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5 votes
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How to robustly and numerically expand a $k$-order polynomial in two variables defined on a polygon domain?

The problem that you are seeing is a well known problem of these basis functions: $$\phi_{m,n}(x,y) = x^{m}y^{n}$$ I quote from here: The reason why the coefficient matrix is nearly singular and ...
Mithridates the Great's user avatar
5 votes
Accepted

Projection method FVM poisson part, adding source term

The smooth solution turned out to have BC's applied in the following way: Walls and inlet: $\frac{\partial p}{\partial n}=0$ Outlet: $p=0$ Actually thought that we need only one value of P to pin, not ...
2Napasa's user avatar
  • 362
4 votes

How can I make sure the flow is divergence-free when I use moving mesh?

It's important to realize that the original velocity field $\mathbf v_0$ is also not divergence free in a pointwise sense. Rather, it is only divergence free when tested with the pressure test ...
Wolfgang Bangerth's user avatar
4 votes

Incompressible Navier-Stokes equations: Is projection method exact?

Note: I originally posted an answer that I was not 100% pleased with, so I have revised it heavily. 1/20/2017 The projection method is not in an exact approximation to the full system in general. ...
A.Vigs's user avatar
  • 71
4 votes

Efficiently removing projection to subspace without having an orthogonal basis

What about simply computing the projection of $w$ along the complement of $V$? $$ \tilde w = [I - V(VV^T)^{-1}V^T]w, $$ where $V$ is the matrix that has the basis vectors of $V$ as columns. This $\...
Jan's user avatar
  • 3,418
3 votes

Questions about iterative projection methods in Saad book

In order to understand these results, you need to know how minimization and projection problems are connected. Namely, Let $\mathbb V$ be a subspace of $\mathbb C^n$ and take $y \in \mathbb C^n$; ...
56th's user avatar
  • 901
3 votes
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Minimizing linear objective on intersection of convex sets

See this recent paper on an extension of stochastic gradient descent that could be used on your problem: https://arxiv.org/abs/1511.03760 You could also apply Dykstra's algorithm (or any other ...
Brian Borchers's user avatar
2 votes

Motivation behind Galerkin method

While this question is old and has been answered by plenty of smart people, I just want to jot down the intuition I use to explain the Galerkin method to people. The goal in our situation is to find ...
spektr's user avatar
  • 4,248
2 votes
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Efficiently removing projection to subspace without having an orthogonal basis

Converted to an answer from my comments to Jan's answer. To fix dimensions and notation, let us say that $V$ is a $m\times n$ matrix with the vectors $v_i$ as columns. As Jan notes, we have to ...
Federico Poloni's user avatar
2 votes

Projection onto the set of Orthogonal matrices

Some thoughts on a particular case of low-rank matrices ($k=\text{rank}(M)\ll n)$. Here, I can suggest some economical version of SVDs: Rand-SVD Here is the R package documentation that also ...
Anton Menshov's user avatar
  • 8,672
2 votes
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Efficiently compute a projection matrix from Householders reflectors

Fortunately, LAPACK provides routines to deal with the $\mathbf Q$ factor from the $\mathbf A = \mathbf Q \mathbf R$ decomposition, [dgeqrf]. To find the projection of an arbitrary $\mathbf B$ onto ...
rchilton1980's user avatar
  • 4,896
1 vote

Efficient projection onto the kernel of a matrix

The complement of the null space, $Z^\perp$, is of course spanned by the rows of the matrices $A_i$. If $Nm$ is relatively small, you can compute an orthonormal basis of $Z^\perp$ by running the Gram-...
Wolfgang Bangerth's user avatar
1 vote
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How is the integral of a projection over an element $T$ computed in practice? (deal.II related)

You need to compute the projection $\Pi(f-cU_h)$ as a first step. This is a finite element field, so you can create a global field (in deal.II lingo: Create the corresponding finite element and a <...
Wolfgang Bangerth's user avatar
1 vote

Project to nearest point on convex polytope

Assume that $y$ is not in the polyhedron (it is easy to check whether it is, and we know that the distance is zero in that case). If $y$ is outside then the closest point will be on the surface of the ...
Abdullah Ali Sivas's user avatar
1 vote
Accepted

projective reconstruction from orthogonal views

Thinking about it even more, I feel like the problem resembles triangulation of lines under orthographic camera model. It is possible and useful to visualize the setting in 3D: The blue vector is ...
Tolga Birdal's user avatar
  • 2,229
1 vote

projective reconstruction from orthogonal views

The problem seems quite ill posed so I will assume some level of over-determinism is there. First, note that your vector, $\mathbf{z}$ lives on a hypersphere. Thus, it is indeed on a $(k-1)$ ...
Tolga Birdal's user avatar
  • 2,229
1 vote
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Wanting an explanation of the variables in Iterative PCA algorithm

Section 2.1 in the paper explains that PCA takes a list of vectors $X = [X^{(0)} | X^{(1)} | ... | X^{(N-1)}]$ and maps it to $T = [T^{(0)} | T^{(1)} | ... | T^{(N-1)}]$ So I presume $T$ contains ...
sav's user avatar
  • 257
1 vote

How can I make sure the flow is divergence-free when I use moving mesh?

What sort of field is it? You've said "divergence free", but do you mean harmonic (zero divergence and curl), solenoidal (non-curl), or a mix of both? The distinction is important because it will ...
Sean Lake's user avatar
  • 143
1 vote

$L_2$ projection with integer constraints and prescribed sum

Presuming you have a suitable solver (CPLEX, GUROBI, MOSEK, SCIP, many others), you can solve this as a Mixed-Integer Quadratic Program (by squaring the objective) or as a Mixed-Integer Second Order ...
Mark L. Stone's user avatar
1 vote

Projection of vector field on to a gradient field

To give some background to the answer given by LutzL, here is how you'd set up your linear system to start with. You want to find an image $L$ whose gradient $\nabla L$ is as close as possible to your ...
Roberto's user avatar
  • 11
1 vote
Accepted

Implicit projection method with inflow boundary conditions

tldr: Reformulate the projection and avoid the need for boundary conditions on the pressure. I think you are misinterpreting the projection scheme. In all formulations that I know, the pressure is ...
Jan's user avatar
  • 3,418
1 vote

Implicit projection method with inflow boundary conditions

The most obvious choice is setting $\mathbf{u}^* = \mathbf{u}^{n+1}$ on the boundary. This is true if your velocity is described by Dirichlet BCs, however, if your velocity BCs are described by ...
Charles's user avatar
  • 619
1 vote

Projecting a vector field onto a H(div) space

In the interior of cells, the Raviart-Thomas functions are continuous. As a consequence, the normal component is of course also continuous and the jump is zero. That may not be the case at places ...
Wolfgang Bangerth's user avatar

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