Questions tagged [multigrid]
An approach to solving systems of equations by projecting the problem from a fine scale representation onto a coarser one. A coarse representation generally has fewer unknowns, making it faster to solve than the original problem. The coarse solution can then be projected back onto the finer problem as an initial guess of the solution to the finer problem.
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Multigrid/Two-grid method restriction and prolongation of residual
Starting from the problem $Au=f$, I'm not sure that I understand why a coarse grid solution is implemented on the coarse grid residual $r_c^{(k)}=P^Tr^{(k)}$, with $r^{(k)}=f-Au^{(k)}$ and $P^T$ ...
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Interpolation and Restriction operators in Multigrid
I saw in several places that interpolation operator ($P$) and restriction operator ($P^T$) are usually transposes of each other (up to multiplication by a constant).
As I understood it related to ...
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Eigenvectors of Laplacian
I am studying introduction to Multigrid methods.
In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as
$w_k(x_i) = \sin(k \pi x_i),$
where $x_i$ is a ...
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Algebraic Multigrid on fine mesh vs. just starting with coarse mesh?
So if I've understood it correctly, the Algebraic Multigrid Method (AMG) basically takes a fine mesh, coarsens it, solves the coarse mesh and projects the solution back on the fine mesh.
Wouldn't it ...
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Geometric multigrid transfer operators constraints
I have been reading up on constraints used to construct prolongation and restriction operators for geometric multigrid. One such constraint is that the prolongation matrix is the transpose of the ...
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Prolongation and restriction operators in multigrid for high order PDEs
If I have the Poisson equation $\Delta u = f$ a standard transfer operator (for a regular grid) is the full weighting/bilinear interpolation scheme:
$$K = \frac{1}{4}\begin{bmatrix}\frac{1}{4} & \...
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Restriction in (geometric) multigrid for vectors of non-even length
Naive restriction operators in geometric multigrid that I have seen are typically implemented as a convolution and a subsequent averaging of every two entries in a vector $v^h$. For example:
$$\tilde{...
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Differences between Two-Grid Correction and V-Cycle Scheme
I found in A Multigrid Tutorial: Second Edition by William L. Briggs, Van Emden Henson, Steve F. (here) the following Schemes:
Two-Grid Correction Scheme, p. 37:
And V-Cycle Scheme, p. 40
I want ...
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Multigrid method: linear solver and modified residual
I am trying to better understand the FAS multigrid algorithm for Euler equation in FV discretization. The usage of the modified residual (the residual with forcing) inside the different cases:
...
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Multigrid: Gauss–Seidel eigenvalues and eigenvectors
I am trying to figure out the questions from Multigrid Tutorial by Briggs. However I am stuck on these two questions.
https://www.researchgate.net/publication/...
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Algebraic multigrid as solver and as preconditioner
My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
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Construction of Prolongation and Restriction Operator for Geometric Multigrid (2D-FEM): Resulting in a Decreasing Solution
Consider the following problem,
$$ -\Delta u(x) = f(x), \qquad x \in \Omega \\ u(x) = 0,\qquad x \in \partial \Omega$$
with $\Omega = [0,1]\times [0,1]$ being the domain and $\partial \Omega$ being ...
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Multigrid Reduction In Time Convergence
I am trying to solve a 2D dynamic linear elasticity model parallel in time using Xbraid. The spatial domain is [0,1]x[0,1] and time domain [0,1]. For time integration I am using a backward Euler ...
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Parallel In Time with Multigrid
I am trying to solve the linear finite element equation $M\ddot{u}+Ku=F(t)$, where $M$ is the mass matrix ,$K$ the stiffness matrix and $F(t)$ the external load vector, parallel in time using XBraid ...
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coarsening coefficient matrixes (A2h, A4h...) for geometric multigrid method in 2-D/3-D
I am learning about multigrid methods from the textbook section 6.3 Multigrid Methods, which shows a geometric multigrid algorithm for 1-D examples in detail, including how to build restriction/...
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Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time?
I'm solving a system of linear equations obtained from the FEM discretization of a simple linear elasticity problem on a cube with zero displacements at one plane and a load on the opposite one. The ...
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Calculating coarse grid matrix in geometric multigird
The coarse grid matrix is calculated via RAP where R,P are the restriction and interpolation matrix,respectively.By checking a typical MG algorithm
I want to ask how to calculate efficiently coarse ...
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Full approximation scheme - smoothers - literature recomendation
I would like to use full approximation scheme(FAS) for solving nonlinear PDE. I am looking into practical implementation of FAS in parallel(MPI) settings. I noticed the most common/effective smoother ...
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Residual of Poisson equation with periodic boundaries
I am trying to write a multigrid solver for Poisson's equation, $-\Delta u=f$, on the unit square, $\Omega=(0,1)^2$ with periodic boundaries. My primary source has been Multigrid by Trottenberg, ...
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How the number of pre-smoothing and post-smoothing steps affect the asymtotic convergence rate of geometrical Multigrid?
Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?
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How coarsening rate affects MG convergence?
If we use an aggresive coarsening how does this affect the convergence of a MG cycle?Is the convergence slower?Do we need more cycles in this case?
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Why post-smoothing in MG is needed?
In pre-smoothing we eliminate high frequencies,but what about post-smoothing?Is it used for the remain high frequencies that pre-smoothing didn't eliminate?
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Avoid matrix multiplication in algebraic multigrid method
Currently when I try to solve a linear algebra system of the form of $A x =b$ I use the algebraic multigrid method. The algebraic multigrid method uses a Galerkin product to form the coarse grid ...
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What kind of problem or matrices are suitable for multigrid method?
For Poisson or Convection-diffusion equation as follows:
$$
-\Delta u=f,\qquad u|_\Omega = g.
$$
or
$$
-\Delta u +\vec{w}.\nabla u=f,\qquad u|_\Omega = g.
$$
using FDM or FEM discretization, we can ...
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How to implement Geometric Multigrid in non-rectangular grids?
It is quite easy to implement multigrid on a rectangular grid but what about an non-rectangular?How to coarse a non-rectangular grid and apply multigrid(assume an easy non-rectangular grid capital ...
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Multilevel minimization - boundary conditions
I am interested in minimizing
$$min_{x \in R^{n^l}} f^l(x),$$
where $f^l(x)$ is nonlinear objective function arising from discretization of PDE. I would like to use nonlinear multilevel minimization ...
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How suitable is multigrid method for time-dependent PDEs?
For elliptic PDEs (Poisson-type), the multigrid method is very sufficient, but how about time-dependent problems (i.e parabolic or hyperbolic PDEs)?
Is it efficient to solve such problems using a ...
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Can the standard multigrid performance be used for time-dependent PDEs?
Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...
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Can a direct method like Thomas be used in a multigrid method as a smoother?
As far as I know, multigrid uses stationary iterative methods as smoothers (i.e GS), but can we use a direct method also?
For example, in case we have a tridiagonal system (for example 1D heat ...
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Is there any method to incorporate minor changes into solved meshes to speed convergence in particle-in-cell solvers?
I'm trying to perform a $10^6$ timestep electrostatic particle-in-cell simulation on a rather large mesh, with very limited computational resources (a single GPU). Because of the large number of ...
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How do multigrid approaches deal with Gibbs phenomenon?
I know (from https://scicomp.stackexchange.com/a/31339/20545, among others) that I need a certain mesh density in FEM, else I might get non-physical oscillations in my solution.
How do multigrid ...
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Null space for smoothed aggregation algebraic multigrid
I do not really get the point of null space usage for creating the prolongation operator for smoothed aggregation algebraic multigrid. I know what the null space is per definition and I know that the ...
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Algebraic multigrid for complex valued matrices
Assume one uses the classical AMG with Ruge-Stuben coarsening and direct interpolation for solving real valued problems. How can this approach be recycled to also solve complex valued problems like ...
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Multigrid preconditioner for conjugate gradient methods
When solving $A*x=b$ using preconditioned conjugate gradient methods one has to solve $z=K^{-1}*r$ for the preconditioning where $K$ is the preconditioner of $A$ and $r$ is the residual vector.
...
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Converting mass density to point mass approximation on a grid
In an nbody gravity simulation, instead of doing exact(all-pair brute force) solution, I added masses of each body into cells of a 3D grid(each cell is just a float value having a mass value). Then ...
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Neumann-Neumann boundary intersection
I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
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Algebraic multigrid in PETSc
Consider a potential/poisson equation on a very large, complicated geometry. Currently, an self-written FEM and linear solvers from NumPy are used. Performance is, of course, not good enough for ...
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For which problems Krylov subspace methods are preferred over multigrid methods?
As multigrid methods are known to have grid independent convergence rates with $O(N)$ computational cost, then why would one be interested in using Krylov subspace methods at all, for which ...
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V-cycle Multigrid for 2D transient heat transfer on a square plate using finite difference
I'm currently developing a program to solve 2D transient state heat conduction on a square plate using the V-cycle multigrid. Althought my program is able to reach the steady state solution, it's ...
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What is the default smoother for the "PCMG" preconditioner in PETSc?
For a large parallel sparse matrix (mpiaij type matrix) in my code, I was experimenting with various preconditioners to see which one would do best with GMRES/BiCGSTAB. I tried the PCMG ...
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Can I use multigrid to solve linear algebra problems that do not arise from a differential equation?
Can the Multigrid method be applied to solve a linear system of equations in the form $Ax=b$, that is not necessarily related to differential equations?
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Geometric Multigrid for Conform and Non–″ Elements: Restriction Operators
First of all, let me set up some notations.
Suppose we have a hierarchy of meshes $\mathcal{T}_0 \subset \mathcal{T}_1 \subset \dots \subset \mathcal{T}_n$. For simplicity I restrict myself here to $...
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What does it take to prove that a multigrid algorithm scales linearly with system size?
It is my understanding that the multigrid solution techniques are generally the preferable method to solve large Poisson problems. Now assume I have written a multigrid solver that is tailored to my ...
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Multigrid for Robin boundary conditions
I have a question regarding the treatment of Robin boundary conditions in a multigrid solver. I am solving the Poisson equation in $\Omega=(0,1)^2$ with Robin boundary conditions on the boundary,
$$- \...
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Algorithm for Adaptive Mesh Refinement
I am trying to implement Adaptive Mesh Refinement. I am not a Mathematics/Computational Science person so I will try to write the algorithm in a simpler way. I will be grateful if experts can comment ...
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Python environments for AMG and Gauss Seidel as solvers instead of preconditioners
I am working on block preconditioning and seemingly it is common to write customised Krylov solvers for them. Within each solver, the individual block linear system with preconditioners are ...
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Does algebraic multigrid reuse its coarse grids?
Admittedly, I'm new to the subject, so this is probably a really simple question.
Let's assume I want to solve the (large sparse) linear system Ax = b multiple times with algebraic multigrid and b ...
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Robust smoothers for geometric multigrid
I'm searching for robust smoothers for geometric multigrids.
By robust I mean:
Effective for high order approximations (say spectral element, spectral Discontinuous Galerkin),
Parallel (suitable for ...
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How to define residual in multigrid approach?
I wish to solve the two-dimensional Navier Stokes equations using multigrid method on a collocated grid using the Predictor-Corrector method mentioned below. But first, let me elaborate on what I had ...
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Algebraic Multigrid: Why does the product of interpolation and restriction not result in something with norm 1?
I'm currently working with "A Multigrid Tutorial" by Briggs et al, Chapter 8.
The construction of the interpolation operator is given as:
Then construction of restriction operator and fine grid ...
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