Questions tagged [multigrid]

Questions about multigrid methods for numerically solving differential equations.

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Find the number of rectangles not containing the shaded square [closed]

I wish to find the number of rectangles that don't contain the shaded square: Image of the grid: I used the way of finding the total number of rectangles and subtracting the rectangles containing the ...
Đỗ Quốc Khánh's user avatar
1 vote
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Creating nonuniform grids for FDM with multiple points of concentration

If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use: $$ S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S $$ where $c=\...
THAT'S MY QUANT MY QUANTITATIV's user avatar
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A mathematical expression/ model for a algorithm

I have created a simple algorithm in Python to create an ensemble deep-learning model to improve the accuracy of prediction. It's similar to a grid search method which I used to find the perfect ...
Navod Neranjan Thilakarathne's user avatar
1 vote
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W-cycle multigrid Python

I'm having troubles on solving a PDE by means of multigrid method in Python. I particular I have to implement a W-cycle. My function takes in input $uh$ as a starting vector, defined as follows, and ...
Onofrio Olivieri's user avatar
1 vote
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Laplace equation with a strange Neumann boundary condition

We want to solve the Laplace equation on the above grid. But, I really have no idea about how to start. I think "discretization" is needed and the computer can solve that. Am I right? The ...
Bob Dobbs's user avatar
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1 answer
344 views

What exactly is a multigrid preconditioner?

Background: In preconditioned Krylov subspace (KSP) methods (e.g. PCG, PGMRES etc.), a matrix $\boldsymbol{M \approx A}$ called preconditioner is required so that a new set of linear systems, e.g. $\...
Freewill's user avatar
1 vote
1 answer
116 views

How are interpolation operators derived for multigrid

I am trying to construct transfer operators $I^H_h, \, I^h_H$ for multigrid where $H \ne 2h$. I have gone through Briggs' tutorial, Hemker's paper, Hackbush's book, Trottenberg's book, but the details ...
lightxbulb's user avatar
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Connection two-grid, multigrid and finite elements

In some literature for the two grid method the finite element method is given as an introduction to the topic. But why is it done that way and where is the connection between these methods? The finite ...
marco31's user avatar
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What does it mean that a laplacian of directed graph has full rank?

Suppose we have a directed graph $G=(V, E)$ with $N=|V|$ nodes. Define normalized graph laplacian as $L=I-AD^{-1}$ where $A$ is a adjacency matrix of $G$ and $D$ is a degree matrix of $G$. I'm ...
port trum's user avatar
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1 answer
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Quantization to minimize multiplicative error

Consider a random variable $X$ taking values in $[0, 1]$ with unknown distribution. Consider the set $S_n = \left \{\frac{1}{n}, \frac{2}{n}, ..., \frac{n-1}{n} \right \}$ to be used to quantize $X$. ...
rims's user avatar
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How is a matrix connected to a grid?

I have a hard time finding information about and understanding how a matrix (adjacency matrix) is connected to a grid used in numerical analysis. What would the nodes be and are the matrix weighted or ...
The dickmaster's user avatar
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31 views

Minimum Error Solutions to Restricted Linear Systems

Suppose I have a linear system $Ax=b$ where $A_{n \times n}$ is square and full-rank. I want to restrict the space of solutions by a matrix $P_{n \times m}$ where $m << n$ and P has column rank $...
Amir Vaxman's user avatar
1 vote
0 answers
129 views

Solving the integral with norm vector over bounded region

I have to solve an integral like this: $$ d = \iint\limits_R{||x||^{-2/3} dA} $$ And the problem I have is described like this: *there is a disk region R having an area of 100 m² and in this ...
BigEl's user avatar
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Making largest line in 10 × 10 grid.

How many blocks can you pass through at most in a 10 × 10 grid. The Rules are 1. U cannot go over a line 2. You cannot lift the pencil. 3. You cannot allow the blocks you have passed through to make a ...
Haroon Arfan's user avatar
1 vote
0 answers
83 views

Understanding of multigrid method

I am struggling understanding Multi Grid methods. I would like to develop an example of two-grid methods applying to f.e. this equation: $$-u''(x)+u(x)=x, \qquad u(0)=u(1)=0$$ I am able to create a ...
danielleontiev's user avatar
1 vote
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Multigrid with different stencil

I have a code for MultiGrid in 1D. The code solves the Poisson equation with the central difference approximation. $uL-2u+uR=f$. I would like to generalize this approach and use it for the $uL-a u+...
JimBamFeng's user avatar
1 vote
1 answer
270 views

- Optimization - Standard Grid Search

I'm struck into an portfolio opt. problem and the paper I'm replicating (or, better, trying to) is using a "Standard Grid Search". Since I never encountered it before, I would like to ask you about: ...
L.D.Damono's user avatar
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1 answer
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Number of rectangle of size x,y which cover a cell p,q

I have a rectangle of size n*m (Where n is a number of rows and m is the number of columns) which is divided into of size 1*1. I want to calculate the number of the rectangle of size P*Q which ...
ivan R's user avatar
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What does "smooth" mean? (Numerical Analysis)

I know a notion of smoothness for functions, say, in $\mathbb{R}^n$, which simply means of class $C^\infty$. But in studying Numerical Analysis I sometimes read the term smooth for discrete functions, ...
artful_dodger's user avatar
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Is the application of the multigrid method correct?

We consider the boundary problem \begin{align*}-&y''(x)=0 \ \ \text{ for } \ x\in \Omega:=(0,\pi) \\ &y(0)=y(\pi)=0\end{align*} Let's pick the grid size $n=5$. I want to apply the 2-grid ...
Mary Star's user avatar
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1 vote
0 answers
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ADMM Heuristic or rather stick to Branch and Bound?

I'm intending on giving a solution to this MIQP, which should be faster than the one of solving the exact problem with GUROBI or CPLEX.(It muss not give always the global solution!) The main ...
M.Zhobro's user avatar
1 vote
1 answer
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How many ways can blocks be arranged in a grid

I have a 16x16 grid to fill up with 4 different sized blocks listed below: 16x16 16x8 8x16 8x8 How many different combinations of 16x16 grids can I make while filing the grid up completely? Just to ...
YAHsaves's user avatar
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2 answers
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Looks like a magic square, but can't solve it, please help.

A friend sent me this from her game I tried many combinations and I cannot solve it for the life of me. 9 squares starting from each corner is (top left 61, top right 61, bottom left 69, bottom ...
JoPic's user avatar
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1 answer
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Finding the value of a point in 3D domain depending on neighboring points

I have some data on 3D models of soil's moisture content, the model is in the finite-element grid. I have converted the program outputs to a table in four columns x, y, z, Theta where Theta is the ...
Mohammad ElNesr's user avatar
2 votes
0 answers
63 views

Can L, the square lattice on the plane, be partitioned into finitely many subsets that (up to translation) are contained in a rotated version of L?

I ran into this interesting problem while thinking about some stats models the other day. The context was efficient estimation of anisotropic covariance structures in geostatistical models, but it ...
user19904's user avatar
1 vote
0 answers
380 views

multigrid 2-d restriction and interpolation

How to build a restriction or interpolation matrices for 2-d problem for V-cycle multigrid. I read a lot of references about it, they only shows the coefficients matrix like following. $$ \...
kappasalt's user avatar
1 vote
0 answers
75 views

Coarse grid correction

Let $A_h \in \mathbb{R}^{n \times n}$ be the matrix corresponding to a finite element discretization of some nonselfadjoint, bounded and indefinite bilinear form corresponding to a second order ...
Astor's user avatar
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2 votes
0 answers
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For which problems Krylov subspace methods are preffered 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 ...
EngDR's user avatar
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1 answer
163 views

Idea for multi-grid in non-uniform mesh

I am new in multi-grid method. However, all tutorials I read are 2D Poisson equation on uniform mesh. In practices, the mesh is often stretched, but have no idea how to implement multi-grid method on ...
Yongxin's user avatar
1 vote
0 answers
256 views

Coupling Boundary Condition of one PDE with source term of another PDE

We have a system of equations, wherein the BC of one PDE is coupled with the source term of another PDE. We have a regular 2D unit grid in x and y. There are two PDEs to be solved The first PDE (...
Dr Krishnakumar Gopalakrishnan's user avatar
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0 answers
100 views

Transform cos to e function

What are the steps in order to transform the cosine function to the exponential function: $$ \left[\cos \left(\frac{k \pi} N\right)\right]^n \approx e ^ {\frac{-n}2 \left(\frac{k \pi} N \right)^2} $$...
hr0m's user avatar
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3 votes
1 answer
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How to make Galerkin coarse operator in multigrid?

In multigrid, the coarse operator is given by $$A^{2h}=RA^hP$$ where $R$ is a restriction and $P$ is a interpolation. In 1D case, I can implement it easily with or without making explicitly matrix $A^...
jakeoung's user avatar
  • 1,261
0 votes
1 answer
58 views

Plot my own function 'in two variables'

I have matlab function in two variables say $function_{something}(t)$ where $t$ is of size $2 \times k$, this is to allow to evaluate $k$ times in two values. So my output is then a $k \times 1$ ...
Nadori's user avatar
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1 answer
536 views

A mathematical expression for "grid search"?

I've got a question whether there is a mathematical expression for a grid search? I have two parameters a and b in [0;1]. Depending on the values of a and b, I get a value for my function (the value ...
user217484's user avatar
1 vote
2 answers
3k views

12 column grid, how to calculate for columns(5,7,8,9,10,11)?

I am terrible at math, this is css/sass related, but it's mainly a math question. I feel like the answer is very easy. You can see for example col-1 is ...
Michael Joseph Aubry's user avatar
0 votes
1 answer
194 views

What is it called when we interpolate a point INTO a grid...

Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find the value, $v$ (unknown) of a point that exists at some arbitrary co-ordinate $(x,y)$ ...
Spacey's user avatar
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1 vote
0 answers
35 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
pdevar's user avatar
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3 votes
0 answers
1k views

Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ \begin{...
disaster's user avatar
  • 123
0 votes
0 answers
73 views

Addressing/traversing an infinite 2D grid using a Z-line?

I'm looking for a method to map an infinite 2D grid using a line, so that I would have just one integer from which I would compute the X and Y. I know something like that exists, but can't recall the ...
Ondra Žižka's user avatar
4 votes
1 answer
859 views

Multigrid tutorial/book

I was reading Press et. al., "Numerical Recipes" book, which contain section about multigrid method for numerically solving boundary value problems. However, the chapter is quite brief and I would ...
Libor's user avatar
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