26
votes
Motivation behind Galerkin method
When I studied the finite element method in graduate school, this notion of multiplying by a weight function was also very alien to me. Eventually, I did find a nice (albeit non-rigorous) analogy ...
Paul♦
- 11.8k
5
votes
Accepted
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 ...
- 2,252
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 ...
- 316
4
votes
Accepted
Incompressible Navier-Stokes equations: Is projection method exact?
The projection scheme is indeed analytic, provided that the initial conditions are incompressible. You can derive it by taking the divergence of the evolution equation for $u$.
Finding the ...
- 159
4
votes
Approximation properties of FEM projections operators on a boundary
If you use a sharper trace inequality you get
$$\begin{aligned}\|u-Pu\|_{0,\partial \Omega}^2 &\lesssim \|u-Pu\|_0 \|u-Pu\|_1 \\
&\lesssim h^{k+1} h^{k}|u|^2_{k+1} \end{aligned} $$
implying
$$\...
- 1,822
4
votes
Efficiently removing projection to subspace without having an orthogonal basis
What about simply computing the projection of $w$ along the complement of $V$?
$$
\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 $\...
- 3,398
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 ...
- 50.2k
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 $y \in \mathbb C^n$; ...
- 861
3
votes
Accepted
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 ...
- 17.6k
3
votes
How to project a 0 genus mesh model on a sphere?
Here are some elements of answers to the three questions and references to alternative methods for spherical parameterization:
1. How to compute a voxelization of a given model ?
What it means:
...
- 2,165
3
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. ...
- 61
3
votes
Motivation behind Galerkin method
Let's say you want to solve the Laplace equation, $-\Delta u = f$. Ideally, of course, you'd like to find a function $u$ so that the residual is zero: $r(u) = -\Delta u - f = 0$. But $u$ is an ...
- 50.2k
3
votes
Active Elements in Projected Newton's Method?
The answer to your first question is no -- the active set does not only grow, it can also shrink, even if your objective function is convex. An example is if you start at some point on the boundary, ...
- 50.2k
2
votes
How to project a vector into the H(div) space (in the context of finite elements)?
The projection you consider is local on every cell. With the spaces you cite, your condition 1 requires test functions of degree $k-1$ on each edge (constant, linear, and quadrature along the edge); ...
- 50.2k
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 ...
- 3,673
2
votes
Motivation behind Galerkin method
Boris Grigoryevich wants you not to be able to create residuals with the same functions you used to create the solution.
- 1,373
2
votes
Accepted
Projection Method: Boundary condition on intermediate velocity field
What you are observing is odd-even decoupling between the velocity and pressure. This is a well-known problem on collocated grids. At any point on the grid, the gradient of the pressure is calculated ...
- 36
2
votes
Accepted
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 ...
- 8,553
2
votes
Projection of vector field on to a gradient field
In the matrix case, the least square solution of an overdetermined inconsistent system $Ax=b$ requires the solution of $(A^TA)x=A^Tb$. The projection $A(A^TA)^{-1}Ab$ reduces $b$ to the closest vector ...
- 3,501
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 ...
- 8,287
2
votes
Accepted
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 ...
- 4,286
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 ...
- 2,291
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 ...
- 2,139
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 $(k-1)$ ...
- 2,139
1
vote
Accepted
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 ...
- 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 ...
- 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 ...
- 1,972
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 ...
- 3,398
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 ...
- 599
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 ...
- 50.2k
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