14
votes
Accepted
$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)
Yes, this is the standard Aubin-Nitsche (or duality) trick. The idea is to use the fact that $L^2$ is its own dual space to write the $L^2$-norm as an operator norm
$$\|u\|_{L^2} = \sup_{\phi\in L^2\...
- 12k
11
votes
Convergence rate vs convergence order
$\|x^{(k)}-x^*\|$ is the error in the $k$th term, call it $E_k$. For a "good" numerical method, we want the approximation to get closer and closer to the desired result so $E_k$ has to decrease to ...
- 326
10
votes
Accepted
Analytical convergent sequence and numerical divergent sequence
Jean-Michel Muller, et. al., "Handbook of Floating-Point Arithmetic 2nd ed.", Birkhäuser 2018, gives the following example due to Muller, specifically constructed to deliver incorrect results with ...
- 1,320
10
votes
Accepted
How does gmres method iteration behave for this non-diagonalizable matrix?
Unfortunately, convergence of GMRES does not have a clear dependence on the distribution of eigenvalues.
It was proved by Greenbaum, Ptak and Strakos in 1996 that you can construct examples with an ...
- 10k
9
votes
Accepted
Accurate computation of Gauss-Kuzmin entropy
It's fairly easy to evaluate, to do this expand the logs in Taylor series in $x=(k+1)^{-2}$:
$$ \log_2(1-x) = \frac{-1}{\log 2}\sum_{m\geq1}\frac{x^{m}}{m}$$
$$ \log_2(-\log_2(1-x)) = \frac{\log x}{\...
- 11.4k
9
votes
Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$, $l_2$ and $l_{\infty}$)?
In finite dimensional spaces (say, in $\mathbb R^n$), all norms are equivalent and as a consequence, if something converges with a specific rate in one norm, it also converges with the same rate in ...
- 52.4k
8
votes
Accepted
Lack of quadratic convergence in Newton's method
Yes: See Higham's book "Accuracy And Stability of Numerical Algorithms", second edition, chapter 25: Nonlinear Systems and Newton's Method.
In particular, see the section on the "limiting residual" ...
- 2,055
7
votes
Accepted
When and why is `r./sum(r)` not a good way to renormalize a vector in PageRank computation?
The two normalization formulas result in two different algorithms, that they both "normalize" a vector is not so relevant.
As an example, consider the following transition matrix:
$$M = \begin{...
- 11.4k
7
votes
Accepted
Error in result of finite-difference approximation when refining
High order methods are not necessarily always more accurate than low order methods; they simply have a more rapid convergence rate. For example, if you take the Taylor expansion of a function at a ...
- 3,062
7
votes
Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method
The issues you're running into now are not a failing of Newton-Raphson, but a question of coupling. You're doing iterated sequential coupling -- solving each equation sequentially and then iterating ...
- 396
7
votes
Accepted
Global convergence in trust region algorithm
Standard terminology in nonlinear (i.e., derivative-based) optimization is that "global convergence" means "convergence to a stationary point no matter where you start from" (where the limit may in ...
- 12k
7
votes
Which absolute and/or relative stopping criteria do use for Newton's method?
Some of the confusion comes about because mathematicians often consider artificial functions that happen to have size $O(1)$ whereas practitioners use functions and variables that just happen to be ...
- 52.4k
7
votes
Accepted
Convergence-test for ODE approximates wrong limit
The Euler method has global error order 1, not 2
To get step size $h=1/N$, you need $N$ steps, which gives $N+1$ nodes in the time subdivision. Currently you compare sequences with step sizes $\frac1{...
- 4,829
6
votes
Accepted
Finding rate of convergence by curve fitting in Matlab
First of all, rates of convergence are usually given in the form
$$ \|u-u_h\| \leq C N^\alpha,$$
rather than equality. Furthermore, rates are asymptotic, i.e., only have to hold for $N\to \infty$. ...
- 12k
6
votes
Scaling/nondimensionalization for numerical optimization
One thing that makes your non-dimensionalized ODE confusing is that you use the same symbols for dimensionalized and nondimensionalized variables, even though they are different variables.
Consider ...
- 11.4k
6
votes
Analytical convergent sequence and numerical divergent sequence
In my opinion, an example could be the calculation of the minimal solution of a three term recurrence relation (TTRR) $$y_{n+1} +a_{n}y_{n}+b_{n}y_{n-1} = 0, \quad n=1,2,3,\ldots$$.
For example, the ...
- 6,109
6
votes
Accuracy issues with Arpack in Julia for eigenvalues of smallest magnitude
Looks like everything is working relatively OK? Your matrix is of order 1e10, so residuals of 1e-4 are actually close to machine precision. The convergence criterion is indeed violated, but not by ...
- 196
6
votes
Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$, $l_2$ and $l_{\infty}$)?
Both your questions have been addressed in the answer of @WolfgangBangerth. Here is an applied example of the behaviour of the different norms in the context of polynomial approximation (which is for ...
- 2,937
5
votes
CFL condition for variable coefficients
The CFL number (or simply Courant number) is defined locally:
$\nu = u(x,t) \frac{\Delta t}{\Delta x}$.
The velocity may vary in time and space and the grid may be non-uniform (in $x$ and/or $t$). ...
- 16.4k
5
votes
Accepted
Convergence rate Jacobi/Gauss-Seidel with mesh resolution
The convergence rate that is mentioned here is in the sense that the error in iteration $k$ and $k-1$ are related by
$$
\| x^{(k)} - x^\ast\| \le r \| x^{(k-1)} - x^\ast\|,
$$
which implies that
$$
...
- 52.4k
5
votes
How does gmres method iteration behave for this non-diagonalizable matrix?
Estimates for GMRES convergence based on eigenvalue distribution often implicitly assume that the matrix is normal. Sometimes the convergence rate is still provable in an asymptotic sense in the non-...
- 3,243
5
votes
Accepted
Why FEM for incompressible materials is ill-posed?
For incompressible materials, as the extent of incompressibility increases, the bulk modulus approaches infinity (for the isotropic case). This is what causes ill-conditioning.
Consider the case of ...
- 810
4
votes
Error in result of finite-difference approximation when refining
You've chosen a very specific example where the function in question has a single extremely narrow peak and is almost zero everywhere else in the domain. This is an example of a stiff problem where ...
- 4,571
4
votes
What causes periodic humps in residual plots?
These humps occur in the context of optimization when the solution is approached but we then go past it.
This frequently occurs in gradient descent with momentum, where they appear regularly on plots ...
4
votes
Accepted
What should I put on the paper to show the correctness and convergence of my solution?
You should definitely do a mesh refinement study (some conferences and journals require this). You should compare the results of this study to a known, analytical solution. You can either get this ...
- 10.9k
4
votes
Accepted
convergence of unconstrained convex optimization
What about
$$ F(x,y) = \frac{x^2}{y} + \frac{\epsilon}{x^2}, \qquad \nabla F(x,y) = \Big( \frac{2x}{y}-\frac{2\epsilon}{x^3}, -\frac{x^2}{y^2} \Big), \qquad x,y>0$$
with the sequence $(x_k,y_k) = (...
- 11.4k
4
votes
Accepted
$O(h^2)$ convergence for Elliptic PDE
You appear (and correct me if I am wrong, i do not wish to presume) to be confused on the notion of what is an order of convergence.
The order of convergence is the order at which your approximate ...
- 1,137
4
votes
Does the convergence of finite element have limit?
The overall error of a finite element solution often has many components. For example, these may include (i) the discretization error due to a finite mesh and/or finite polynomial degree, (ii) the ...
- 52.4k
4
votes
First-order ODE scheme implementation giving less than first-order convergence?
I cannot tell you whether that is the cause for your diminished order, but it is asking for trouble to add other approximations than just the one you want to do on purpose. In the current context, ...
- 52.4k
4
votes
Accepted
First-order ODE scheme implementation giving less than first-order convergence?
Let me modify slighly your ODE to see the essence of the problem. Let
$$\left\{\begin{array}{l}y'=-y \\ y(0) = 1 \end{array} \right. \qquad t\in[0,T]\tag{*}$$
the ODE to be solved. Everyone knows that ...
- 1,618
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