Slide 8
Slide 8 text
Estimation form data
Ulam’s method
reference measure m ∈ M+
(X), partition X =
i
Xi
(m-essentially disjoint), reduced space X := {X1, . . . , XN}
Markov matrix P over X: Pi,j
:= m(Xj ∩F−1(Xi ))
m(Xj )
,
estimate by sampling
slow convergence if support of m is high-dimensional
Modern variants
Markov state models
reaction coordinates, transition manifolds, . . .
Estimate adjoint Koopman operator K
basis functions (ψ1, . . . , ψM
) : X → R, estimate K in subspace
spanned by (ψa
)a
, based on samples (xi , yi
= F(xi
))N
i=1
ψa
(yi
) = (Kψa
)(xi
) ≈ b
Ka,bψb
(xi
)
least squares approximation for coefficients Ka,b
:
min
K
i,a
ψa
(yi
) −
b
Ka,bψb
(xi
) 2
wide variety of choices for (ψa
)a
, dictionary learning, kernel
methods, . . . ⇒ Koopmanism
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