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0
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0answers
17 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} * (\...
0
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0answers
20 views

monitor functions for mesh generation: error estimate by FD or by FEM?

I am using a local truncation error estimate as the monitor function for adaptive mesh refinement that comes from a finite difference(FD) scheme and its values are available only at nodal points. ...
0
votes
3answers
196 views

$\exp(\ln(x))-x\neq0$?

Analytically, the expression $$\exp(\ln(x))-x \enspace,$$ should give 0. However, in Matlab, it does not. x = linspace(1, 10, 10); exp(log(x)) - x; for $x \in ...
4
votes
0answers
39 views

Is resampling more accurate than block average for statistical analysis of data?

I'm working in laboratories where molecular dynamics data are almost always analysed usign block average as stated in the famous Allen and Tildesley book. We divide the datas in blocks of size $M$ on ...
1
vote
2answers
43 views

MATLABs double arithmetic

this is a classical problem, but I need help to pinpoint what I am missing. Problem: In MATLAB (exp(1) + 10^12) - 10^12 gives you a double which equal to e, up to 5 correct digits. But I thought ...
1
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0answers
20 views

Gradient convergence on a checkerboard domain in finite element

I've got an elliptic problem in a domain with two materials in a checkerboard arrangement. Both materials are highly dissimilar, being one much more compliant than the other. If I use classic linear ...
0
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0answers
18 views

Methods for discriminate model characteristics

Reading this recent trend, I was wondering if there are available methods to detect the qualitative features of experimental data, in order to get some clue about how to model those data. For instance,...
0
votes
1answer
90 views

What should I put on the paper to show the correctness and convergence of my solution?

I am using FEM to do an assignment on a heat conduction problem on a complex domain, which needs me to get the variation of the temparature distribution subject to the variation of boundary conditions,...
11
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1answer
200 views

What are differences between 'a priori' and 'posteriori' error estimate in numerical analysis?

I have learnt about Finite Element Method (also a little on other numerical methods) but I don't know what are exactly definition of these two errors and differences between them?
3
votes
1answer
111 views

Convergence of the second derivative of the finite element solution

Let $u_h$ be the finite element solution of a fourth order equation (like biharmonic equation), using polynomial degree two. If the convergence rate of $u_h$ is $2$, what is the convergence rate of ...
4
votes
0answers
74 views

discrete definitions of curl $\nabla \times F$?

I have some data defined in an array (an image) and I need to find the curl of a certain function. Wikipedia has an integral definition of curl that I like, maybe it can be discrete. $$ \nabla \...
3
votes
1answer
88 views

Using kalman filter when samples don't have time index

Assume $X$ and $N$ are two sets of observations from two different normal distribution, where $X$ represents clean data and $N$ represents noise; and $A$ a projection matrix of a filter and the ...
8
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2answers
297 views

Difference between l2 norm and L2 norm

What is the difference between the $l^2$ norm and the $L^2$ norm. I can not find a definitive reference. Wikipedia uses them interchangeably.
0
votes
1answer
105 views

How to avoid the round-off errors in the larger calculations?

Now I need to sum up more than one thousands of terms and then make the four-dimmensional integral in my Fortran program. I found that there are some numerical errors. Can you give me some suggestions ...
0
votes
1answer
109 views

Calculation of error

I have written a code in which I find the approximation of the solution of this elliptic problem. I calculated the error using the following part of code: http://pastebin.com/7b5mmuRW but I get the ...
1
vote
2answers
129 views

How can one describe the accuracy of a Runge-Kutta method?

I am solving a nonlinear ODE with a regular singularity using MATLAB ODE45 or ODE113. I am wondering what precision and accuracy they have and what one can say about the numerical error. The idea ...
3
votes
1answer
122 views

GSL linear algebra LU/determinant precision

I am working with symmetric matrices of order $n \times n$ where $n \leq 50$. The diagonal elements of my matrices are a fixed number $d$ and the off diagonal elements are limited to two small numbers ...
1
vote
1answer
45 views

problem about simulating recurrence relation

We have the recurrence relation: $5x_{n+1}-x_n=\frac{1}{3}$ $x_0=\frac{1}{12}$ solution: $y_h=(\frac{1}{5})^nC$ $y_p: 5A-A=\frac{1}{3} $ $ A=\frac{1}{12}$ $y=(\frac{1}{5})^nC+\frac{1}{12}$ ...
1
vote
1answer
106 views

Evaluate numerical error estimates

I am developing a finite element simulation and want to evaluate the errors in $H^1$ and $L_2$ norms. The problem is the classical Poisson equation, with Dirichlet B.C.: $$-\Delta u=f\mbox{ in }\...
0
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1answer
59 views

Using numerical integration to calculate Fourier series' coefficients

I am using Fourier series to find the analytical solution to the 2D heat equation. The problem is that the integrals which are used to calculate the coefficients of the series cannot be solved ...
0
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0answers
12 views

Assigning new values based on original guesstimates and ranking / ordering?

Lets say we have two things as input, $N$ scalars (measurements) that we know are erroneous to some degree (i.e. the correct values are somewhat similar). In addition, we also have a roughly more ...
1
vote
1answer
278 views

Correct way of computing norm $L_2$ for a finite difference scheme

I am computing the rate of convergence of my finite difference scheme in norm $L_2$. Which is the correct way to compute it? This: \begin{align} L_2 &= \frac{1}{N}\sqrt{\sum_{j=1}^N(u^{...
2
votes
1answer
543 views

Why a finite difference scheme would give second order of accuracy in norm L2 but 1.5 with L1 (while 1 with Linf)?

My finite difference scheme for the 2D Euler equations is second order accurate in theory, since all the terms are second order accurate, with the advective terms being third order. So I expect a rate ...
0
votes
1answer
56 views

Best path for estimation

I have a Cartesian grid (100x100) in which some of the points are known (30 out of 10,000) and the rest are unknown. I want to use the known points and estimate the other cells. Is there any ...
1
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0answers
39 views

Error analysis in evaluation of daub4 scaling function

Let the Daubechies 4-coefficient scaling function $\phi\in C_{0}([0,3])$ be defined by \begin{align} \phi(x) &= \frac{1+\sqrt{3}}{4} \phi(2x) + \frac{3+\sqrt{3}}{4}\phi(2x-1) + \frac{3-\sqrt{3}}{...
5
votes
1answer
109 views

Quantify integration error of scipy ode / ODEPACK

I am trying to integrate a 2nd order ODE with potential several singularities using the lsoda solver wrapped in scipy.integrate.ode(). I would like to put an error bar on the solution or at least ...
10
votes
2answers
131 views

Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them? (...
2
votes
1answer
115 views

Error of interpolating polynomial

$f(x)= \frac{1}{1+x^2}$ and when I computed the interpolating polynomial of 5 equally spaced points in [-5,5] I got $ p(x)= 0.0053x^4 -0.1711x^2 +1$ Now I need to estimate the error in the ...
7
votes
2answers
125 views

diagonalization of matrix - omitting small matrix elements

I was wondering whether there is some theorem that allows me to put an upper bound on the error introduced by omitting small matrix elements from a matrix before diagonalization. Let's assume we ...
2
votes
3answers
157 views

Compute accuracy order as mesh gets refined?

I have implemented a FVM code and now I need to plot the accuracy of the method as the mesh gets refined. Having a very fine mesh, my idea is to compare what is the error between the coarser and fine ...
0
votes
1answer
91 views

How to give a simple estimation of errors for results obtained from 4th order Runge-Kutta

This question is a follow up of another one I have asked a while ago. I have successfully implemented my problem using odeint library and I get the results I expect. However I would like to give an ...
1
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0answers
69 views

Iterative algorithm prove precise conditions for convergence

Question: Consider the iterative improvement algorithm below. Starting with $Az_i = r_i$ and $(A + E)\hat{z}_i = r_i$ derive a formula showing how the absolute error in the $(i + 1)^{st}$ iterate $\...
1
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2answers
99 views

When are two vectors considered “close”?

I want to check numerically if a certain vector relation like $$ \alpha_1v_1+...+\alpha_kv_k=c \ (1)$$ holds (where $v_i,c$ are vectors of $100$ or more components). For this, I use least squares ...
5
votes
2answers
252 views

Machine precision and local error

I'm working with an RKF45 integrator that I have programmed using CUDA C++ on my GPU and am pondering a few questions as I'm trying to track down some issues with my code. I'm using double ...
4
votes
4answers
156 views

Measurement error library

Is there a python library that would keep track of uncertainty in measured data? i.e. if I put in a figure of a±b is there an easy way to track the propagation of error through calculations.
0
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0answers
40 views

Error estaimate on points

Let $\Omega$ is a domain for example square, and $N$ points is uniformly distributed on this domain. I want to solve a Poisson equation on this domain using a numerical method. The exact solution is ...
3
votes
1answer
225 views

How do error estimates scale for multidimensional cubature?

Suppose I have a quadrature method with a theoretical error estimate that scales with the number of points as $e(N)$: $$\int_a^b f(x)\mathrm{d}x = \sum_{i=1}^N w_i f(x_i) + \mathcal{O}\bigl(e(N)\bigr)...
11
votes
1answer
873 views

What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?

I am solving a system of two coupled PDE's in two spatial dimensions and in time computationally. Since the function evaluations are expensive, I would like to use a multistep method (initialised ...
1
vote
1answer
154 views

Error propagation on GSL eigenvalues computation

The problem comes from the need to estimate the error propagation of the spectral norm computation of a square matrix $A$ of which I know the components' $A_{ij}$ absolute error. The fundamental step ...
13
votes
4answers
384 views

Estimating hardware error probability

Say I run a supercomputer computation on 100k cores for 4 hours on http://www.nersc.gov/users/computational-systems/edison/configuration, exchanging about 4 PB of data over the network and performing ...
4
votes
0answers
69 views

Methods for integrating black box functions on a non-uniform grid

If i have some function expressed as points on a non-uniform grid (I'm specifically interested in logarithmic grids, but general results are also interesting), and I want to integrate it, I believe ...
3
votes
1answer
56 views

a posteriori error estimation for skewed elements

I'm working with error estimates for Poisson's equation of the form $$\mathcal{E}^2_T = h_T^2\|-\Delta u - f\|_{L^2(T)} + \sum_{e\in \partial T} h_e\|n\cdot \nabla u\|_{L^2(e)}$$ where $T$ is an ...
0
votes
0answers
45 views

Approximating forward Error function

and i have i question. i was given equation $$f(x) = 0$$ $$f(x) = cos(\frac{x}{50}) - \frac{1}{\sqrt{2}}$$ and the approximation root $$x_a $$ such that $$\vert f(x_a)\vert < \epsilon =0.001$$....
6
votes
1answer
218 views

Proving convergence of adaptive finite elements - min res FEM?

There's a body of work out there dealing with the discrete convergence of adaptive finite element methods using error estimators. Most deal with proving the property $\|u-u_{k+1}\|_U \leq (1-\alpha) \...
2
votes
1answer
326 views

Error norm in finite difference calculation

I've used an explicit finite difference scheme to model the 1D time dependent temperature distribution in a friction weld. I want to now verify the consistency and convergence of my algorithm. I have ...
6
votes
1answer
245 views

Computing $\sin(\pi/2)$ numerically

I'm trying to understand the types of numerical errors, to do this I want to calculate $\sin(\pi/2)=1$ numerically. To do this I use the Taylor series of $\sin(x)$ in 0: $$\sin(x)=x-\frac{x^3}{6}+\...
2
votes
2answers
106 views

Normalizing error when data passes through zero

Given a time-dependent measure of error, $e(t)$, and a time-dependent function to normalize it with, $f(t)$, one would report the normalized error as $$E(t) = \left|\frac{e(t)}{f(t)}\right|.$$ ...
4
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2answers
584 views

How can you calculate percent error in tensor approximations?

I have a matrix A which is an approximation to the known matrix B. Both matrices are square, 3x3 matrices and, in this case, are ...
3
votes
1answer
32 views

Known Algorithm to compute errors of given nature

I have three sets of data, measured by three different devices: A,B and C of air balloon whose fall is influenced by wind and ...
8
votes
2answers
304 views

Bounding the relative error of derivative given relative error of the function

Suppose a function $f$ can be computed such that the bound on the relative error is $R$ i.e. $f^-(x) = f(x)(1+r)$ where $f^-$ and $f$ are respectively the computed and exact value $f$ and $|r| \leq ...