Skip to main content
11 votes
Accepted

What happens when I use a conjugate gradient solver with a symmetric positive semi-definite matrix?

You first have to make sure that your system is solvable: this happens iff the right-hand side $b$ is orthogonal to the kernel of $A$. If $A$ has a dimension-1 kernel spanned by $v$, you need to have $...
Federico Poloni's user avatar
9 votes
Accepted

Correct eigenfunctions of Laplace operator by Finite Differences

You should specify the eigenvalues you want with which="SM", for example. Check the following snippet. I also changed the solver, since your system is symmetric. <...
nicoguaro's user avatar
  • 8,524
8 votes
Accepted

Solving Ax = b with sparse A and sparse b

When looking at the solution of your system, you will find that almost all entries of $x$ are nonzero although the right-hand side is "sparse". Hence, whatever algorithm you use, it'll have to visit ...
Nico Schlömer's user avatar
8 votes
Accepted

3D laplacian operator

Yes, that finite difference is correct. You can obtain it using a finite difference in each direction for the Laplace operator in each coordinate. \begin{align} \nabla^2 u =& \frac{\partial^2 u}{\...
nicoguaro's user avatar
  • 8,524
6 votes

Eigenvectors of Laplacian

They're on Wikipedia, for instance, in a page with the slightly unclear name of "Eigenvalues and eigenvectors of the second derivative".
Federico Poloni's user avatar
5 votes

Factorize laplacian in terms of first derivative matrix

It is sufficient if you consider a $D$ that uses forward or backward differences with reflecting boundaries: \begin{equation} D_f = \frac{1}{h}\begin{bmatrix} -1 & 1 & & \\ & \ddots &...
lightxbulb's user avatar
  • 2,197
5 votes

Computing the Fiedler vector of a large, sparse graph

Disclosure: This is my master's supervisor's work, he and his co-authors are pretty well-known in the field. TRACEMIN-Fiedler is a parallel algorithm to compute the Fiedler vector of large graphs ...
Abdullah Ali Sivas's user avatar
4 votes
Accepted

Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?

The "issue" seems to have been that there is a discrepancy between the used transforms. The DCT transforms discussed in the linked paper are orthogonal, i.e. $U^{-1}=U^T$, where the even one ...
lightxbulb's user avatar
  • 2,197
4 votes

Numerical estimation of eigenfunctions of Laplacian

Fundamentally, here are the building blocks of what you are asking for: Consider solving the problem $$ -\Delta u = f $$ in a domain $\Omega$ with boundary values $u=g$ on $\partial\Omega$. ...
Wolfgang Bangerth's user avatar
3 votes

preconditioner for Laplace "without" boundary values

Here is at least an idea, whether it works is a different question. Let's say you sort unknowns so that you have the ones in the interior of the domain first, and then all those at the boundary. Then ...
Wolfgang Bangerth's user avatar
3 votes
Accepted

Discrete Laplacian operator with finite element discretization

Weak Laplacian Let $u,v\in H^2(\Omega)$ then using the divergence theorem you have the identity $$\int_{\Omega} v \Delta u = \int_{\partial\Omega} v \partial_{\vec{n}} u - \int_{\Omega} \nabla v \cdot ...
lightxbulb's user avatar
  • 2,197
3 votes
Accepted

Kronecker product representation of the finite difference laplacian

Maybe this isn't a helpful response but the reason this happens for the matrix form of Laplacians is because this actually happens for the true infinite-dimensional Laplacians in some settings. In ...
whpowell96's user avatar
  • 2,636
3 votes

Kronecker product representation of the finite difference laplacian

This is my attempt at providing some intuition. Everything I state might be obvious, moreover it doesn't have much to do with physics, so this could be a non-answer. I will ignore boundary conditions. ...
Eman Yalpsid's user avatar
3 votes

Factorize laplacian in terms of first derivative matrix

@lightxbulb's answer gives the correct factorization already, but since you mention failed attempts with the Cholesky factorization, let me describe a method to discover the factorization numerically, ...
Federico Poloni's user avatar
3 votes

Solving Ax = b with sparse A and sparse b

It is unclear to me from your question whether the answer is of theoretical or practical interest. I'll address both. For this to be of practical importance, either your system would need to have a ...
Bill Greene's user avatar
  • 6,144
3 votes

What happens when I use a conjugate gradient solver with a symmetric positive semi-definite matrix?

To at least partially answer the question: You have to make sure the system is solvable in the first place. As mentined in the accepted answer, "the system $Ax=b$ is solvable iff $b \bot Ker(A)$&...
Jakub Homola's user avatar
3 votes

Access optimized data structure for representing integer lattice

In essence, you are asking whether you can enumerate the integer lattice sites within your domain from $1$ to $N$ in such a way that accessing the east/west/north/south neighbors of a location $n$ ...
Wolfgang Bangerth's user avatar
2 votes

Access optimized data structure for representing integer lattice

You can probably speed things up a little bit by storing the array in 4x4 square subarrays so that each of them fit in cache line (64 bytes = 4x4 32-bit integers). This changes the probability ...
Ark-kun's user avatar
  • 131
2 votes

How avoid square shape with Laplacian operator in reaction diffusion calculations?

There are finite difference stencils specifically designed to have rotational symmetry. For example, instead of the standard second order stencil $$ \frac{1}{h^2} \begin{bmatrix} & 1 & \\ 1 &...
Steven Roberts's user avatar
2 votes
Accepted

preconditioner for Laplace "without" boundary values

As others have pointed out, (algebraic) multigrid can actually be a good preconditioner in this scenario. Below is a proof-of-concept implementation with scikit-fem and pyamg. It shows that pyamg's ...
Nico Schlömer's user avatar
2 votes
Accepted

Eigenvectors of Laplacian

You can find a discussion about the topic (and the derivation) in chapter V of: R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I (1989). doi:10.1002/9783527617210
Juan M. Bello-Rivas's user avatar
2 votes

Reference request: graph Laplacian approximation for domains/manifolds

In a plane, you have something that resembles the graph Laplacian in the form of the 5-point stencil. You can derive the 5-point stencil as an approximation of the finite element matrix by using a ...
Wolfgang Bangerth's user avatar
1 vote
Accepted

Solving the wave equation for a circular membrane in polar cordinates

The issue was with the initial conditions and how I was understanding them. The problem was the following. When you are solving the differential equation for the wave propagation on a circular ...
Manuel Borra's user avatar
1 vote

Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions

For this particular finite-element approach, Dirichlet (zero) boundary conditions can be implemented by zeroing-out the Laplacian operator at boundaries; Neumann (zero derivative) boundary conditions ...
MRule's user avatar
  • 153
1 vote

ON the Kronecker product form of the laplacian matrix

I think your question is too general. In 1D the discrete linear FE operator often recovers the linear centered FD one, however this is not always the case. For DG you additionally have consider the ...
ConvexHull's user avatar
  • 1,379
1 vote

Numerical estimation of eigenfunctions of Laplacian

The solution of $$ \nabla^2 f(r,\theta,\phi) = 0 $$ can be written in terms of eigenfunctions of the Laplacian. As you said, the expansion coefficients then should be chosen as to satisfy the ...
davidhigh's user avatar
  • 3,167
1 vote

preconditioner for Laplace "without" boundary values

If you are not enforcing any boundary conditions, you should remove the nullspace of the problem (to make sure that there is a unique solution, MatSetNullSpace can be used to achieve this in PETSc) ...
Abdullah Ali Sivas's user avatar
1 vote

How to discretize Laplacian near refinement boundary

I think one option is to interpolate $u$ to the left face of (3,1) first. Then you can use a modified Laplacian stencil to take into account the half-distance between $u$ at (3,1) and $u$ on the face.
Charles's user avatar
  • 619

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