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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18
votes
5answers
3k views

What is the advantage of multigrid over domain decomposition preconditioners, and vice versa?

This is mostly aimed for elliptic PDEs over convex domains, so that I can get a good overview of the two methods.
17
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2answers
1k views

Which libraries have good high-level support for multigrid?

I'm planning to use multigrid to calulate some eigenvalues and vectors, and I noticed PETSc has high-level support for multigrid. The PETSc documentation says that this part of PETSc should not be ...
15
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3answers
2k views

multigrid method to solve PDE

I need simple explanation of the Multigrid Method or some literature about this. I am familiar with iterational methods including BiCGStab,CG,GS,Jacobi and preconditioning, but I am a beginner with ...
15
votes
1answer
2k views

Can a Krylov subspace method be used as a smoother for multigrid?

As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate ...
14
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1answer
598 views

Is there a multigrid algorithm that solves Neumann problems and has a convergence rate independent of the number of levels?

Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann ...
13
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3answers
4k views

Is the Thomas algorithm the fastest way to solve a symmetric diagonally dominant sparse tridiagonal linear system

I am wondering if the Thomas algorithm is the fastest way (provably?) to solve a symmetric diagonally dominate sparse tridiagonal system in terms of algorithmic complexity (not looking for ...
13
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1answer
1k views

How is Krylov-accelerated Multigrid (using MG as a preconditioner) motivated?

Multigrid (MG) may be used to solve a linear system $Ax=b$ by constructing an initial guess $x_0$ and repeating the following for $i=0,1..$ until convergence: Compute the residual $r_i = b-Ax_i$ ...
13
votes
3answers
322 views

Is it usual to have no convergence checking in Multigrid?

I just read Chapter 3 in "A Multigrid Tutorial" by Briggs/Henson/McCormick, link. The text is about Multigrid cycles such as V-cycle, mu-cycle, FMG. What caught my eye: In most iterative procedures ...
12
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2answers
491 views

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 ...
12
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1answer
2k views

How exactly does the *full* multigrid algorithm run?

So I understand (or at least I believe I do) how a V-cycle runs. I've written in Matlab the 1-D, recursive version of a V-cycle. However, when I ran my code for FMG, my solution wasn't converging. I ...
12
votes
3answers
372 views

In what application cases are additive preconditioning schemes superior to multiplicative ones?

In both domain decomposition (DD) and multigrid (MG) methods, one may compose the application of the block updates or coarse corrections as either additive or multiplicative. For pointwise solvers, ...
11
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4answers
4k views

Algebraic Multigrid Code

I would like to understand more details about the implementation of Algebraic Multigrid Methods (AMG). I have been reading "A Multigrid Tutorial", which is quite good and explain all the details of ...
11
votes
1answer
2k views

How can one parallelize a multigrid method for solving a linear system of equations?

As I understand it, the multigrid method solves a linear system by solving a coarser version of the same problem (there by eliminating low frequency error) then projecting back to the fine grid to ...
10
votes
3answers
3k views

How to construct a prolongation and restriction operator for an algebraic multigrid solver?

I am trying to solve a linear system of equations that is sparse, but lacks any kind of banded structure. I have heard that there is a way to extend the principles of a multigrid solver for implicit ...
9
votes
3answers
481 views

FAS-multigrid slower than linear defect correction?

I have implemented a V-Cycle multigrid solver using both a linear defect correction (LDC) and full approximation scheme (FAS). My problem is the following: Using LDC the residual is reduced by a ...
9
votes
2answers
287 views

Multigrid on “not perfectly rectangular” grid

Multigrid introductions normally use a rectangular grid. Interpolation of values is then straight forward: Just interpolate linearly on the edge between two adjacent nodes of the coarse grid to find ...
9
votes
1answer
391 views

Increasing V-cycles for constant Coarsest Grid Size and increasing Fine Grid size

Problem statement I implemented geometric multigrid for $-\nabla^{2}=f$ where $f=\frac{3\pi^{2}}{4}sin \frac{\pi x}{2} sin \frac{\pi y}{2} sin \frac{\pi z}{2}$ on $\Omega \in [0,1]$ on a unit cube. ...
7
votes
2answers
120 views

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 ...
7
votes
1answer
247 views

Is multigrid useful for finding all eigenvalues and eigenvectors of a differential equation, or only the lowest eigenvalues?

I've been considering using a multigrid method to calculate the eigenvalues of a particular PDE. I know that multigrid is extremely good at finding the least eigenvalues and their associated ...
6
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1answer
294 views

Question about the smoothing operators in multigrid methods for nonlinear PDEs

Suppose we are dealing with a nonlinear problem, say $$ A u := L u + G(u) = f $$ the nonlinearity of the operator $A$ is the polynomial type, ie, $L$ is a linear operator, and $G(u) = u^k$, or ...
6
votes
1answer
311 views

Periodic BC for Multigrid in MD

I know that this question might be very specific and maybe nobody will know the answer, but this is probably the only community where I could find an answer: So, as part of my master's project, I am ...
6
votes
2answers
741 views

How to use grid sizes that are not powers of two in Geometric Multigrid

I am trying to solving a poisson equation in structured grid with Geometric Multigrid method. However, when coarsening the fine grid, I simply double the grid spacing at each successive level. That ...
6
votes
1answer
190 views

How to perform multigrid technique when relaxation methods don't converge?

It is well known that, when a system of linear equations is obtained from discretization of partial differential equation, the solution process can be accelerate significantly by multigrid technique. ...
6
votes
1answer
382 views

Full Multigrid Prolongation Operator

I am looking into full multigrid, FMG, and several sources, including these slides, that a lot of people are referring to, state that the prolongation operator used in FMG the first time you visit a ...
5
votes
1answer
968 views

restriction and interpolation in multigrid method

I need detailed explanation of the formula below A2=I1*A1*I2 I suppose this formula computes matrix A2 on a coarse grid and here A1 is original matrix on fine ...
5
votes
2answers
284 views

I'm having trouble debugging multigrid. What to do?

I've spent far too much time coding and debugging multigrid. While I clearly can't post all of my code as it would be silly to ask someone to go through all that code, is there anything I should pay ...
5
votes
1answer
116 views

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 ...
5
votes
1answer
243 views

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 ...
4
votes
1answer
93 views

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 ...
4
votes
1answer
259 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
4
votes
0answers
159 views

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 $...
3
votes
3answers
306 views

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 ...
3
votes
1answer
870 views

Full Multigrid convergence is too slow. What could possibly be causing it?

I've coded full multigrid in Matlab and it doesn't seem to be converging fast enough. When I increase the number of grids or the number of iterations, it converges to the analytical solution. But FMG ...
3
votes
1answer
133 views

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?
3
votes
1answer
177 views

stabilizing advection-diffusion with multi-grid?

If one chooses to discetize the advection-diffusion (AD) equation using the standard Galerkin finite element method, stability issues may arise in cases of high Peclet number (i.e., high advection to ...
3
votes
1answer
437 views

Linear Algebra / Numerical Solution Of Matrix With Nullspace

I have a question relating to linear algebra. We have a fluid solver that solves the poisson equation for pressures. When there are areas of the domain that are entirely enclosed by Neumann ...
3
votes
1answer
939 views

Prolongation/Restriction Operator in Multigrid

In Multigrid, using Poisson's equation, does the equality below always hold regardless of what type of boundary conditions you use? $$ R= c\cdot I^T, \text{ for some constant }c $$ where $R$ and $I$ ...
3
votes
1answer
99 views

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 ...
3
votes
2answers
662 views

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 ...
3
votes
2answers
429 views

Full Multigrid Performance for Poisson's equation using Higher Order Compact scheme as a Gauss Seidel smoother

I have a question regarding the FMG (Full Multi Grid) performance while computing Poisson's equation using Higher Order Compact discretization. I am using a sixth order compact scheme to discretize ...
3
votes
1answer
123 views

explicitly forming coarse matrices with polynomial smoothing AMG

I've been reading about the algebraic multigrid algorithm and came across polynomial smoothers in this paper. It's my understanding that usually the coarse-level matrices $A_H = I_h^HA_hI_H^h$ are ...
3
votes
1answer
158 views

Challenges in implementing Algebraic Multigrid on millions of processors

I just implemented an Algebraic Multigrid solver for a Mixed Dirichlet-Neumann Boundary Value problem and was surprised to see the speed-up as compared to a simple iterative solver for a large problem ...
3
votes
0answers
92 views

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 ...
3
votes
0answers
501 views

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 ...
3
votes
0answers
184 views

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, $$- \...
3
votes
1answer
226 views

Can F-cycle substitue FMG for update of existent solution?

I have a nicely working multigrid solver, which I use for solving the Poisson equation from an electrostatic problem. I solve this equation first without any charges, and then many times with a slowly ...
2
votes
1answer
65 views

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?
2
votes
2answers
91 views

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 ...
2
votes
1answer
146 views

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 ...
2
votes
1answer
75 views

Solving new linear system that comes from an $p$ enrichment

Let's say I am solving a simple Poisson problem using a Mixed (DG) finite element method. If we use orthogonal polynomials as basis functions we can write the finite-dimensional linear system as $$ ...