bayescg¶

probnum.linalg.bayescg(A, b, x0=None, maxiter=None, atol=None, rtol=None, callback=None)[source]¶

Conjugate Gradients using prior information on the solution of the linear system.

In the setting where \(A\) is a symmetric positive-definite matrix, this solver takes prior information on the solution and outputs a posterior belief over \(x\). This code implements the method described in Cockayne et al. 1.

Note that the solution-based view of BayesCG and the matrix-based view of problinsolve() correspond 2.

Parameters

A (array-like or LinearOperator, shape=(n,n)) – A square linear operator (or matrix). Only matrix-vector products \(Av\) are used internally.
b (array_like, shape=(n,) or (n, nrhs)) – Right-hand side vector or matrix in \(A x = b\).
x0 (array-like or RandomVariable, shape=(n,) or or (n, nrhs)) – Prior belief over the solution of the linear system.
maxiter (int) – Maximum number of iterations. Defaults to \(10n\), where \(n\) is the dimension of \(A\).
atol (float, optional) – Absolute residual tolerance. If \(\lVert r_i \rVert = \lVert Ax_i - b \rVert < \text{atol}\), the iteration terminates.
rtol (float, optional) – Relative residual tolerance. If \(\lVert r_i \rVert < \text{rtol} \lVert b \rVert\), the iteration terminates.
callback (function, optional) – User-supplied function called after each iteration of the linear solver. It is called as callback(xk, sk, yk, alphak, resid, **kwargs) and can be used to return quantities from the iteration. Note that depending on the function supplied, this can slow down the solver.

References

1: Cockayne, J. et al., A Bayesian Conjugate Gradient Method, Bayesian Analysis, 2019, 14, 937-1012
2: Bartels, S. et al., Probabilistic Linear Solvers: A Unifying View, Statistics and Computing, 2019