Questions tagged [error-estimation]

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147 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 ...
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49 views

What error expect in practice for numerical method

I've implemented an Adams Bashforth method with 4 steps, and I was checking the global error of the solution I obtained. I expected the error to be about $O(\Delta t^4)$, so $O(10^{-10})$ for a $\...
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1answer
49 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 ...
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1answer
65 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), ...
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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 ...
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1answer
61 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{ ...
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1answer
70 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) ...
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44 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 ...
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1answer
142 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=...
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37 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 ...
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1answer
46 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 ...
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94 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 ...
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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 ...
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1answer
122 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
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1answer
45 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, $$ \...
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1answer
143 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, $$||v||_{H^1}^2 = ||\nabla v||_{2}^2 + ||v||_2^2 $$ ...
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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 ...
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1answer
143 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 ...
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76 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_{...
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1answer
106 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)=...
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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 $...
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1answer
99 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 ...
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2answers
214 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
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1answer
252 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(...
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166 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{{\...
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1answer
60 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 ...
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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
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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 ...
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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 ...
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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 = ...
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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 ...
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0answers
246 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)...
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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 ...
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3answers
119 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(\...
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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 ...
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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 ...
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1answer
108 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
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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 ...
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2answers
914 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?
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1answer
83 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 ...
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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 <...
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0answers
109 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....
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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 ...
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1answer
192 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 ...
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0answers
36 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 ...
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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? ...
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2answers
349 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)...
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0answers
39 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 ...
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0answers
117 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. ...
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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 ...