# Questions tagged [projection]

For questions about implementing mappings from a set of elements (points, vectors, etc) to some subset of these elements.

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### How is the integral of a projection over an element $T$ computed in practice? (deal.II related)

I'm studying an error estimator for the equation $\nabla\cdot(\beta u) + cu = f$ and it contains the following term $$||f - cU_h - \Pi(f-c U_h) ||_T$$ where : $\Pi$ is the local orthogonal $L^2$ ...
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### Matlab - Equality between 2 Fisher matrices constructed in a different way

I want to know if, on a Fisher matrix, the projection operation (with a Jacobian matrix) commutes with a matricial inversion operation. The 2 ways to build these 2 matrices are: 1) First method: 1.1) ...
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### Do the projection (with Jacobian) and marginalisation (inversion of matrix and remove a row/column and reinversion) commute?

I try to check the equality or the inequality between 2 Fisher matrices. The goal is too see if the projection (with Jacobian) and marginalisation (inversion of matrix and remove a row/column and ...
1 vote
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### Why does including the pressure in this FVM for Stokes 2nd Problem lead to wrong solutions?

I'm trying to learn how to use finite volume methods and I want to solve a more general case of Stokes' second problem i.e. an infinite half-plane oscillating harmonically with no-slip boundary ...
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### Projection method FVM poisson part, adding source term

The idea of the method is to decompose the Navier-Stokes equation into the solenoidal and irrotational parts. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
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### Projecting nodal solution to gauss points with certain accuracy

I am having a problem that was also mentioned at the accepted answer to this question by Wolfgang Bangerth. I need to calculate, as it was specified at the question at the link, F1 integral and for ...
1 vote
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### Lumped mass matrices for higher-order finite elements for CFD

Given that some of the mass lumping techniques, for example, row-sum lumping does not produce practically viable lumped mass matrices for all the element shapes, what are the techniques used for mass ...
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### Efficiently compute a projection matrix from Householders reflectors

Let $A \in \mathbb{R}^{m \times n}$ where $m \geq n$. Let $B$ and $\tau$ be the result of applying LAPACK's dgeqrfp method (R on the upper right triangle, and the ...
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### projective reconstruction from orthogonal views

This is a problem from projective geometry. Suppose I have a vector $z \in R^k$ of unit length $\| z \| =1$ inside a $k$-dimensional hypercube. I don't know its value but do know its projection upto ...
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### Projection onto the set of Orthogonal matrices

Let $M \in \mathbb{R}^{n \times n}$ and denote the set of Orthogonal matrices by \mathcal{O}_{n} = \left\lbrace Q \in \mathbb{R}^{n \times n} \colon QQ^{T} = \mathbb{I}_{n} \right\...
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### Geometric interpretation of lemma

I am currently studying eigenvalue problems. I already worked through the minimax-principles and seen why $\lambda_{h, m} \geq \lambda_m$, when comparing the eigenvalues of a discretization and the ...
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### Padua-type pointset for functions singular on line $x=y$

The Padua points $\mathrm{Pad}_{n} \subset [-1,1]^{2}$ are a unisolvent pointset with optimal growth of Lebesgue constant, described in detail here. With some work they can be used to generate a ...
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### Wanting an explanation of the variables in Iterative PCA algorithm

I've been trying to implement the CPU GS-PCA algorithm in this paper . The code starts on page 28 I have a program written a script in python which gives the same output as the C++ program in the ...
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I am reading Chapter 5 of Saad's iterative methods book, and I don't understand section 5.2.1 about the two propositions of optimality results. In the statements of the propositions, what does it mean ...
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### Stable Method of orthogonal projection onto a subspace with the help of Moore-Penrose inverse,

Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v$. Below is ...
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### Backward stable algorithm to get orthogonal projection onto the column space of a matrix

I have to find the orthogonal projection of a vector $b$ onto the matrix $A$ of size $m \times n$. In my application, I don't have the luxury of calculating the QR factorization. All I have are ...
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### How can I make sure the flow is divergence-free when I use moving mesh?

I am using projection method and P2/P1 finite element method to solve the incompressible Navier-Stokes equations while the mesh is constantly adapted as the body moves (edge swapping, splitting and ...
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### $L_2$ projection with integer constraints and prescribed sum

Suppose I am given a vector $v^0\in\mathbb{R}^n$ and integers $k,\ell\in\mathbb{Z}$. Assuming $k$ is close to zero (e.g. $0\leq k\leq5$), is there an algorithm for solving the following integer ...
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### How does this Constrained Minimization algorithm work?

I don't fully understand the subsection 3.2 Constrained minimization of this paper. In particular, I don't understand the first step "Register active set" and the definition of projection $P(x)$. ...
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### Efficiently removing projection to subspace without having an orthogonal basis

I have a number of vectors $v_1, …, v_n$ and another vector $w$, that all are linearly independent, but not orthogonal. Let $V := \mathop{\text{span}}(v_1, …, v_n)$. I need to remove $w$’s projection ...
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### How to project a vector into the H(div) space (in the context of finite elements)?

Say I have a simple elliptic PDE: $$-\nabla\cdot(K\nabla p) = f \;\;\;\text{in}\;\Omega$$ with the appropriate boundary conditions. I solve for $p$ using a FEM (a discontinuous Galerkin method to ...
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### Active Elements in Projected Newton's Method?

To those who are familiar with the projected Newton's method or projected gradient method... We consider a constrained optimization problem with simple bounds. Particularly, minimize f(x) subject to ...
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### Motivation behind Galerkin method

I have a question about Galerkin method. I don't understand why the Galerkin method weights the residual by the shape functions and sets it equal to zero. I want to know what is reason of this. Why we ...
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### Treatment of Neumann (Traction) boundary conditions using projection methods

I am looking to solve the incompressible Navier-Stokes equations in 3D, using an inflow boundary condition specifying a velocity: $\mathbf{u} = \mathbf{g}_0 \,\, \forall \,\, \mathbf{x} \in \Gamma_u$ ...
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### Projection Method: Boundary condition on intermediate velocity field

I'm trying to solve variable density and viscosity Navier-Stokes equation using lagged pressure projection method. I'm solving for cavity problem as a test case now (once I get projection right, I ...
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### Projection of vector field on to a gradient field

Say I have a vector field with non-zero curl, therefore the potential function depends on the path I choose to integrate. In this paper the authors proposed to project the vector field into a gradient ...
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### Mass-conservative reprojection (on a sphere)

I have a 2D distribution of mass on a sphere given as a matrix of masses in latitude-longitude grid cells. I need these masses projected to another grid on the same sphere with different location of ...
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### Lid driven cavity flow problem to solve with the help of projection method and use coding in fortran 90 [closed]

I want to solve Lid driven cavity flow with the help of projection method i want to make own fortran program to solve two dimensional unsteady flow. I propose to solve the equations in the following ...