Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.

A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2).

The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel,[1][2] who programmed it on the Z4,[3] and extensively researched.[4][5]

The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear equations and black-box objective functions.

Description of the problem addressed by conjugate gradientsEdit

Suppose we want to solve the system of linear equations

 

for the vector  , where the known   matrix   is symmetric (i.e., AT = A), positive-definite (i.e. xTAx > 0 for all non-zero vectors   in Rn), and real, and   is known as well. We denote the unique solution of this system by  .

Derivation as a direct methodEdit

The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the residuals and conjugacy of the search directions. These two properties are crucial to developing the well-known succinct formulation of the method.

We say that two non-zero vectors u and v are conjugate (with respect to  ) if

 

Since   is symmetric and positive-definite, the left-hand side defines an inner product

 

Two vectors are conjugate if and only if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation: if   is conjugate to  , then   is conjugate to  . Suppose that

 

is a set of   mutually conjugate vectors with respect to  , i.e.  . Then   forms a basis for  , and we may express the solution   of   in this basis:

 

Left-multiplying by   yields

 
 

This gives the following method[4] for solving the equation Ax = b: find a sequence of   conjugate directions, and then compute the coefficients αk.

As an iterative methodEdit

If we choose the conjugate vectors   carefully, then we may not need all of them to obtain a good approximation to the solution  . So, we want to regard the conjugate gradient method as an iterative method. This also allows us to approximately solve systems where n is so large that the direct method would take too much time.

We denote the initial guess for x by x0 (we can assume without loss of generality that x0 = 0, otherwise consider the system Az = bAx0 instead). Starting with x0 we search for the solution and in each iteration we need a metric to tell us whether we are closer to the solution x (that is unknown to us). This metric comes from the fact that the solution x is also the unique minimizer of the following quadratic function

 

The existence of a unique minimizer is apparent as its second derivative is given by a symmetric positive-definite matrix

 

and that the minimizer (use Df(x)=0) solves the initial problem is obvious from its first derivative

 

This suggests taking the first basis vector p0 to be the negative of the gradient of f at x = x0. The gradient of f equals Axb. Starting with an initial guess x0, this means we take p0 = bAx0. The other vectors in the basis will be conjugate to the gradient, hence the name conjugate gradient method. Note that p0 is also the residual provided by this initial step of the algorithm.

Let rk be the residual at the kth step:

 

As observed above,   is the negative gradient of   at  , so the gradient descent method would require to move in the direction rk. Here, however, we insist that the directions   be conjugate to each other. A practical way to enforce this is by requiring that the next search direction be built out of the current residual and all previous search directions. The conjugation constraint is an orthonormal-type constraint and hence the algorithm can be viewed as an example of Gram-Schmidt orthonormalization. This gives the following expression:

 

(see the picture at the top of the article for the effect of the conjugacy constraint on convergence). Following this direction, the next optimal location is given by

 

with

 

where the last equality follows from the definition of   . The expression for   can be derived if one substitutes the expression for xk+1 into f and minimizing it w.r.t.  

 

The resulting algorithmEdit

The above algorithm gives the most straightforward explanation of the conjugate gradient method. Seemingly, the algorithm as stated requires storage of all previous searching directions and residue vectors, as well as many matrix-vector multiplications, and thus can be computationally expensive. However, a closer analysis of the algorithm shows that   is orthogonal to  , i.e.   , for i ≠ j. And  is  -orthogonal to   , i.e.   , for  . This can be regarded that as the algorithm progresses,   and   span the same Krylov subspace. Where   form the orthogonal basis with respect to standard inner product, and   form the orthogonal basis with respect to inner product induced by  . Therefore,   can be regarded as the projection of   on the Krylov subspace.

The algorithm is detailed below for solving Ax = b where   is a real, symmetric, positive-definite matrix. The input vector   can be an approximate initial solution or 0. It is a different formulation of the exact procedure described above.

 

This is the most commonly used algorithm. The same formula for βk is also used in the Fletcher–Reeves nonlinear conjugate gradient method.

RestartsEdit

We note that   is computed by the gradient descent method applied to  . Setting   would similarly make   computed by the gradient descent method from  , i.e., can be used as a simple implementation of a restart of the conjugate gradient iterations.[4] Restarts could slow down convergence, but may improve stability if the conjugate gradient method misbehaves, e.g., due to round-off error.

Explicit residual calculationEdit

The formulas   and  , which both hold in exact arithmetic, make the formulas   and   mathematically equivalent. The former is used in the algorithm to avoid an extra multiplication by   since the vector   is already computed to evaluate  . The latter may be more accurate, substituting the explicit calculation   for the implicit one by the recursion subject to round-off error accumulation, and is thus recommended for an occasional evaluation.[6]

Computation of alpha and betaEdit

In the algorithm, αk is chosen such that   is orthogonal to  . The denominator is simplified from

 

since  . The βk is chosen such that   is conjugate to  . Initially, βk is

 

using

 

and equivalently

 

the numerator of βk is rewritten as

 

because   and   are orthogonal by design. The denominator is rewritten as

 

using that the search directions pk are conjugated and again that the residuals are orthogonal. This gives the β in the algorithm after cancelling αk.

Example code in MATLAB / GNU OctaveEdit

function x = conjgrad(A, b, x)
    r = b - A * x;
    p = r;
    rsold = r' * r;

    for i = 1:length(b)
        Ap = A * p;
        alpha = rsold / (p' * Ap);
        x = x + alpha * p;
        r = r - alpha * Ap;
        rsnew = r' * r;
        if sqrt(rsnew) < 1e-10
              break
        end
        p = r + (rsnew / rsold) * p;
        rsold = rsnew;
    end
end

Numerical exampleEdit

Consider the linear system Ax = b given by

 

we will perform two steps of the conjugate gradient method beginning with the initial guess

 

in order to find an approximate solution to the system.

SolutionEdit

For reference, the exact solution is

 

Our first step is to calculate the residual vector r0 associated with x0. This residual is computed from the formula r0 = b - Ax0, and in our case is equal to

 

Since this is the first iteration, we will use the residual vector r0 as our initial search direction p0; the method of selecting pk will change in further iterations.

We now compute the scalar α0 using the relationship

 

We can now compute x1 using the formula

 

This result completes the first iteration, the result being an "improved" approximate solution to the system, x1. We may now move on and compute the next residual vector r1 using the formula

 

Our next step in the process is to compute the scalar β0 that will eventually be used to determine the next search direction p1.

 

Now, using this scalar β0, we can compute the next search direction p1 using the relationship

 

We now compute the scalar α1 using our newly acquired p1 using the same method as that used for α0.

 

Finally, we find x2 using the same method as that used to find x1.

 

The result, x2, is a "better" approximation to the system's solution than x1 and x0. If exact arithmetic were to be used in this example instead of limited-precision, then the exact solution would theoretically have been reached after n = 2 iterations (n being the order of the system).

Convergence propertiesEdit

The conjugate gradient method can theoretically be viewed as a direct method, as in the absence of round-off error it produces the exact solution after a finite number of iterations, which is not larger than the size of the matrix. In practice, the exact solution is never obtained since the conjugate gradient method is unstable with respect to even small perturbations, e.g., most directions are not in practice conjugate, due to a degenerative nature of generating the Krylov subspaces.

As an iterative method, the conjugate gradient method monotonically (in the energy norm) improves approximations   to the exact solution and may reach the required tolerance after a relatively small (compared to the problem size) number of iterations. The improvement is typically linear and its speed is determined by the condition number   of the system matrix  : the larger   is, the slower the improvement.[7]

If   is large, preconditioning is commonly used to replace the original system   with   such that   is smaller than  , see below.

Convergence theoremEdit

Define a subset of polynomials as

 

where   is the set of polynomials of maximal degree  .

Let   be the iterative approximations of the exact solution  , and define the errors as  . Now, the rate of convergence can be approximated as [4][8]

 

where   denotes the spectrum, and   denotes the condition number.

Note, the important limit when   tends to  

 

This limit shows a faster convergence rate compared to the iterative methods of Jacobi or Gauss–Seidel which scale as  .

No round-off error is assumed in the convergence theorem, but the convergence bound is commonly valid in practice as theoretically explained[5] by Anne Greenbaum.

Practical convergenceEdit

If initialized randomly, the first stage of iterations is often the fastest, as the error is eliminated within the Krylov subspace that initially reflects a smaller effective condition number. The second stage of convergence is typically well defined by the theoretical convergence bound with  , but may be super-linear, depending on a distribution of the spectrum of the matrix   and the spectral distribution of the error.[5] In the last stage, the smallest attainable accuracy is reached and the convergence stalls or the method may even start diverging. In typical scientific computing applications in double-precision floating-point format for matrices of large sizes, the conjugate gradient method uses a stopping criteria with a tolerance that terminates the iterations during the first or second stage.

The preconditioned conjugate gradient methodEdit

In most cases, preconditioning is necessary to ensure fast convergence of the conjugate gradient method. The preconditioned conjugate gradient method takes the following form:[9]

 
 
 
 
repeat
 
 
 
if rk+1 is sufficiently small then exit loop end if
 
 
 
 
end repeat
The result is xk+1

The above formulation is equivalent to applying the conjugate gradient method without preconditioning to the system[10]

 

where

 

The preconditioner matrix M has to be symmetric positive-definite and fixed, i.e., cannot change from iteration to iteration. If any of these assumptions on the preconditioner is violated, the behavior of the preconditioned conjugate gradient method may become unpredictable.

An example of a commonly used preconditioner is the incomplete Cholesky factorization.[11]

The flexible preconditioned conjugate gradient methodEdit

In numerically challenging applications, sophisticated preconditioners are used, which may lead to variable preconditioning, changing between iterations. Even if the preconditioner is symmetric positive-definite on every iteration, the fact that it may change makes the arguments above invalid, and in practical tests leads to a significant slow down of the convergence of the algorithm presented above. Using the Polak–Ribière formula

 

instead of the Fletcher–Reeves formula

 

may dramatically improve the convergence in this case.[12] This version of the preconditioned conjugate gradient method can be called[13] flexible, as it allows for variable preconditioning. The flexible version is also shown[14] to be robust even if the preconditioner is not symmetric positive definite (SPD).

The implementation of the flexible version requires storing an extra vector. For a fixed SPD preconditioner,   so both formulas for βk are equivalent in exact arithmetic, i.e., without the round-off error.

The mathematical explanation of the better convergence behavior of the method with the Polak–Ribière formula is that the method is locally optimal in this case, in particular, it does not converge slower than the locally optimal steepest descent method.[15]

Vs. the locally optimal steepest descent methodEdit

In both the original and the preconditioned conjugate gradient methods one only needs to set   in order to make them locally optimal, using the line search, steepest descent methods. With this substitution, vectors p are always the same as vectors z, so there is no need to store vectors p. Thus, every iteration of these steepest descent methods is a bit cheaper compared to that for the conjugate gradient methods. However, the latter converge faster, unless a (highly) variable and/or non-SPD preconditioner is used, see above.

Conjugate gradient method as optimal feedback controller for double integratorEdit

The conjugate gradient method can also be derived using optimal control theory.[16] In this approach, the conjugate gradient method falls out as an optimal feedback controller,

 
for the double integrator system,
 
The quantities   and   are variable feedback gains.[16]

Conjugate gradient on the normal equationsEdit

The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations ATA and right-hand side vector ATb, since ATA is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGNR).

ATAx = ATb

As an iterative method, it is not necessary to form ATA explicitly in memory but only to perform the matrix-vector and transpose matrix-vector multiplications. Therefore, CGNR is particularly useful when A is a sparse matrix since these operations are usually extremely efficient. However the downside of forming the normal equations is that the condition number κ(ATA) is equal to κ2(A) and so the rate of convergence of CGNR may be slow and the quality of the approximate solution may be sensitive to roundoff errors. Finding a good preconditioner is often an important part of using the CGNR method.

Several algorithms have been proposed (e.g., CGLS, LSQR). The LSQR algorithm purportedly has the best numerical stability when A is ill-conditioned, i.e., A has a large condition number.

Conjugate gradient method for complex Hermitian matricesEdit

The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations   for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose using the MATLAB/GNU Octave style. The trivial modification is simply substituting the conjugate transpose for the real transpose everywhere. This substitution is backward compatible, since conjugate transpose turns into real transpose on real-valued vectors and matrices. The provided above Example code in MATLAB/GNU Octave thus already works for complex Hermitian matrices needed no modification.

See alsoEdit

ReferencesEdit

  1. ^ Hestenes, Magnus R.; Stiefel, Eduard (December 1952). "Methods of Conjugate Gradients for Solving Linear Systems" (PDF). Journal of Research of the National Bureau of Standards. 49 (6): 409. doi:10.6028/jres.049.044.
  2. ^ Straeter, T. A. (1971). "On the Extension of the Davidon–Broyden Class of Rank One, Quasi-Newton Minimization Methods to an Infinite Dimensional Hilbert Space with Applications to Optimal Control Problems". NASA Technical Reports Server. NASA. hdl:2060/19710026200.
  3. ^ Speiser, Ambros (2004). "Konrad Zuse und die ERMETH: Ein weltweiter Architektur-Vergleich" [Konrad Zuse and the ERMETH: A worldwide comparison of architectures]. In Hellige, Hans Dieter (ed.). Geschichten der Informatik. Visionen, Paradigmen, Leitmotive (in German). Berlin: Springer. p. 185. ISBN 3-540-00217-0.
  4. ^ a b c d Polyak, Boris (1987). Introduction to Optimization.
  5. ^ a b c Greenbaum, Anne (1997). Iterative Methods for Solving Linear Systems. doi:10.1137/1.9781611970937. ISBN 978-0898713961.
  6. ^ Shewchuk, Jonathan R (1994). An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (PDF).
  7. ^ Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). Philadelphia, Pa.: Society for Industrial and Applied Mathematics. pp. 195. ISBN 978-0-89871-534-7.
  8. ^ Hackbusch, W. (2016-06-21). Iterative solution of large sparse systems of equations (2nd ed.). Switzerland: Springer. ISBN 9783319284835. OCLC 952572240.
  9. ^ Barrett, Richard; Berry, Michael; Chan, Tony F.; Demmel, James; Donato, June; Dongarra, Jack; Eijkhout, Victor; Pozo, Roldan; Romine, Charles; van der Vorst, Henk. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (PDF) (2nd ed.). Philadelphia, PA: SIAM. p. 13. Retrieved 2020-03-31.
  10. ^ Golub, Gene H.; Van Loan, Charles F. (2013). Matrix Computations (4th ed.). Johns Hopkins University Press. sec. 11.5.2. ISBN 978-1-4214-0794-4.
  11. ^ Concus, P.; Golub, G. H.; Meurant, G. (1985). "Block Preconditioning for the Conjugate Gradient Method". SIAM Journal on Scientific and Statistical Computing. 6 (1): 220–252. doi:10.1137/0906018.
  12. ^ Golub, Gene H.; Ye, Qiang (1999). "Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration". SIAM Journal on Scientific Computing. 21 (4): 1305. CiteSeerX 10.1.1.56.1755. doi:10.1137/S1064827597323415.
  13. ^ Notay, Yvan (2000). "Flexible Conjugate Gradients". SIAM Journal on Scientific Computing. 22 (4): 1444–1460. CiteSeerX 10.1.1.35.7473. doi:10.1137/S1064827599362314.
  14. ^ 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. doi: 10.1016/j.procs.2015.05.241
  15. ^ Knyazev, Andrew V.; Lashuk, Ilya (2008). "Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning". SIAM Journal on Matrix Analysis and Applications. 29 (4): 1267. arXiv:math/0605767. doi:10.1137/060675290. S2CID 17614913.
  16. ^ a b Ross, I. M., "An Optimal Control Theory for Accelerated Optimization," arXiv:1902.09004, 2019.

Further readingEdit

External linksEdit