Preconditioner Explained
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.
Preconditioning for linear systems
In linear algebra and numerical analysis, a preconditioner
of a matrix
is a matrix such that
has a smaller
condition number than
. It is also common to call
the preconditioner, rather than
, since
itself is rarely explicitly available. In modern preconditioning, the application of
, i.e., multiplication of a column vector, or a block of column vectors, by
, is commonly performed in a
matrix-free fashion, i.e., where neither
, nor
(and often not even
) are explicitly available in a matrix form.
Preconditioners are useful in iterative methods to solve a linear system
for
since the
rate of convergence for most iterative linear solvers increases because the
condition number of a matrix decreases as a result of preconditioning. Preconditioned iterative solvers typically outperform direct solvers, e.g.,
Gaussian elimination, for large, especially for
sparse, matrices. Iterative solvers can be used as
matrix-free methods, i.e. become the only choice if the coefficient matrix
is not stored explicitly, but is accessed by evaluating matrix-vector products.
Description
Instead of solving the original linear system
for
, one may consider the
right preconditioned system
and solve
for
and
for
.
Alternatively, one may solve the left preconditioned system
Both systems give the same solution as the original system as long as the preconditioner matrix
is nonsingular. The left preconditioning is more traditional.
The two-sided preconditioned systemmay be beneficial, e.g., to preserve the matrix symmetry: if the original matrix
is real symmetric and real preconditioners
and
satisfy
then the preconditioned matrix
is also symmetric. The two-sided preconditioning is common for
diagonal scaling where the preconditioners
and
are diagonal and scaling is applied both to columns and rows of the original matrix
, e.g., in order to decrease the dynamic range of entries of the matrix.
The goal of preconditioning is reducing the condition number, e.g., of the left or right preconditioned system matrix
or
. Small condition numbers benefit fast convergence of iterative solvers and improve stability of the solution with respect to perturbations in the system matrix and the right-hand side, e.g., allowing for more aggressive
quantization of the matrix entries using lower
computer precision.
The preconditioned matrix
or
is rarely explicitly formed. Only the action of applying the preconditioner solve operation
to a given vector may need to be computed.
Typically there is a trade-off in the choice of
. Since the operator
must be applied at each step of the iterative linear solver, it should have a small cost (computing time) of applying the
operation. The cheapest preconditioner would therefore be
since then
Clearly, this results in the original linear system and the preconditioner does nothing. At the other extreme, the choice
gives
which has optimal
condition number of 1, requiring a single iteration for convergence; however in this case
and applying the preconditioner is as difficult as solving the original system. One therefore chooses
as somewhere between these two extremes, in an attempt to achieve a minimal number of linear iterations while keeping the operator
as simple as possible. Some examples of typical preconditioning approaches are detailed below.
Preconditioned iterative methods
Preconditioned iterative methods for
are, in most cases, mathematically equivalent to standard iterative methods applied to the preconditioned system
For example, the standard
Richardson iteration for solving
is
Applied to the preconditioned system
it turns into a preconditioned method
Examples of popular preconditioned iterative methods for linear systems include the preconditioned conjugate gradient method, the biconjugate gradient method, and generalized minimal residual method. Iterative methods, which use scalar products to compute the iterative parameters, require corresponding changes in the scalar product together with substituting
for
Matrix splitting
A stationary iterative method is determined by the matrix splitting
and the iteration matrix
. Assuming that
is
symmetric positive-definite,
is
symmetric positive-definite,
- the stationary iterative method is convergent, as determined by
,the
condition number
is bounded above by
Geometric interpretation
For a symmetric positive definite matrix
the preconditioner
is typically chosen to be symmetric positive definite as well. The preconditioned operator
is then also symmetric positive definite, but with respect to the
-based
scalar product. In this case, the desired effect in applying a preconditioner is to make the
quadratic form of the preconditioned operator
with respect to the
-based
scalar product to be nearly spherical.
[1] Variable and non-linear preconditioning
Denoting
, we highlight that preconditioning is practically implemented as multiplying some vector
by
, i.e., computing the product
In many applications,
is not given as a matrix, but rather as an operator
acting on the vector
. Some popular preconditioners, however, change with
and the dependence on
may not be linear. Typical examples involve using non-linear
iterative methods, e.g., the
conjugate gradient method, as a part of the preconditioner construction. Such preconditioners may be practically very efficient, however, their behavior is hard to predict theoretically.
Random preconditioning
One interesting particular case of variable preconditioning is random preconditioning, e.g., multigrid preconditioning on random coarse grids.[2] If used in gradient descent methods, random preconditioning can be viewed as an implementation of stochastic gradient descent and can lead to faster convergence, compared to fixed preconditioning, since it breaks the asymptotic "zig-zag" pattern of the gradient descent.
Spectrally equivalent preconditioning
The most common use of preconditioning is for iterative solution of linear systems resulting from approximations of partial differential equations. The better the approximation quality, the larger the matrix size is. In such a case, the goal of optimal preconditioning is, on the one side, to make the spectral condition number of
to be bounded from above by a constant independent of the matrix size, which is called
spectrally equivalent preconditioning by
D'yakonov. On the other hand, the cost of application of the
should ideally be proportional (also independent of the matrix size) to the cost of multiplication of
by a vector.
Examples
Jacobi (or diagonal) preconditioner
The Jacobi preconditioner is one of the simplest forms of preconditioning, in which the preconditioner is chosen to be the diagonal of the matrix
Assuming
, we get
It is efficient for
diagonally dominant matrices
. It is used in analysis softwares for beam problems or 1-D problems (EX:-
STAAD.Pro)
SPAI
The Sparse Approximate Inverse preconditioner minimises
where
is the Frobenius norm and
is from some suitably constrained set of
sparse matrices. Under the Frobenius norm, this reduces to solving numerous independent least-squares problems (one for every column). The entries in
must be restricted to some sparsity pattern or the problem remains as difficult and time-consuming as finding the exact inverse of
. The method was introduced by M.J. Grote and T. Huckle together with an approach to selecting sparsity patterns.
[3] Other preconditioners
External links
Preconditioning for eigenvalue problems
Eigenvalue problems can be framed in several alternative ways, each leading to its own preconditioning. The traditional preconditioning is based on the so-called spectral transformations. Knowing (approximately) the targeted eigenvalue, one can compute the corresponding eigenvector by solving the related homogeneous linear system, thus allowing to use preconditioning for linear system. Finally, formulating the eigenvalue problem as optimization of the Rayleigh quotient brings preconditioned optimization techniques to the scene.[4]
Spectral transformations
By analogy with linear systems, for an eigenvalue problem
one may be tempted to replace the matrix
with the matrix
using a preconditioner
. However, this makes sense only if the seeking
eigenvectors of
and
are the same. This is the case for spectral transformations.
The most popular spectral transformation is the so-called shift-and-invert transformation, where for a given scalar
, called the
shift, the original eigenvalue problem
is replaced with the shift-and-invert problem
. The eigenvectors are preserved, and one can solve the shift-and-invert problem by an iterative solver, e.g., the
power iteration. This gives the
Inverse iteration, which normally converges to the eigenvector, corresponding to the eigenvalue closest to the shift
. The
Rayleigh quotient iteration is a shift-and-invert method with a variable shift.
Spectral transformations are specific for eigenvalue problems and have no analogs for linear systems. They require accurate numerical calculation of the transformation involved, which becomes the main bottleneck for large problems.
General preconditioning
To make a close connection to linear systems, let us suppose that the targeted eigenvalue
is known (approximately). Then one can compute the corresponding eigenvector from the homogeneous linear system
. Using the concept of left preconditioning for linear systems, we obtain
, where
is the preconditioner, which we can try to solve using the
Richardson iteration
The ideal preconditioning
is the preconditioner, which makes the
Richardson iteration above converge in one step with
, since
, denoted by
, is the orthogonal projector on the eigenspace, corresponding to
. The choice
is impractical for three independent reasons. First,
is actually not known, although it can be replaced with its approximation
. Second, the exact
Moore–Penrose pseudoinverse requires the knowledge of the eigenvector, which we are trying to find. This can be somewhat circumvented by the use of the Jacobi–Davidson preconditioner
T=(I-\tildeP\star)(A-\tildeλ\starI)-1(I-\tildeP\star)
, where
approximates
. Last, but not least, this approach requires accurate numerical solution of linear system with the system matrix
, which becomes as expensive for large problems as the shift-and-invert method above. If the solution is not accurate enough, step two may be redundant.
Practical preconditioning
Let us first replace the theoretical value
in the
Richardson iteration above with its current approximation
to obtain a practical algorithm
A popular choice is
using the
Rayleigh quotient function
. Practical preconditioning may be as trivial as just using
T=(\operatorname{diag}(A))-1
or
T=(\operatorname{diag}(A-λnI))-1.
For some classes of eigenvalue problems the efficiency of
has been demonstrated, both numerically and theoretically. The choice
allows one to easily utilize for eigenvalue problems the vast variety of preconditioners developed for linear systems.
Due to the changing value
, a comprehensive theoretical convergence analysis is much more difficult, compared to the linear systems case, even for the simplest methods, such as the
Richardson iteration.
External links
Preconditioning in optimization
In optimization, preconditioning is typically used to accelerate first-order optimization algorithms.
Description
For example, to find a local minimum of a real-valued function
using
gradient descent, one takes steps proportional to the
negative of the
gradient
(or of the approximate gradient) of the function at the current point:
The preconditioner is applied to the gradient:
Preconditioning here can be viewed as changing the geometry of the vector space with the goal to make the level sets look like circles.[5] In this case the preconditioned gradient aims closer to the point of the extrema as on the figure, which speeds up the convergence.
Connection to linear systems
The minimum of a quadratic functionwhere
and
are real column-vectors and
is a real
symmetric positive-definite matrix, is exactly the solution of the linear equation
. Since
, the preconditioned
gradient descent method of minimizing
is
This is the preconditioned Richardson iteration for solving a system of linear equations.
Connection to eigenvalue problems
The minimum of the Rayleigh quotientwhere
is a real non-zero column-vector and
is a real
symmetric positive-definite matrix, is the smallest
eigenvalue of
, while the minimizer is the corresponding
eigenvector. Since
is proportional to
, the preconditioned
gradient descent method of minimizing
is
This is an analog of preconditioned Richardson iteration for solving eigenvalue problems.
Variable preconditioning
In many cases, it may be beneficial to change the preconditioner at some or even every step of an iterative algorithm in order to accommodate for a changing shape of the level sets, as in
One should have in mind, however, that constructing an efficient preconditioner is very often computationally expensive. The increased cost of updating the preconditioner can easily override the positive effect of faster convergence. If
, a
BFGS approximation of the inverse hessian matrix, this method is referred to as a
Quasi-Newton method.
Sources
- Book: Axelsson, Owe . Iterative Solution Methods . 1996 . Cambridge University Press . 978-0-521-55569-2 . 6722.
- Book: D'yakonov, E. G. . Optimization in solving elliptic problems . 1996 . CRC-Press . 978-0-8493-2872-5 . 592 .
- Book: Saad, Yousef . Yousef Saad . Henk . van der Vorst . Henk van der Vorst . amp . 2001 . Iterative solution of linear systems in the 20th century . §8 Preconditioning methods, pp 193–8 . Numerical Analysis: Historical Developments in the 20th Century . C. . Brezinski . L. . Wuytack . . 0-444-50617-9 .
- Book: van der Vorst
, H. A.. Iterative Krylov Methods for Large Linear systems . Cambridge University Press, Cambridge . 2003 . 0-521-81828-1.
- Book: Chen, Ke . Matrix Preconditioning Techniques and Applications . 2005 . Cambridge University Press . 978-0521838283 . Cambridge . 61410324.
Notes and References
- Web site: An Introduction to the Conjugate Gradient Method Without the Agonizing Pain . Jonathan Richard . Shewchuk . August 4, 1994 .
- Henricus Bouwmeester, Andrew Dougherty, Andrew V Knyazev. Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods. Procedia Computer Science, Volume 51, Pages 276-285, Elsevier, 2015. https://doi.org/10.1016/j.procs.2015.05.241
- M. J. . Grote . T. . Huckle . amp . 1997 . Parallel Preconditioning with Sparse Approximate Inverses . . 18 . 3 . 838–53 . 10.1137/S1064827594276552 .
- Preconditioned eigensolvers - an oxymoron?. Electronic Transactions on Numerical Analysis. 7 . 104–123. 1998. Knyazev . Andrew V. .
- Book: Himmelblau, David M. . Applied Nonlinear Programming . New York . McGraw-Hill . 1972 . 0-07-028921-2 . 78–83 .