Hot answers tagged

6 votes

How to compute numerically the $H^{1/2}$ norm of a function

I think what you are referring to is $\|h^{-1/2}g\|_{0,\partial D}$. The point is that it can be though of as the 'discrete' $H^{1/2}$ norm. It comes down to the so called 'inverse inequalities' where ...
knl's user avatar
  • 2,076
6 votes
Accepted

Minimize distance between curves

Let's assume you have a set of abscissas $x_i$ and two sets of function values on these grid points $f_i, g_i$ representing the functions $f$ and $g$. As mentioned in the comments, you'll need a model ...
davidhigh's user avatar
  • 3,127
5 votes
Accepted

Residual norm of PDE discretization: correspondence in the continuous problem?

TLDR If you're using scalar products in FEM/FVM discretizations, use the mass-matrix scalar product, not $\ell_2.$ or If you're solving FEM/FVM systems with Krylov methods, precondition with the ...
Nico Schlömer's user avatar
4 votes
Accepted

Fastest way to calculate the $2$-norm (or an upper bound for the $2$-norm) of the inverse of a matrix $A\in \mathbb{C}^{N\times N}$

You might want to use the fact that: $$ ||A||_2=\sigma_\max(A) $$ where $\sigma_\max$ is the largest singular value. If you are interested in details, this Math SO question should be interesting. ...
Anton Menshov's user avatar
  • 8,652
4 votes
Accepted

$L^2$-error in FEM: how to compute integral over reference element?

You can write $$\int_{0}^1 (f - \Pi_h^1(f))^2 dx = \sum_{i=0}^{N_e} \int_{e} (f - \Pi_h^1(f))^2 dx$$ where $e = [x_k, x_{k+1}]$ is an interval (triangle as you say). Method A (simple, no reference ...
Dr_Sam's user avatar
  • 1,246
3 votes
Accepted

Does the loss function in a deep neural network act as a norm?

Connection between Mean squared error and $L_2$ norm: First the $L_2$ norm of a vector $\boldsymbol{x}\in\mathbb{R}^n$ is defined as \begin{align*} \lVert \boldsymbol{x}\rVert_2=\left(\sum\limits_{i=1}...
Pepe's user avatar
  • 459
3 votes
Accepted

Physical interpretation of L2 norm of heat equation solution

In these slide there are some comments about the energy. At pag 4 it focus on the fact that this energy is not a physical energy, but it is a mathematical tool. At pag 8 it observes that: From ...
Mauro Vanzetto's user avatar
3 votes
Accepted

Frobenius norm of a binary matrix

The Frobenius norm of a binary matrix is the square root of the number of non-zero elements. Let the point (0.,...0) be the origin, and let's say the vec'd binary matrix elements are the coordinates ...
Mark L. Stone's user avatar
3 votes
Accepted

About the discrete $H^1$ norm

Assuming 1-D and equidistant gridpoints with spacing $h$ and some form of homogenous boundary conditions, we can use $\|\nabla v\|^2\approx -h\sum_{i=1}^nv(x_i)D_2v(x_i)$, where $D_2$ is a finite ...
whpowell96's user avatar
  • 2,259
2 votes

2-norm and infinty norm of a system in controls

You probably know that matrix norms can be defined by the vector norms in the following way: \begin{equation} ||A||:= \max_{x\neq 0} \frac{||Ax||}{||x||} \end{equation} for a matrix $A$. So you just ...
Pepe's user avatar
  • 459
2 votes
Accepted

Finding weighted average of curves

I promised you an answer in the other question, and was just about to edit it in. Now I see you spend another 100 points as a bounty ... seems quite a serious topic to you. I'll post my promised ...
davidhigh's user avatar
  • 3,127
2 votes

Minimize distance between curves

Here is a simple solution. Find the curve $(x_m, y_m)$ with the largest domain $x$. In your case it is (x2, y2). Assign it to be the main curve and shift all other ...
Vladislav Gladkikh's user avatar
1 vote

Minimize distance between curves

Taking a function as reference $f_r$ the scale-translation transformations for each remaining functions can be handled by minimizing $$ E(a,b,r) = \sum_{k\ne r}^{m}\sum_{j=1}^n \left(f_k(j)a_k+b_k - ...
Cesareo's user avatar
  • 166
1 vote
Accepted

How I could calculate L2 norm of an unstructured grid?

How are you on a uniform unstructured grid? Are you in 1-D or 2-D? You're missing a lot of detail. This expression of the norm that you found is area weighted. If you're 1D then multiplying by $\Delta ...
EMP's user avatar
  • 2,079
1 vote

Residual norm of PDE discretization: correspondence in the continuous problem?

I am digressing slightly to address how the equation is discretized before going to the solution. General methods for numerically solving PDEs: Use pseudospectral methods: Approximate the solution ...
Prithvi Thakur's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible