94 views
What are the advantages and disadvantages of using norm error control in the MATLAB ODE suit?
In MATLAB's ODE suit, there seem to be two basic methods of controlling the Local Truncation Error (LTE) of the ODE which the user can choose from, namely:
The absolute error control (default), ...
4
votes
2answers
83 views
Dividing functions over a wide range-
I try to solve a system of coupled equations, where a very nasty division operation occurs. In fact, I need to compute a derivative of two exponential decaying functions. Let's illustrate this with ...
1
vote
1answer
83 views
How to compute the $L^{2}$ error of the gradient in the Finite Element Method
Let $\Omega\subset \mathbb{R}^{2}$ and $\tau_{h} = \{\Omega_{k}\}_{k=1}^{N}$ be a triangulation of $\Omega$. The $L^2$ error for a FEM approximation $u_{h}$ is given by:
$ || u-u_{h} ||_{L^2} = \sqrt{ ...
3
votes
1answer
73 views
Cauchy Lorentzian simulation on FFT with oscillation
Recently I do simulation on Lorentzian Function with FFT
Lorentzian Function is 2a/(x**2+a**2)
...
0
votes
0answers
57 views
Error in Monte Carlo integration
I am looking for a concise description to help me understand the error for Monte Carlo Integration using Uniform Sampling and Importance Sampling
For Importance Sampling I have that the error is just ...
3
votes
1answer
282 views
Calculating Error for Poisson Equation using Successive Over-Relaxation technique, Python
I am trying to solve the Poisson Equation
$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 32(x(x-1) + y(y-1))$
for a 61x61 grid using Python3 with boundary conditions being $T=...
1
vote
1answer
54 views
Recovery of smoothed continuous stresses using the Z^2 error estimator
I have trouble implementing the $Z^2$ error estimator (The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique by Zienkiewicz and Zhu).
For this, I am ...
2
votes
0answers
157 views
Implementation of Z^2 error estimator in Abaqus for adaptive mesh refinement
Currently, I am working on a remeshing routine for my simulations (Abaqus 6.14-1) using python scripts. The simulation deals with the Brinell indentation test and as the remeshing software I use Gmsh ...
1
vote
1answer
65 views
Solving the pulse propagation using four different FDTD methods gives four different results - Which to trust?
I'd like to simulate the propagation of a pulse, and have different options for solving that.
On the one hand, I can use the non-linear schrödinger equation
$$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp ...
0
votes
1answer
138 views
Question on comparing the accuracy of numerical schemes
This is a follow up to my previous post here
I'm solving the following 1D transport equation .
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
3
votes
1answer
54 views
Nonlinear least squares resolution matrix
For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem,
$$
\...
2
votes
1answer
260 views
About the discrete $H^1$ norm
I need to understand what is the right expression for the "discrete $H^1$-norm of a function v(which of course need to be in $H^1$).
By definition, $$||v||_{H^1}^2 = ||\nabla v||_{2}^2 + ||v||_2^2 $$
...
2
votes
1answer
52 views
Correct relative error for Comparison of levelsets outside a constrained region
I am working on the estimation of tumor extent outside of the region which is visible in MRT/CT/DTI images. I want to compare two methods, which approximate the tumor density profile.
Lets say the ...
0
votes
1answer
196 views
The proper way to assess the error of Jacobi iteration (for 2D Poisson equation)?
Motivation: I'm using 2D regular grid (it's actually a quadtree but I can still treat it as a finite difference thing if I weight-average the solution over smaller scale cells for the purpose of ...
1
vote
0answers
82 views
Determine truncation error of PDE discretization
The equation is $$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$
I'm discretizing this PDE using FVM as follows:
$0=x_0=x_{1/2}<x_1<x_{...
0
votes
1answer
121 views
Actual global error vs theoretical global error: How to combine theory with practice
I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation:
$y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
1
vote
1answer
70 views
Correct weighting in least squares fitting
I am trying to fit some data points $d_i$ to a non-linear model function $m_i$, which depends on a number of fit parameters $f_k$ (I want to determine these) and also on some known, constant values $...
2
votes
1answer
106 views
Step size updating scheme adaptive embedded RK methods
If I have a RK method $y$ of order $p$ and a RK method $z$ of order $p-1$ I have read I can estimate the local error as $r_{n+1} = y_{n+1} - z_{n+1}$. First of all I don't see how this estimates the ...
3
votes
2answers
242 views
Asymptotic error of forward Euler
I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....
1
vote
1answer
314 views
Lower bound for bilinear form in FEM
I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries.
I did some research and found:
$$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
5
votes
0answers
214 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
61 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
45 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
178 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
95 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
52 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
82 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
338 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
58 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
123 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
85 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 ...
