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Questions tagged [multigrid]

Questions about multigrid methods for numerically solving differential equations.

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25 views

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+...
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Discretization and grids approach

In a text about multigrid methods, there is the following in the introductory section. $L_h=-\Delta_h\equiv \text{standard five-point } O(h^2) \text{ approximation of the PDE operator }L\text{ as ...
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45 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: ...
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1answer
28 views

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 ...
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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, ...
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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 ...
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effect of relaxation factor for Jacobi smoother in multigrid solver

I've been reading about multigrid solvers and they mention that using Jacobi as a smoother with a relaxation factor can have a difference in the performance of the algorithm. I never really understood ...
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42 views

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 ...
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1answer
194 views

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 ...
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2answers
45 views

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 ...
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1answer
29 views

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 ...
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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 ...
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126 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. $$ \...
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56 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 ...
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52 views

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 ...
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1answer
102 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 ...
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109 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 (...
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44 views

Transform cos to e function

What are the steps in order to transform the cosine function to the exponential function: $$ \left[\cos \left(\frac{k \pi} N\right)\right]^n \approx e ^ {\frac{-n}2 \left(\frac{k \pi} N \right)^2} $$...
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1answer
665 views

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^...
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1answer
43 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 $k \times 1$ ...
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1answer
102 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 ...
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2answers
1k 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 ...
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1answer
82 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 $(x,y)$ ...
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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 ...
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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{...
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59 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 ...
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1answer
454 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 ...