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Questions tagged [error-estimation]

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

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Understanding proof of the error bound for Simpson's quadrature rule

I have found the following proof of the error bound for Simpson's quadrature rule: Using Newton's interpolation method, we derive a cubic polynomial $p_3(x)$ that interpolates $f(x)$ at the points $a, ...
codeing_monkey's user avatar
1 vote
1 answer
123 views

How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE

Setting I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system: $$\...
user1313292's user avatar
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1 answer
49 views

numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?

I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
Millemila's user avatar
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1 vote
0 answers
101 views

Can I reduce my simulation error with a staggered grid, postprocessing and compatibility equation feedback?

What I did Using the finite difference method, I solved with a certain amount of error the following system of hyperbolic partial differential equations in cylindrical coordinates (the problem is ...
FriendlyNeighborhoodEngineer's user avatar
0 votes
0 answers
34 views

Order of Error - Confusion: Clarifying Constraints on Constants and Determining Order of Error

I'm struggling to determine the order of error when considering the error value denoted by $\text{err}$ in relation to the variable $h$. Specifically, I aim to ascertain the value of $x$ in the ...
Ferran Gonzalez's user avatar
1 vote
2 answers
125 views

How to estimate the stage error for Runge kutta method

Consider an ordinary differential equation (ODE) in the form $u_t=g(t,u(t))$ and apply the explicit Runge-Kutta method, as defined by the following Butcher tableau: $$ \mathrm{RK}(s,p):\begin{array}{c|...
Owen Jun's user avatar
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0 answers
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What is the minimum error achievable using gaussian process emulation?

I am interested in using Gaussian processes as emulators for other computational models, and I would like to characterize the expected numerical precision of the emulator. Specifically, how small can $...
user9794's user avatar
  • 465
4 votes
2 answers
443 views

Understanding the diffusion error of numerical schemes

In chapter 7 of Numerical Computation of Internal and External Flows (Second Edition) (https://www.sciencedirect.com/science/article/abs/pii/B9780750665940500497) the author describes the diffusion ...
F_B's user avatar
  • 111
0 votes
1 answer
61 views

Reverse engineering phase shift and numerical damping

I've been trying to validate the physics behind a particle system framework, but I'm having some difficulties. A particle system is a set of lumped masses connected by spring-damper elements. Linear ...
AlexBatch's user avatar
4 votes
1 answer
172 views

The error propagation in calculating the inverse using a matrix decomposition

I have been trying to calculate the matrix inverse of some large matrix with entries ranging by orders of magnitude. I tried to use the matrix decomposition to simplify the computation, where a matrix ...
ShoutOutAndCalculate's user avatar
1 vote
1 answer
169 views

Storing Raw Simulation Data or Truncated Data?

I have a simulation that can generate quite a bit of data when it runs, for example $650\cdot 400 \cdot 400$ floating point numbers. Without compression, that's a few gigabytes worth if I want to save ...
cgbsu's user avatar
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1 answer
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Finite difference approximation error

I was reading Scientific Computin, An Introductory Survey, by Michael Heath. In the Example 1.11, he madr a Finite Difference Aproximation, with the usual approxination : $f’(x)\neq \frac{f(x+h)-f(x)}{...
RES's user avatar
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1 vote
0 answers
119 views

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, $k\le d$, ...
Yaroslav Bulatov's user avatar
1 vote
1 answer
80 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 ...
Mohamed Abdelhamid's user avatar
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0 answers
43 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 ...
Auberron's user avatar
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0 answers
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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 ...
Natasha's user avatar
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-1 votes
1 answer
66 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 ...
Lilla's user avatar
  • 259
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1 answer
94 views

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 ...
420's user avatar
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7 votes
1 answer
360 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 ...
quantumflash's user avatar
1 vote
1 answer
88 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 ...
Lilla's user avatar
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3 votes
0 answers
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 ...
vsz's user avatar
  • 131
2 votes
1 answer
138 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 ...
alligator's user avatar
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2 votes
1 answer
195 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 ...
H. Zhou's user avatar
  • 123
-1 votes
1 answer
51 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. ...
Vishal Jain's user avatar
0 votes
2 answers
93 views

calculating the Laplacian of the field variable in estimating the local residual error in the finite element method

to perform adaptive refinement in the finite element method according to the explicit residual method, the quantity $$\eta_K^2=h_K^2\left\lVert r\right\rVert_{L_2(K)}^2+h_K\left\lVert R\right\rVert_{...
Masa's user avatar
  • 194
0 votes
1 answer
55 views

Numerical Error source when dealing with integer series

I am currently trying to compute the value of the first Fibonacci number recursively. the idea is as follow: Compute $f_{n}$ and $f_{n-1}$ for $n = 2,...,100$, Compute $f_k$ for $k = n−2, n−3, \dots, ...
devCharaf's user avatar
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8 votes
0 answers
105 views

How do we approximate the numerical error a numerical scheme (e.g Runge Kutta, Euler etc) makes without having access to an analytical solution?

So I recently encountered this question in my head while taking my Scientific Computing class, where the lecturer talked about computing numerical error of a scheme. My guess would be that we take a ...
KARTIK BALI's user avatar
4 votes
1 answer
100 views

Roundoff errors in FEM computations - generalized eigenvalues

This is a continuation of my previous question. I am trying to effectively compute a bound for the roundoff errors in some FEM computation (2d polygons, triangular meshes). Below I will write some of ...
Beni Bogosel's user avatar
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2 votes
2 answers
606 views

Different sources of error in Finite Element computations

Consider the problem $-\Delta u = f$ in $\Omega$, with $u=0$ on $\partial \Omega$. Suppose that $\Omega$ is a polygon and that we approximate the solutions to the previous problem using Lagrange ...
Beni Bogosel's user avatar
  • 1,067
3 votes
1 answer
204 views

How to measure the error of Finite Element approximation in satisfying the PDE?

In Galerkin methods, we seldom can measure the accuracy of an approximation by tracking the value of the residual. For example, take the wave equation: $ u_{tt} = u_{xx}$, $(x,t) \in (0,L) \times (0,T)...
Snifkes's user avatar
  • 131
1 vote
1 answer
331 views

Question about step in the proof of standard discrete trace inequality

I'm studying from Guermond lecture notes available at https://www.math.tamu.edu/~guermond/M661_FALL_2019/chap12.pdf (see Lemma 12.8( Discrete trace inequality).) Consider the simple case $p=r$, i.e. ...
FEGirl's user avatar
  • 405
4 votes
0 answers
85 views

Global reconstruction defined elementwise in a-posteriori error estimator

This question is a follow-up of this previous one. In "Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems" by Georgoulis et al., an error estimator is ...
FEGirl's user avatar
  • 405
0 votes
1 answer
109 views

How is the integral of a projection over an element $T$ computed in practice? (deal.II related)

I'm studying an error estimator for the equation $\nabla\cdot(\beta u) + cu = f$ and it contains the following term $$||f - cU_h - \Pi(f-c U_h) ||_T$$ where : $\Pi$ is the local orthogonal $L^2$ ...
FEGirl's user avatar
  • 405
2 votes
1 answer
5k views

Error in Simpson's 3/8 rule is higher than that of Simpson's 1/3 rule

For a given function $f(x)$, I have tried to find its numerical integral using Simpson's 1/3 and Simpson's 3/8 rules. I then compare the solution from the numerical quadratures to the analytical ...
justauser's user avatar
  • 145
3 votes
0 answers
72 views

Typo in a-priori error estimate in a Discontinuous Galerkin paper

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
FEGirl's user avatar
  • 405
1 vote
1 answer
118 views

Finding the source of numerical instability in a electrostatic problem solved by conformal mapping

I'm using conformal mapping to solve a 2D electrostatic problem (calculating the potential $u(x,y)$ in the plane). Let $C_1$ and $C_2$ be two circles at an electric potential $U_1$ and $U_2$, ...
Pedro H. N. Vieira's user avatar
3 votes
1 answer
160 views

Proof of R. Verfürth paper on adaptive mesh and bubble functions

I'm studying adaptive meshes, and my professor wrote the following property for a bubble function ( see this scicomp post for the definition I'm using)$b_T$ defined on a triangle $T$. $$||b_T \phi ||_{...
FEGirl's user avatar
  • 405
1 vote
1 answer
114 views

Classical global estimate for $H^1$ error

I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$. $$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
FEGirl's user avatar
  • 405
3 votes
1 answer
336 views

How to find the optimum finite difference method for derivatives?

Related to: What are the negatives of using higher order finite diference schemes? Problem: I have some discrete data of a trajectory $x_t$ with errors $\delta x_t$ of a physical system sampled at ...
Puco4's user avatar
  • 131
4 votes
1 answer
161 views

Computing the residual in a Dual Weighted Residual (DWR) method

I am in the process if computing the Dual-Weighted Residual (DWR) for a linear PDE with a linear functional but I am struggling with the residual part of the calculation. For example suppose we want ...
gnikit's user avatar
  • 141
1 vote
2 answers
244 views

Stability condition FCTS method

The FTCS method comes from the discretization of a diffusion PDE like this: $$ a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t} $$ If I have the ...
LongJohn's user avatar
0 votes
1 answer
53 views

Round-off error step choice

In the Numerical Recipes in section 5.7.- Numerical derivatives the choice of the step size $h$ in the numerical derivative should lead to a difference between $x$ and $x+h$ representable by an exact ...
LongJohn's user avatar
0 votes
1 answer
227 views

Taylor expansion round-off error

In the Numerical Recipes in section 5.7.- Numerical derivatives it's introduced de roundoff error of: $$ f^{\prime}(x) \approx \frac{f(x+h)-f(x)}{h} $$ as (with $h$ an "exact" number): $$ \...
LongJohn's user avatar
2 votes
1 answer
65 views

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 ...
Zebx's user avatar
  • 101
1 vote
1 answer
318 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 (...
FEGirl's user avatar
  • 405
0 votes
1 answer
218 views

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 ...
Zebx's user avatar
  • 101
2 votes
1 answer
229 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 ...
Zebx's user avatar
  • 101
0 votes
1 answer
233 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 ...
Brain Stroke Patient's user avatar
0 votes
1 answer
232 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), ...
kostas1335's user avatar
4 votes
2 answers
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 ...
Hamilcar's user avatar
  • 141