Questions tagged [saddlepoint]
For questions about finding saddlepoints, critical points of a function which are not local extrema.
14
questions
3
votes
1
answer
56
views
Matrices that achieve worst-case $LDL^T$ element growth
Matrices that achieve the worst case $\rho_n = 2^{n-1}$ element growth in LU factorization with partial pivoting are known; see e.g. Theorem 9.7 in
Higham, Nicholas J., Accuracy and stability of ...
- 11.1k
3
votes
1
answer
346
views
Distributed Lagrange multiplier approach to impose constraint in Poisson equation
I'm trying to understand how Lagrange multipliers are applied in order to impose constraints in PDEs. Consider $B \subset \Omega$. For instance, a square inside another square domain $\Omega$. Let's ...
- 67
5
votes
1
answer
248
views
Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$
Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem:
$$M \dot{u} = f- Au -B^...
- 281
5
votes
0
answers
153
views
About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation
I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$:
$$A u + B^t p = f$$
$$Bu = g$$
where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
- 281
2
votes
0
answers
179
views
Understanding inf-sup conditions for classical saddle point problems
I'm studying the inf-sup conditions for saddle point problems. I'm referring to the usual one $$\begin{cases}Au + B^t p = f \\Bu=g \end{cases}$$ In the book I'm using (Ern - Guermond: Theory and ...
- 281
2
votes
0
answers
138
views
Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method
Consider the time-dependent Navier-Stokes equation
$$u_t + (u \cdot \nabla) u - \Delta u + \nabla p = f$$
$$\operatorname{div}(u)=0$$
Looking at deal.ii tutorials, I've notice that there are ...
- 309
1
vote
0
answers
37
views
constrained zero-sum two person game
Finding the saddle point of a constrained zero-sum two-person game is equivalent to a resolution of primal-dual programs (with bi-linear objective function).
I am looking for a free solver to compute ...
- 11
1
vote
2
answers
133
views
Parallelisation strategies for mixed FE formulations
Mixed FE formulations with LBB-stable elements require two different meshes for the primary and the constraint variables, for example, displacement and pressure. With continuous approximation for the ...
- 875
3
votes
0
answers
66
views
Some formulations of domain coupling lead to saddle point problems. Is this merely an artifact?
Background I wanted to learn how to couple FEM and BEM (for the Poisson equation), because I wanted to better understand how open boundary conditions look like. Therefore I worked through the relevant ...
- 2,141
3
votes
0
answers
170
views
Solving saddle point problem having non-invertible top-left block with a PETSc nested matrix
My system is a symmetric FE problem with lagrange multipliers:
$Z=\begin{pmatrix}A & C^T \\ C & 0\end{pmatrix}$
The matrix $A$ is positive semi-definite, non-invertible. The whole matrix is ...
- 151
6
votes
1
answer
367
views
Iterative linear solver for "ugly" saddle point system
I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern ...
1
vote
1
answer
155
views
Simplest way to precondition Uzawa iteration
I have a diffusion problem with an internal circular dirichlet constraint and a side condition which shall enforce a certain global volume integral.
$\nabla(D \nabla u(x)) = 0$
outer boundary ...
- 2,627
0
votes
1
answer
171
views
Saddle point formulation and minima
Given two Hilbert spaces $X$ and $Y$ and $a \colon X \times X \to \mathbb{R}$ as well as $b \colon X \times M \to \mathbb{R}$, both being continuous bilinear forms with $a(\cdot, \cdot)$ being ...
- 155
2
votes
0
answers
83
views
How can I numerically solve a saddle point problem with repeated constraints?
I am interested in numerically solving the following constrained minimization problem; Find the value of $x\in \mathbb{R}^n$ that minimizes $f$ where
$f\colon \mathbb{R}^n\to \mathbb{R}$ is defined ...
- 1,000