Keep in mind that you are only solving to a tighter tolerance on the mesh that you are currently using, and it is often more reasonable to refine the mesh. Failed to find a solution. With iterative methods, you always update your old guess and get hopefully a bit closer to the true solution.
So if you have a very large matrix, but you can relatively quickly compute its application to a vector for instance, because the matrix is very sparse or has some other kind of structureyou can use this to your advantage.
History[ edit ] Probably the first iterative method for solving a linear system appeared in a letter of Gauss to a student of his. However there is usually no point in making the tolerance too tight since the inputs to your model, such as Direct and iterative method properties, are often not accurate to more than a couple of digits.
We will introduce both of these methods and look at their general properties and relative performance, below. The big advantage of the iterative solvers is their memory usage, which is significantly less than a direct solver for the same sized problems.
The direct solvers differ primarily in their relative speed. Typically, the solution you get will then be close to the exact one.
The book presents other case studies using the iterative methods to solve monoenergetic transport and nonlinear network flow multidimensional boundary-value problems.
However, in the presence of rounding errors this statement does not hold; moreover, in practice N can be very large, and the iterative process reaches sufficient accuracy already far earlier. If you are going to change the relative tolerance, we generally recommend making the tolerance tighter in increments of one order of magnitude and comparing solutions.
This number comes into play with the numerical methods used to solve systems of linear equations. Krylov subspace methods[ edit ] Krylov subspace methods work by forming a basis of the sequence of successive matrix powers times the initial residual the Krylov sequence.
The book cites case studies involving iterative methods applications, including those concerning only three particular boundary-value problems. From the point of view of the solution, it is irrelevant which one of the direct solvers you choose, as they will return the same solution.
A typical convergence graph for an iterative solver is shown below: If you are solving a problem that does not have a solution such as a structural problem with loads, but without constraints then the direct solvers will still attempt to solve the problem, but will return an error message that looks similar to: Although COMSOL never directly computes the condition number it is as expensive to do so as solving the problem we do speak of the condition number in relative terms.
Different physics do require different iterative solver settings, depending on the nature of the governing equation being solved. Convergence of Krylov subspace methods[ edit ] Since these methods form a basis, it is evident that the method converges in N iterations, where N is the system size.
The book explains different general methods to present computational procedures to automatically determine favorable estimates of any iteration parameters, as well as when to stop the iterative process. The default iterative solvers are chosen for the highest degree of robustness and lowest memory usage, and do not require any interactions from the user to set them up.
These methods are compared in terms of CPU time, size of the matrices and number of iterations, for the same systems of equations with complex symmetric and indefinite matrices.
With a direct method, you have to do a certain amount of work, and then you obtain your solution. COMSOL will automatically detect the physics being solved as well as the problem size, and choose the solver — direct or iterative — for that case.
In the case of not even symmetric matrices methods, such as the generalized minimal residual method GMRES and the biconjugate gradient method BiCGhave been derived.
If you are working on problems that are not as well-conditioned, then the convergence will be slower. This tolerance can be made looser, for faster solutions, or tighter, for greater accuracy on the current mesh. When solving such a system of linear equations on a computer, one should also be aware of the concept of a condition numbera measure of how sensitive the solution is to a change in the load.
Iterative solvers, on the other hand, would be things like Gauss-Seidel iteration, SOR successive over-relaxationKrylov subspace methods like conjugate gradients, or multilevel methods. For well-conditioned problems, this convergence should be quite monotonic.
The two direct methods are based on a Crout variant, the LDLt factorization, but they are implemented with two different sparse storage modes, one being the standard skyline, and the other being the compressed sparse row.
The second big difference is that, for a direct method, you generally need to have the entire matrix stored in memory. The tolerance must always be greater than a number that depends on the machine precision 2.
The prototypical method in this class is the conjugate gradient method CG which assumes that the system matrix A is symmetric positive-definite. The theory of stationary iterative methods was solidly established with the work of D.
The text explains polynomial acceleration procedures for example, Chebyshev acceleration and conjugate gradient acceleration which can be applied to certain basic iterative methods or to the successive overtaxation SOR method.
The construction of preconditioners is a large research area.Solving the Linear System Matrix • Linear system matrix • Direct methods • Iterative methods • Suggestions on solver selection.
Choosing the linear system solver • COMSOL chooses the optimized solver and its settings based on the chosen space dimension, physics and study type. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after a single application of Gaussian elimination.
View Notes - Direct and Iterative Methods from PED PE00 at Petronas Technology University. UNIVERSITI TEKNOLOGI PETRONAS PCB RESERVOIR MODELLING AND SIMULATION MAY Matrix Solvers: Direct and. COMSOL will automatically choose a direct or iterative solver when solving linear systems of equations.
Learn more about these solvers here: Solutions to Linear Systems of Equations: Direct and Iterative Solvers Contrary to direct solvers, iterative methods approach the solution gradually, rather than in one large. For large periods, the iterative method is always convergent and requires (a lot) less storage than the direct methods.
In these cases, the iterative method shows this significant advantage over the direct methods tested. A Direct Solution of Linear Systems Iterative methods are msot useful in solving large sparse system.
2. One advantage is that the iterative methods may not require any extra storage and hence are more practical. 3. One disadvantage is that after solving Ax = b1, one must start over again from.Download