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 ...
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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 ...
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5 votes
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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 ...
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  • 316
4 votes
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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 ...
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  • 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 $$\...
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  • 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 $\...
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  • 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 ...
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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 $y \in \mathbb C^n$; ...
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  • 861
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 ...
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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: ...
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  • 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. ...
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  • 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 ...
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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, ...
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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); ...
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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 ...
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  • 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.
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2 votes
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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 ...
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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 ...
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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 ...
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  • 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 ...
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  • 8,287
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 ...
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  • 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 ...
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1 vote
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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 ...
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  • 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 $(k-1)$ ...
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  • 2,139
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 ...
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  • 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 ...
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  • 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 ...
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1 vote
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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 ...
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  • 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 ...
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  • 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 ...
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