gpflow.models.cglb#

Classes#

gpflow.models.cglb.NystromPreconditioner#

class gpflow.models.cglb.NystromPreconditioner(A, LB, sigma_sq)[source]#

Bases: object

Preconditioner of the form $Q = (Q_{f} f + σ ² I) ⁻ ¹$ , where L is lower triangular with :math: LLᵀ = Kᵤᵤ $A = σ ⁻ ² L ⁻ ¹ K ᵤ ₓ$ and $B = A A ᵀ + I = L ᵦ L ᵦ ᵀ$

Parameters

A (Tensor) –
LB (Tensor) –
sigma_sq (float) –

Functions#

gpflow.models.cglb.cglb_conjugate_gradient#

gpflow.models.cglb.cglb_conjugate_gradient(K, b, initial, preconditioner, cg_tolerance, max_steps, restart_cg_step)[source]#

Conjugate gradient algorithm used in CGLB model. The method of conjugate gradient (Hestenes and Stiefel, 1952) produces a sequence of vectors $v_{0}, v_{1}, v_{2}, . . ., v_{N}$ such that $v_{0}$ = initial, and (in exact arithmetic) $K v_{n} = b$ . In practice, the v_i often converge quickly to approximate $K^{- 1} b$ , and the algorithm can be stopped without running N iterations.

We assume the preconditioner, $Q$ , satisfies $Q ≺ K$ , and stop the algorithm when $r_{i} = b - K v_{i}$ satisfies $| | r ᵢ ᵀ | |_{Q ⁻ ¹ r}^{2} = r ᵢ ᵀ Q ⁻ ¹ r ᵢ <= ϵ$ .

Parameters

K (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) – Matrix we want to backsolve from. Must be PSD. Shape [N, N].
b (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) – Vector we want to backsolve. Shape [B, N].
initial (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) – Initial vector solution. Shape [N].
preconditioner (NystromPreconditioner) – Preconditioner function.
cg_tolerance (float) – Expected maximum error. This value is used as a decision boundary against stopping criteria.
max_steps (int) – Maximum number of CG iterations.
restart_cg_step (int) – Restart step at which the CG resets the internal state to the initial position using the currect solution vector $v$ . Can help avoid build up of numerical errors.

Return type

Tensor

Returns

v where v approximately satisfies $K v = b$ .