# Questions tagged [error-estimation]

For questions about determining the error caused by specific computational procedures, approximations, or numerical representations.

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### Which dense matrices are hard to invert?

Suppose I'm solving $Ax=b$ for dense $m\times d$ matrix $A$. For which $A$ is this hard to do? More concretely, is there any work on estimating the error after $k$ steps of iterative solver, $k\le d$, ...
1 vote
77 views

### Geometrically nonlinear finite element problem and mesh distortion

In my research on topology optimization of fluid-structure interaction problems (2D), I am using a geometrically nonlinear model to represent the structure. Body/surface forces are extracted from the ...
41 views

### How to compare the bias between the two contour plots?

I wish to compare the value of the normalized concentration (c+), between two contour plots (one is experimental and another is a simulation). If I don't have a c+ value from the same point on both ...
55 views

### Finding optimal values from multiple parameter estimation runs

I've performed a parameter estimation repeat (i.e. 1000 parallel runs with the same initial values of parameters). I am trying to estimate ~20 parameters using measurements from experiments. After ...
64 views

### Convergence of FEM on curved boundaries, and inhomogenous boundary data

In a smooth domain in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ let's consider $-\Delta u = f$ with $u=g$ on a part of the boundary and $\partial_\nu u = w$ on another part of the boundary, which is far ...
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### finding discretization error in Burger equation

I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method ...
305 views

### How does non-dimensionalization improve the behavior of ODE solvers?

I have a set of coupled ODEs that I'm solving numerically. The independent variable is time and runs from values of $10^{15}$ to $10^{17}$ in units of seconds. The state variables in their usual ...
1 vote
77 views

### A priori estimates in finite elements for inhomogeneous heat equation

Consider the problem $$\partial_t u-\Delta u = f\\ u(\Sigma_1)=f_D\\ \partial_\nu u (\Sigma_2)=f_N\\u(0)=u_0$$ where the sides of the space-time cylinder $\Sigma_i$ are disjoint (one of them could be ...
60 views

### How to maximize effectiveness of 2 check bits overseeing 6 data bits?

Having a single parity bit gives a Hamming distance of 2, so only one single bit can be corrupted to reliably detect an error. The next step I know of would be a Hamming code, which uses at least 3 ...
130 views

### "A posteriori" estimates for finite difference methods

Suppose I have a PDE in a rectangular domain that I am solving numerically via a finite difference method. How do I answer the question, "How fine do I need to make the grid to so that my ...
182 views

### Accuracy loss in single-precision Euclidean norm computation

I do hydrodynamics simulations with Fortran and recently I met with this issue: I have a single-precision array b of length ...
46 views

### What is the ERRCON parameter in rkqs?

Ive take a course in computational physics and was asked to implement some numerical methods to solve ODES. I was reading up on the algorithms described in the textbook: NUMERICAL RECIPES IN FORTRAN. ...
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1 vote
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### 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
279 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 (...
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### 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 ...
228 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 ...
229 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 ...
200 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), ...
89 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
523 views

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: $|| u-u_{h} ||_{L^2} = \sqrt{ ... 3 votes 1 answer 106 views ### Cauchy Lorentzian simulation on FFT with oscillation Recently I do simulation on Lorentzian Function with FFT Lorentzian Function is 2a/(x**2+a**2) ... 0 votes 0 answers 133 views ### Error in Monte Carlo integration I am looking for a concise description to help me understand the error for Monte Carlo Integration using Uniform Sampling and Importance Sampling For Importance Sampling I have that the error is just ... 3 votes 1 answer 857 views ### Calculating Error for Poisson Equation using Successive Over-Relaxation technique, Python I am trying to solve the Poisson Equation$\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
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 ...