# Questions tagged [preconditioning]

For questions regarding design and implementation of preconditioners for solving linear systems.

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### Why is my iterative linear solver not converging?

What can go wrong when using preconditoned Krylov methods from KSP (PETSc's linear solver package) to solve a sparse linear system such as those obtained by discretizing and linearizing partial ...
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### How to directly compute the inverse of an ill-conditioned dense matrix

I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try? ...
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Suppose that I have a large (on the order of 10^6 unknowns) 3D scalar Poisson system which I would like to precondition. The boundary conditions have been treated so that the system is SPD. I.e., \... 18 votes 5 answers 3k views ### What is the advantage of multigrid over domain decomposition preconditioners, and vice versa? This is mostly aimed for elliptic PDEs over convex domains, so that I can get a good overview of the two methods. 13 votes 2 answers 2k views ### Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use? My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow): \begin{pmatrix}A & B^T \\ B & C\end{pmatrix} where C = 0 is the typical case (I have even ... 8 votes 2 answers 488 views ### Why does conjugate gradient work with this nonsymmetric preconditioner? In this previous thread the following multiplicative way to combine symmetric preconditioners P_1 and P_2 for the symmetric system Ax=b was suggested: \begin{align} P_\text{combo}^{-1} :=& ... 3 votes 2 answers 464 views ### Why iterative method: AMG preconditioned PCG is slower than Matlab direct method 'A\b'? Recently, I have met a question that a saying goes that for large linear system: iterative methods are required because of memory problem of direct methods. But when I implement some experiments ... 2 votes 0 answers 105 views ### Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method Consider the time-dependent Navier-Stokes equationu_t + (u \cdot \nabla) u - \Delta u + \nabla p = f\operatorname{div}(u)=0$$Looking at deal.ii tutorials, I've notice that there are ... 15 votes 2 answers 518 views ### Is there any way to do "double preconditioning" Question: Suppose that you have two different (factored) preconditioners for a symmetric positive definite matrix A:$$A \approx B^TB$$and$$A \approx C^TC,$$where the inverses of the factors B, ... 7 votes 3 answers 1k views ### role of initial guess for iterative linear solver Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ... 5 votes 2 answers 296 views ### Preconditioner for the GMRES method in the Uzawa algorithm I'm trying to solve \begin{equation}\left\{ \begin{split} \frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\frac1\rho\nabla p&=f\;\;\;\text{in }\Lambda\\ u&=0\;\;\;\text{on }\partial\... 7 votes 1 answer 654 views ### Which preconditioning for large linear elasticity problem? The problem I want to solve is the displacement formulation of the linear elasticity :$$ \nabla \cdot \sigma = 0 \quad \text{in} \quad \Omega \\ \sigma = \lambda ( \nabla \cdot u ) I + \mu (\nabla \...
I am a physicist with limited numerical methods knowledge and I am trying to speed up the inversion of a very ill-conditioned problem ($rcond>10^{30}$). The same sparse square matrix is used ...