G= · 2020-04-24 · 1950 1970 1990 2010 9 9.2 9.4 9.6 9.8 10 0° 60°E 120°E 180° 120°W 60°W...
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Introduction into Dynamic mode Decomposition CD MD ) × time series Xj • & • • & • , 0 • @ I I I I I I I I I I in st t trajectory - wise picture : Jc = fax ) Kool - - Sco 1- finite dimensional . . - nonlinear Koopman - operator : observable ICI ) X-p c) = . if exit ) ) - infinite dimensional Kt : Koopman operator t linear Dynamic Mode Decomposition X = ( Tifton Tigh . . . .ie , ¥ m ) K' " is a good approximate of the Koopman opoofokg if X ! ( 9¥ . . . , at , .si - - - - Fine . . , ) i ) data sufficiently spare ii ) data suffices the rich E eigenfunctions of Khem X ' . X - s Ko 's X' Xt span of ITH it ) is hardly ever satisfied and also not a priori verifiable Reconstruction of dynamics outwit . II of . ewikstb ; + c. c . K' " ol ; = A ;¢ . w ; - - lnhilst • good for short times C " linear " ) • needs good truncation v • St needs to be adapted to characters - hi timescale
Transcript of G= · 2020-04-24 · 1950 1970 1990 2010 9 9.2 9.4 9.6 9.8 10 0° 60°E 120°E 180° 120°W 60°W...
Introduction intoDynamic
modeDecomposition CD MD )
×
time series Xj
•
&
•
•
&
•
,
0
•
@
I I I I I I I I I I
in
stt
trajectory- wise picture
: Jc = fax ) Kool -
-
Sco
1- finitedimensional
..
-nonlinear
Koopman-
operator: observable ICI ) X-p c) =
. if exit ) )
- infinite dimensionalKt
:
Koopman operatort linear
DynamicMode Decomposition
X = (Tifton Tigh . ..
.
.ie
,
¥m)
K'"
is a good approximate
ofthe
Koopmanopoofokg if
X!
( 9¥. .
.,
at,.si
- - --
- Fine. .
, )i) data sufficiently spare
ii ) datasufficesthe
rich
E eigenfunctions ofKhem
X
'
. X - s Ko's X' Xt span of
ITH
it) is hardlyever satisfied
and
also not a priori verifiable
Reconstruction of dynamics
outwit . II of.
ewikstb;
+ c. c.
K'"
ol;
= A ;¢.
w
;-
- lnhilst
•
good forshort times C
"
linear"
)
• needsgood
truncation v
• St needs to be adapted to characters-
hi timescale
MainMenage
:
fast transitory dynamics
* aan
Transitionstypically
occur on timescales faster
than the timescaleof
the equilibrium dynamics .
A samplingSt chosen to
compute K'M
maynot
be sufficient to resolve the transientdynamics
and hence DMD - reconstructionmay
be
bad .
( Gottwald &Gugole , y
stat Phys .
2019 )
tested for artificialdata
frantee Kurama to - Siu shu's
ly
equation ,and fer
NCE P re
analytesdata
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84°N
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NCEP reanalysis data: Northern Hemisphere
E(tk, r) =1
m
m�1X
l=0
kx(tk+l)� � exp(⌦l�t)bk � c.ckReconstruction error:
1950 1970 1990 20104
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reanalysis DMD reconstruction
large-scale regime change in the 1970s
transition from a predominantly negative NAO phase to a predominantly positive NAO phase
1950 1970 1990 20109
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150
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NCEP reanalysis data: Southern Hemisphere
E(tk, r) =1
m
m�1X
l=0
kx(tk+l)� � exp(⌦l�t)bk � c.ckReconstruction error:
reanalysis DMD reconstruction
large-scale regime change in the 1970sSignificant decrease of frequency of blocking events around the mid 1970s and more
zonal flow
