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0
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0answers
11 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
75 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 &= ...
2
votes
1answer
91 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
53 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
vote
0answers
30 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) + ...
5
votes
1answer
54 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 ...
9
votes
2answers
107 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
89 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
106 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 ...
0
votes
0answers
18 views

Directional/axes errors in structured grid?

Do you know any good article/review out there on the problematics of direction/axis preference coming from structured grid (cartesian ones for example) and depending on the interpolation scheme?
2
votes
3answers
134 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
64 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
vote
0answers
66 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
vote
2answers
88 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 ...
4
votes
3answers
166 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
102 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
votes
0answers
38 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
94 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) + ...
10
votes
1answer
668 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
100 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
326 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
50 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
46 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
41 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 ...
6
votes
2answers
204 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
198 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
198 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: ...
2
votes
2answers
87 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
votes
2answers
391 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
31 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
276 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 ...
5
votes
1answer
84 views

Can we compare the speed of convergence of two different iteration methods of same order looking at their error estimates?

I have a two iterative method for approximating the inverse of given square matrix $A$ whose error terms are given as follows Error estimate of method $1$: $\lVert A^{-1} - X_{k}\rVert \leq ...
4
votes
1answer
143 views

Error bars for pair-correlation function

I have obtained some data from neutron diffraction for some material samples. The "rawest" form of the data is the structure function $S(Q)$. We can choose a variety of different Q-maxes when ...
1
vote
1answer
126 views

How can I quantify the error of FFT-based poisson solvers?

I have an FFT code that solves a particular case of the steady Euler equations where a Poisson equation is solved, what is a good way to quantify the error? Is what I am doing ok? Since I do not have ...
4
votes
2answers
155 views

Problem Condition and Algorithm Stability

Consider 2 mathematical problems: $$ f_1(x) = a - x \\ f_2(x) = e^x -1 $$ The condition number for a function is defined as follows: $$ k(f) = \left| x \cdot \frac{f'}{f} \right| $$ Lets analyze ...
17
votes
2answers
446 views

Why do equi-spaced points behave badly?

Description of experiment: In Lagrange interpolation, the exact equation is sampled at $N$ points (polynomial order $N - 1$) and it is interpolated at 101 points. Here $N$ is varied from 2 to 64. ...
1
vote
1answer
115 views

Error analysis of WENO scheme

I have three questions regarding WENO schemes 1) How to actually compute the smoothness indicators $\beta_j$ for required order of polynomial? Any reference which explains the algorithm will be ...
12
votes
3answers
501 views

Numeric Quadrature with Derivatives

Most numerical methods for quadrature treat the integrand as a black-box function. What if we have more information? In particular, what benefit, if any, can we derive from knowing the first few ...
2
votes
0answers
36 views

Propogated Error in Mesh Interpolation

I am working with a code that solves diffusion/reaction equations on a 2D unstructured mesh. Due to the stiffness of some of the processes, I start with time steps near 1e-13, and end with a final ...
2
votes
1answer
114 views

Diffusion-Transport problem FEM

I was looking at a book of FEM on problems of Diffusion-Transport. $$-div(\mu \nabla u) + b \cdot \nabla u + \gamma u = f \qquad in~\Omega$$ $$u = 0 \qquad in~\partial\Omega\text{ (in the ...
32
votes
4answers
909 views

Scientific standards for numerical errors

In my field of research the specification of experimental errors is commonly accepted and publications which fail to provide them are highly criticized. At the same time I often find that results of ...
5
votes
1answer
129 views

a priori error analysis of cell-centered finite-volume methods

I'm using the cell-centered finite-volume method (for example Morton, Numerical solution of convection-diffusion problems, Chapman&Hall, 1996) to discretize the advection-diffusion equation and ...
2
votes
0answers
78 views

function over conditional probability

I need to create a scoring model out of estimated conditional probability functions for two events, A and B. Let 0.5 be the threshold value. Ideally, the probability is in the interval $[0,0.5)$ for A ...
1
vote
1answer
329 views

How to find the number of principal components that lead to the smallest generalization error?

I am working on a paper part of which is the application of validation rules to find how many principal components give us the least generalization error. The concept goes more or less like this: ...
6
votes
2answers
1k views

When to stop Gauss-Seidel-iterations?

I want to have an estimation, that my solution has an error, let's say less than 1e-8. Usually, I stop the Gauss-Seidel algorithm, when the residual is "small enough" and this is already the problem. ...
7
votes
3answers
2k views

What norm to choose when?

Recently, I saw this question: how to measure the error of a finite difference method I am student of simulation sciences and unfortunately, for me, it's totally unclear, what norm to use in what ...
3
votes
1answer
212 views

Integral average approximation and error bounds

I'm looking into integrals of the form: $$\int_a^b {f(x)g(x)dx}$$ Where $f(x)$ is unknown, but it's integral is: $$\int_a^b {f(x)dx}=F$$ It's been suggested to me that one could approximate this ...
9
votes
2answers
198 views

What about this simple error estimate for linear PDE?

Let $\Omega$ be a convex polygonally bounded Lipschitz domain in $\mathbb R^2$, let $f \in L^2(\Omega)$. Then the solution of the Dirichlet problem $\Delta u = f$ in $\Omega$, $\operatorname{trace} u ...
4
votes
1answer
217 views

finite difference methods and global error

I was going through my notes on different finite difference methods and came across something I don't quite understand. I have code that will calculate an approximate solution we can call this ...