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Questions tagged [saddlepoint]

For questions about finding saddlepoints, critical points of a function which are not local extrema.

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Saddle point system

I am solving a system of the form $$ \begin{pmatrix} A & b^T \\ b & 0 \end{pmatrix} \begin{pmatrix} x \\ \ell \end{pmatrix} = \begin{pmatrix} c\\ 0 \end{pmatrix} $$ Where $A$ is a symmetric ...
Beni Bogosel's user avatar
  • 1,067
3 votes
1 answer

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 ...
Federico Poloni's user avatar
3 votes
1 answer

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 ...
bob_bill's user avatar
5 votes
1 answer

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^...
FEGirl's user avatar
  • 405
5 votes
0 answers

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 $A: V \rightarrow V', B:V \rightarrow Q'$ ...
FEGirl's user avatar
  • 405
2 votes
0 answers

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 ...
FEGirl's user avatar
  • 405
2 votes
0 answers

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 ...
Vefhug's user avatar
  • 309
1 vote
0 answers

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 ...
azra's user avatar
  • 11
1 vote
2 answers

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 ...
Chenna K's user avatar
  • 944
3 votes
0 answers

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 ...
Thomas Klimpel's user avatar
3 votes
0 answers

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 ...
janou195's user avatar
  • 161
6 votes
1 answer

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 ...
Hoarsehinghing's user avatar
1 vote
1 answer

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 ...
MPIchael's user avatar
  • 2,935
0 votes
1 answer

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 ...
mdot's user avatar
  • 155
2 votes
0 answers

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 ...
fred's user avatar
  • 1,000