Questions tagged [error-estimation]
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128
questions
0
votes
1answer
42 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
59 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
82 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
60 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
68 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
36 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 ...
1
vote
1answer
124 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=...
0
votes
0answers
30 views
Error analysis of Modified Lentz's method
In Numerical Recipes, the authors state:
There is at present no rigorous analysis of error propagation in Lentz’s algorithm.
This statement is now ~15 years old, so I wonder has this gap in the ...
1
vote
1answer
36 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
83 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 ...
2
votes
1answer
60 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
$$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp ...
0
votes
1answer
115 views
Question on comparing the accuracy of numerical schemes
This is a follow up to my previous post here
I'm solving the following 1D transport equation .
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
3
votes
1answer
44 views
Nonlinear least squares resolution matrix
For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem,
$$
\...
2
votes
1answer
136 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
122 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
74 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_{...
1
vote
1answer
100 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
66 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
95 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
211 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)....
2
votes
1answer
249 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:
$$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
5
votes
0answers
153 views
$L^2$ norm error estimates of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \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
58 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
39 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
171 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
94 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
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
235 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
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0answers
57 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
115 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
102 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
101 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
845 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
82 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
63 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
108 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 ...
1
vote
1answer
184 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 ...
2
votes
0answers
35 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 ...
5
votes
2answers
1k 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? ...
3
votes
2answers
341 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
36 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
114 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
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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
47 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
70 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 ...
