Referring to Krylov Subspaces and the methods of solutions to linear systems of equations which exploit these spaces.

learn more… | top users | synonyms (1)

6
votes
2answers
121 views

PetSc vs Sundials for serial numerical computations?

I am currently working on a physics problem that turns into a non-linear boundary value problem. I need an efficient numerical solver that I could run on my laptop with i5 dual core CPU. I am ...
3
votes
2answers
198 views

A numerical GMRES example

I'm having trouble understanding how GMRES works. I've read the part in Saad's book and a few others but still I am confused. Can someone provide me a numerical example to understand it better? Or if ...
4
votes
1answer
68 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 ...
6
votes
0answers
74 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} :=& ...
4
votes
0answers
71 views

Conjugate gradient: the 1-norm of the residual

I am trying to solve $Ax=b$ using the conjugate gradient method. However, it is important to me to obtain a bound not only on the usual residual $||b-Ax_k||_2$ but also on the quantity $||b-Ax_k||_1$. ...
2
votes
0answers
57 views

Does Conjugate Residual really have convergence properties similar to that of Conjugate Gradient?

I have coded up a toy implementation of Conjugate Residual and have been testing it. Both wikipedia and the Saad claim that Conjugate Residual and Conjugate Gradient have similar convergence behavior....
2
votes
0answers
75 views

preconditioned Uzawa method with Petsc

I am trying to improve the resolution of a Stokes problem (P2/P1 on unstructured mesh) defined by the matrix $M$: $M= \begin{pmatrix} A_u & 0 & B_u \\ 0 & A_v & B_v\\ B_u^T & B_v^...
4
votes
1answer
146 views

Appropriate iterative linear solver for an eigenvalue problem

I'm trying to solve a generalized eigenvalue problem $$Ax = \lambda Bx, \quad A = A^\top > 0,\; B = B^\top > 0$$ with $\lambda \approx \sigma$ using Rayleigh Quotient Iteration (RQI) (RQI is ...
4
votes
1answer
107 views

Choosing preconditioner for unsymmetric pressure-velocity coupled system

I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one ...
3
votes
1answer
123 views

Trust-region Newton: implementation issue with Conjugate Gradient calculations

UPDATE: The problem turned out to be the step (refer penultimate paragraph below) where I was factoring out a small value from the vectors of the numerator and denominator and then computed dot ...
9
votes
1answer
246 views

How is Krylov-accelerated Multigrid (using MG as a preconditioner) motivated?

Multigrid (MG) may be used to solve a linear system $Ax=b$ by constructing an initial guess $x_0$ and repeating the following for $i=0,1..$ until convergence: Compute the residual $r_i = b-Ax_i$ ...
1
vote
0answers
107 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve Ax=b, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous direction ...
9
votes
0answers
99 views

Guidelines for nested preconditioners

Consider the situation where you want to solve a linear system using a preconditioned Krylov method, but applying the preconditioner itself involves solving an auxiliary system, which is done with ...
7
votes
1answer
256 views

Algorithm to calculate the exponential of an Hessenberg matrix

I am interested in computing the solution of a lage system of ODEs using a krylov method as in [1]. Such method involve functions related to the exponential (the so-called $\varphi$-functions). It ...
11
votes
2answers
403 views

preconditioning a krylov method with another krylov method

In method like gmres or bicgstab it could be attractive to use another krylov method as a preconditioner. After all they are easy to implement in a matrix-free way and in a parallel environment. For ...
6
votes
0answers
163 views

Wanted: sequences of linear systems for recycling Krylov solver analysis

In the solution of sequences of linear systems $$A_ix_i=b_i\quad\text{for}\quad i=1,2,\dots$$ with Krylov subspace methods, data can be recycled from already solved linear systems in order to speed up ...
13
votes
2answers
308 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
2answers
663 views

Convergence/stagnation of BiCGStab(l)

I am solving 3D time-harmonic Maxwell FDFD problems (which result in huge sparse linear systems) using BiCGStab(l). I have tried out a bunch of different methods and for my specific use case, it seems ...
6
votes
1answer
291 views

Are there any libraries out there that implement block Krylov subspace methods?

Question Are there libraries out there that implement block Krylov subspace methods? (I was not able to find any from a simple Google search.) Background Right now, I am working with a code that ...
7
votes
2answers
116 views

Convergence of adaptive finite elements with inexact solves

I'm working on some adaptive discontinuous Galerkin codes for time harmonic wave propagation, currently just Helmholtz, but will be branching out once I have a working prototype in this case. There ...
10
votes
2answers
898 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 ...
5
votes
1answer
189 views

Are there any specialized methods available for solving structurally symmetric sparse linear systems?

When solving $Ax=b$, prior knowledge about $A$'s structure can help in designing an efficient solver which exploits this information (e.g conjugate gradient method is to be used when $A$ is ...
4
votes
3answers
301 views

PRIMA gives an unstable result?

I am working with Modified Nodal Admittance representation of circuits. I am doing Model Order Reduction using PRIMA on MATLAB. I am considering these circuits as Descriptor State-Space systems. I ...
11
votes
4answers
528 views

Calculating determinant while solving $Ax=b$ using CG

I am solving $Ax=b$ for a huge sparse positive definite matrix $A$ using the conjugate gradient (CG) method. It is possible to compute the determinant of $A$ using the information produced during the ...
4
votes
2answers
167 views

Krylov subspace iterative methods in floating point arithmetic

Is there any work that considers Krylov subspace iterative methods in floating point arithmetic? I'm especially interested in how rounding errors influence the convergence and the accuracy of the ...
10
votes
1answer
546 views

Why is pinning a point to remove a null space bad?

A Poisson equation with all Neumann boundary conditions has a single constant dimensional null space. When solving via a Krylov method, the null space can be removed either by subtracting the mean of ...
12
votes
1answer
361 views

What is the current state of polynomial preconditioners?

I wonder what has happened to polynomial preconditioners. I am interested in them, because they appear to be comparatively elegant from a mathematical perspective, but as far as I have read in surveys ...
12
votes
1answer
198 views

How do low rank modifications affect Krylov method convergence?

Say I have a linear system $A x = b$, which converges quickly using a suitable Krylov method (such as CG or GMRES) for all $b$. If $B$ is a matrix with low rank $r$, will the same Krylov method on ...
6
votes
1answer
147 views

Identifying the name/provenance of a technique to find the nullspace vectors of a matrix by random sampling and the conjugate residual method

I have got a large sparse matrix $A \in \mathbb R^{n \times n}$ and I want to find non-trivial elements in the kernel/nullspace of this matrix. How can this be done? I would like to learn more about a ...
19
votes
3answers
599 views

Solving $(G^TA^{-1}G)x = b$ without inverting $A$

I have matrices $A$ and $G$. $A$ is sparse and is $n\times n$ with $n$ very large (can be on the order of several million.) $G$ is an $n\times m$ tall matrix with $m$ rather small ($1 \lt m \lt 1000$) ...
0
votes
1answer
592 views

A Comparison between GMRES, QMR and LU for Dense Matrices

As I see it, there are 3 ways to solve unstructured dense system of equations: GMRES, QMR and LU. Has anyone done a comparison for these three? As far as I know, LU is the preferred choice and it is ...
11
votes
1answer
358 views

How to establish that an iterative method for large linear systems is convergent in practice?

In computational science we often encounter large linear systems which we are required to solve by some (efficient) means, e.g. by either direct or iterative methods. If we focus on the latter, how ...
7
votes
2answers
1k views

Understanding OpenCL performance

I'm using ViennaCL's interface to Eigen as a way to leverage OpenCL. Specifically, I'm using the ::viennacl::linalg::bicgstab_tag with an Eigen sparse matrix. ...
18
votes
3answers
1k views

What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?

As I understand it, there are two major categories of iterative methods for solving linear systems of equations: Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid) Krylov Subspace methods ...
10
votes
1answer
964 views

Can a Krylov subspace method be used as a smoother for multigrid?

As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate ...
5
votes
3answers
774 views

Krylov Subspace Methods for Dense Systems

I am currently researching on the viability of using KS methods for solving large dense systems. What I wish to prove (or disprove) is that methods like CG, BiCG and QMR are as good (if not better) ...
14
votes
2answers
5k views

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 ...