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## Hot answers tagged laplacian

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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 $... • 11.5k 9 votes Accepted ### Correct eigenfunctions of Laplace operator by Finite Differences You should specify the eigenvalues you want with which="SM", for example. Check the following snippet. I also changed the solver, since your system is symmetric. <... • 8,524 8 votes Accepted ### Solving Ax = b with sparse A and sparse b When looking at the solution of your system, you will find that almost all entries of$xare nonzero although the right-hand side is "sparse". Hence, whatever algorithm you use, it'll have to visit ... • 3,126 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}{\... • 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". • 11.5k 5 votes ### Factorize laplacian in terms of first derivative matrix It is sufficient if you consider aD$that uses forward or backward differences with reflecting boundaries: D_f = \frac{1}{h}\begin{bmatrix} -1 & 1 & & \\ & \ddots &... • 2,197 5 votes ### Computing the Fiedler vector of a large, sparse graph Disclosure: This is my master's supervisor's work, he and his co-authors are pretty well-known in the field. TRACEMIN-Fiedler is a parallel algorithm to compute the Fiedler vector of large graphs ... • 2,781 4 votes Accepted ### Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions? The "issue" seems to have been that there is a discrepancy between the used transforms. The DCT transforms discussed in the linked paper are orthogonal, i.e.$U^{-1}=U^T$, where the even one ... • 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$\partial\Omega$. ... • 55.7k 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 ... • 55.7k 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 ... • 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 ... • 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. ... • 151 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, ... • 11.5k 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 ... • 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)&... 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 ... • 55.7k 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 ... • 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 &... • 1,114 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 ... • 3,126 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 • 3,994 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 ... • 55.7k 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 ... 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 ... • 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 ... • 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 ... • 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) ... • 2,781 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.
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