problinsolve¶

probnum.linalg.
problinsolve
(A, b, A0=None, Ainv0=None, x0=None, assume_A='sympos', maxiter=None, atol=1e06, rtol=1e06, callback=None, **kwargs)[source]¶ Infer a solution to the linear system \(A x = b\) in a Bayesian framework.
Probabilistic linear solvers infer solutions to problems of the form
\[Ax=b,\]where \(A \in \mathbb{R}^{n \times n}\) and \(b \in \mathbb{R}^{n}\). They return a probability measure which quantifies uncertainty in the output arising from finite computational resources. This solver can take prior information either on the linear operator \(A\) or its inverse \(H=A^{1}\) in the form of a random variable
A0
orAinv0
and outputs a posterior belief over \(A\) or \(H\). This code implements the method described in Wenger et al. 1 based on the work in Hennig et al. 2. Parameters
A (
Union
[ndarray
,spmatrix
,LinearOperator
,RandomVariable
]) – shape=(n, n) – A square linear operator (or matrix). Only matrixvector products \(v \mapsto Av\) are used internally.b (
Union
[ndarray
,RandomVariable
]) – shape=(n, ) or (n, nrhs) – Righthand side vector, matrix or random variable in \(A x = b\). For multiple right hand sides,nrhs
problems are solved sequentially with the posteriors over the matrices acting as priors for subsequent solves. If the righthandside is assumed to be noisy, every iteration of the solver samples a realization fromb
.A0 (
Union
[ndarray
,spmatrix
,LinearOperator
,RandomVariable
,None
]) – shape=(n, n) – A square matrix, linear operator or random variable representing the prior belief over the linear operator \(A\). If an array or linear operator is given, a prior distribution is chosen automatically.Ainv0 (
Union
[ndarray
,spmatrix
,LinearOperator
,RandomVariable
,None
]) – shape=(n, n) – A square matrix, linear operator or random variable representing the prior belief over the inverse \(H=A^{1}\). This can be viewed as taking the form of a preconditioner. If an array or linear operator is given, a prior distribution is chosen automatically.x0 (
Union
[ndarray
,RandomVariable
,None
]) – shape=(n, ) or (n, nrhs) – Prior belief for the solution of the linear system. Will be ignored ifAinv0
is given.assume_A (
str
) –Assumptions on the linear operator which can influence solver choice and behavior. The available options are (combinations of)
generic matrix
gen
symmetric
sym
positive definite
pos
(additive) noise
noise
maxiter (
Optional
[int
]) – Maximum number of iterations. Defaults to \(10n\), where \(n\) is the dimension of \(A\).atol (
float
) – Absolute convergence tolerance.rtol (
float
) – Relative convergence tolerance.callback (
Optional
[Callable
]) – Usersupplied function called after each iteration of the linear solver. It is called ascallback(xk, Ak, Ainvk, 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 considerably.kwargs (optional) – Optional keyword arguments passed onto the solver iteration.
 Return type
 Returns
x – Approximate solution \(x\) to the linear system. Shape of the return matches the shape of
b
.A – Posterior belief over the linear operator.
Ainv – Posterior belief over the linear operator inverse \(H=A^{1}\).
info – Information on convergence of the solver.
 Raises
ValueError – If size mismatches detected or input matrices are not square.
LinAlgError – If the matrix
A
is singular.LinAlgWarning – If an illconditioned input
A
is detected.
Notes
For a specific class of priors the posterior mean of \(x_k=Hb\) coincides with the iterates of the conjugate gradient method. The matrixbased view taken here recovers the solutionbased inference of
bayescg()
3.References
 1
Wenger, J. and Hennig, P., Probabilistic Linear Solvers for Machine Learning, Advances in Neural Information Processing Systems (NeurIPS), 2020
 2
Hennig, P., Probabilistic Interpretation of Linear Solvers, SIAM Journal on Optimization, 2015, 25, 234260
 3
Bartels, S. et al., Probabilistic Linear Solvers: A Unifying View, Statistics and Computing, 2019
See also
bayescg
Solve linear systems with prior information on the solution.
Examples
>>> import numpy as np >>> np.random.seed(1) >>> n = 20 >>> A = np.random.rand(n, n) >>> A = 0.5 * (A + A.T) + 5 * np.eye(n) >>> b = np.random.rand(n) >>> x, A, Ainv, info = problinsolve(A=A, b=b) >>> print(info["iter"]) 9