Questions tagged [error-estimation]
For questions about determining the error caused by specific computational procedures, approximations, or numerical representations.
175
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How to estimate a finite element function with exponential matrix
I would like to know if there are any similar theorems related to the mass lumping finite element method:
For any function $\boldsymbol{u} \in H^m_{per}(\Omega)$
$$
\|I_h e^{\tau \Delta} \boldsymbol{u}...
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63
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Understanding proof of the error bound for Simpson's quadrature rule
I have found the following proof of the error bound for Simpson's quadrature rule:
Using Newton's interpolation method, we derive a cubic polynomial $p_3(x)$ that interpolates $f(x)$ at the points $a, ...
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1
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137
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How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE
Setting
I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system:
$$\...
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53
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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 ...
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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 ...
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34
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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 ...
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2
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130
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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|...
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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 $...
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2
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453
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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 ...
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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 ...
4
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1
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191
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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
...
1
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1
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172
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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 ...
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108
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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)}{...
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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$, ...
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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 ...
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46
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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 ...
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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 ...
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1
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67
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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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97
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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 ...
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417
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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 ...
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92
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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 ...
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61
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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 ...
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143
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"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 ...
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1
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224
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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 ...
- 123
-1
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1
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60
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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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2
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97
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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_{...
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1
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56
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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, ...
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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 ...
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105
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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 ...
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2
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654
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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 ...
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3
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1
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208
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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)...
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360
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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. ...
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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 ...
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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 $L^2$ ...
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1
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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 ...
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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 ...
- 405
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1
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119
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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 $U_2$, ...
3
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1
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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 $T$.
$$||b_T \phi ||_{...
- 405
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1
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124
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Classical global estimate for $H^1$ error
I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$.
$$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
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399
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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 ...
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163
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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 ...
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2
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257
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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 ...
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54
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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 ...
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1
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253
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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):
$$
\...
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1
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66
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Initial condition precision
Is there a way to have an estimate of the error propagated on an ode numerical solution by the error of the initial conditions? I suppose this depend on the numerical method used and on the problem ...
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1
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329
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FEM solution for Poisson is not exact at nodes
Let's consider the usual Poisson problem for FEM $$-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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1
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225
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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 ...
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1
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232
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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 ...
- 101
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1
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235
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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 ...
0
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1
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247
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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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