Data Driven, Non-Equilibrium Dynamics

Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges

Data Driven, Non-Equilibrium DynamicsHow Warm is it Getting and Other Tales in Uncertainty

JUAN M. RESTREPO

Department of Mathematicsand

Department of Statistics, and Physics of Oceans and Atmospheres

Oregon State University

SIAM Geosciences Meeting, 2017


THREE TIME-DEPENDENT ESTIMATION PROBLEMSGiven a random time series {X(t) ∈ RN : t ≤ t0} (from models,observations, controls):

I Retrodiction:

X(t) : t ≤ t0.

e.g., paleoclimate reconstruction, polluting sourceidentification.

I Nudiction:X(t) : t = t0.

e.g., initial conditions for weather/geodynamics models.I Prediction (no observations used):

X(t) : t > t0.

e.g., weather forecasting.


DATA ASSIMILATION IN GEOSCIENCES AND

ENGINEERING

Combine information derived from data and models....

Bayes Theorem:

P(X|Y) ∝ Likelihood× Prior


LOTS OF DATA IS GOOD!When data fool us...

0 100 200 300 400 500 600−2

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ESTIMATING FROM DATAWhen data fool us...

same data, zoomed in

0 5 10 15 20−2

−1

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ESTIMATING X FROM MODEL

dx = 4x(1− x2)︸︷︷︸−gradV(x)

dt + κdWt︸︷︷︸stochasticity

Double Well V(x)

-1 Stationary Distribution P(X)


DATA IS OFTEN SPARSE IN GEOSCIENCES

The Observations Ym

HOT

COLD


DATA ASSIMILATION IN GEOSCIENCES AND

ENGINEERINGCombine information derived from data and models....

Bayes Theorem:



REALLY GOOD MODEL...



REALLY GOOD DATA...



TIME DEPENDENT DATA ASSIMILATION

Bayes Theorem in Time:

P(X(0 ≤ t ≤ t∗)|Y(tm ≤ t0) ∝ Πmp(Ym|Xm)Πt[p(Xt)]

I How to find (at least first) moments ofP(X|Y) := P(X(0 ≤ t ≤ t∗)|Y(tm ≤ t0), whennonlinear/non-Gaussian?

I How to estimate when X has high dimensions?I How do we find distributions of P(Y|X) and P(X)?I How good are the estimates, for general case?I How do we interpret the outcome?


NONLINEAR/NON-GAUSSIAN EXAMPLE AND

EXTENDED KALMAN FILTER RESULTS1

Time10% uncertainty, ∆t = 1.

1R. Miller, M. Ghil, P. Gauthiez, Advanced data assimilation in stronglynonlinear dynamical systems, J. Atmo. Sci. 51 1037-1056 (1994)


THE EXTENDED KALMAN FILTER RESULTS

Time

20% uncertainty, ∆t = 1.

R. Miller, M. Ghil, P. Gauthiez, Advanced data assimilation in strongly nonlinear dynamical systems, J. Atmo. Sci. 511037-1056 (1994)



Time20% uncertainty, ∆t = 0.25.




Time


Time


Time

20% uncertainty, ∆t = 0.25.

The Good News: you get an estimate.The Bad News: you get an estimate.



APPROACHES ON NONLINEAR/NON-GAUSSIAN

PROBLEMS

I (Variance-minimizer)I KSP, (Kushner, Stratonovich, Pardoux), early 60’s


KSP FILTER AND SMOOTHER RESULTS

G. Eyink, J.M.R., Most Probable Histories, via the Mean Field Variational Approach, J. Stat. Phys. 2001G. Eyink, J.M.R., F. Alexander, A mean field approximation in data assimilation for nonlinear dynamics, Physica D, 2004G. Eyink, J.M.R., F. Alexander, Mean-Field Variational Data Assimilation Using Moment Closures, (unpublished) J. Stat.Phys. 2006


APPROACHES ON NONLINEAR/NON-GAUSSIAN

PROBLEMS

I Variance-minimizerI KSP, (Kushner, Stratonovich, Pardoux), early 60’s

I 4D-Var/Adjoint, Lorenc, Talagrand, Courtier, 80’s, Representer (Bennett)

I Extended Kalman Kalman, Bucy 60’s, EnKF (Evensen, ’92) ,Local/Transform EnKF UMD group ’95, Hybrid EnKF Reich ’05

I Sample-BasedI Particle Filters Crisan, Van Leeuwen, Gordon, Del Moral, ’90s

I Mean Stochastic Sampler (Harlim and Majda, ’10)

I Langevin Sampler (A. Stuart, ’05)

I Path Integral Monte Carlo (JMR ’07. Alexander, Eyink & JMR, ’05)

I OtherI Mean Field Variational (Eyink, JMR, ’01)

I Diffusion Kernel Filter (Krause, JMR, ’09)

I Relative Entropy Minimizer, (Eyink, et al, ’05)

Restrepo, Leaf, Griewank, Circumventing storage limitations in variational data assimilation, SIAM J. Sci Comp, ’95


PIMC THE PATH INTEGRAL MONTE CARLO2

J. Restrepo, A Path Integral Method for Data Assimilation, Physica D, 2007,F. Alexander, G. Eyink, J. Restrepo, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005

2Cartoon from the Abstruse Goose


PIMC THE PATH INTEGRAL MONTE CARLO

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THE PATH INTEGRAL MONTE CARLO SMOOTHER

Π(Q|Y) ∝ e−Uobs(Y,Q)e−Umodel(Q)

J. M.R., A Path Integral Method for Data Assimilation, Physica D, 2007,J.M.R. A Homotopy Path Integral Filter, J. Stat. Phys. 2017, in preparation.F. Alexander, G. Eyink, J. M.R, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005



If Prob[e−Umodel ] ∼ exp(−Z2/D):

dx− f (x, t)dt = [2D(x, t)]1/2dW

is approximated as

qn+1 − qn −∆tf (qn, tn) = [2D(qn, tn)]1/2[Wn+1 −Wn]

n = 0, 1, ...,T − 1 Hence,

Umodel ≈T∑

n=1

[(qn+1−qn−∆tf (qn, tn))>D(qn, tn)−1 (qn+1−qn−∆tf (qn, tn))],



If Prob[e−Uobs(q,Y)] ∼ exp(−Z2/R).

ym −H(qm) = [2R[qm, tm)]1/2ηm

m = 1, 2, . . . ,M.

Udata =

M∑m=1

[(ym −H(qm))> R(qm, tm)−1 (ym −H(qm))],


(GHMC) GENERALIZED HYBRID MARKOV CHAIN

MONTE CARLO

I P(Q|Y) ∝ e−H

I H = V(Q,Y) + K(P).I V(Q,Y) =−Umodel(Q)− Uobs(Q,Y)

I K(P) = −12 P>M−1P

I ∂τQ = G δHδP and

∂τP = −G> δHδQ .

J. M.R., A Path Integral Method for Data Assimilation, Physica D, 2007,J.M.R. A Homotopy Path Integral Filter, J. Stat. Phys. 2017, in preparation.F. Alexander, G. Eyink, J. M.R, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005


DOUBLE WELL OUTCOME

G. Eyink, J.M.R., Most Probable Histories, via the Mean Field Variational Approach, J. Stat. Phys. 2001G. Eyink, J.M.R., F. Alexander, A mean field approximation in data assimilation for nonlinear dynamics, Physica D, 2004G. Eyink, J.M.R., F. Alexander, Mean-Field Variational Data Assimilation Using Moment Closures, (unpublished) J. Stat.Phys. 2006


RESEARCH DIRECTIONS

I Variance Estimation: ”less ad-hoc” estimation, increaseensembles, better experimental design.

I Reduce model uncertainty: better models (couplecomputation/data/model design.

I Scales matter: marginalization, NOT interpolation.

I Forecasting: least-squares is not the only thing we know.I Bias/Trend Errors. trends in multiscale problems are

challenging.


TREND (BIAS) ERRORS LEAD TOBAD ESTIMATION...



THE PREDICTION PROBLEM

Atmospheric CO2 at Mauna Loa Observatory (D. Keeling, and others, Scripps).


THE PREDICTION PROBLEM

Which estimate do we use in the forecast?


FORECASTING USING PIMC

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J. Restrepo, A Path Integral Method for Data Assimilation, Physica D, 2007,F. Alexander, G. Eyink, J. Restrepo, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005


FORECASTING USING PIMC

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In ”Prediction Mode, only Model has a Bearing on Results”


IMPROVING HURRICANE PREDICTIONS

Property Damage ($USD)I Harvey $190B ?I Katrina $108BI Sandy $65BI Ike $30BI Andrew $27BI · · ·


DYNAMIC LIKELIHOOD DATA ASSIMILATION

Use a model for the wave and observations...



Stochastic One-Way Wave Equation:

ut − C(x, t)ux = F(x, t), t > 0, x ∈ [0,L],

u(x, 0) = U(x), x ∈ [0,L],

F(x, t) = f (x, t) + Nf (t), C(x, t) = c(x, t) + Nc(t)

Φ`(0) = U(x`), ` = 1, 2, ...,N

dΦ = f (Φ)dt + A(t)dW(f )t ,

Φ(0) = U(x`),

dx = c(x, t)dt + B(t)dW(c)t ,

x(0) = x`,J.M.R., Dynamic Likelihood Approach to Filtering, Q. J. Roy. Met. Soc, 2017,P. Krause, J.M.R. Using the Diffusion Kernel Filter in Lagrangian Data Assimilation, Mon. Wea. Rev, 2009


KF Likelihood Dynamics Likelihood


DATA:MEASURED •PROPAGATED •

ζn+1 = ∆tc(ζn, tn) + ζn, tn ≥ tm,

Y(ζn+1, tn+1) = Y(ζn, tn),

Rn+1m ≈ An(t)[An(t)]T∆t + Rn, tn ≥ tm,


THE DYNAMIC LIKELIHOOD FILTER

Forecast (Like Kalman Filter):

V = Ln−1〈V〉n−1 + ∆tfn−1, n = 1, 2, . . . ,Nf − 1.

P = Ln−1Pn−1LTn−1 + Qn−1, n = 1, 2, . . . ,Nf − 1.

Multi-analysis (Dynamic Likelihood): project onto state space...

〈V〉n = Vn +Kn∑

m′∈m

(Hnm′Y

nm′ − Vn)δm′,n,

Kn = Pn[Pn +∑

m′∈m

Hnm′R

nm′ [Hn

m′ ]Tδm′,n]−1,

Pn = (I −Kn)Pn.



Exact Model

Dynamic Likelihood Kalman


FEATURE-BASED, LAGRANGIAN DATA BLENDING

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ClassicCont ourAnalysis

ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2


IMPROVING HURRICANE PREDICTIONS

I SHARPENING: Use an L1 estimator.I DISPLACEMENT CORRECTION: Either by adding

constraints, or by doing assimilation in space/time.

S. Rosenthal, S. Venkataramani, J.M.R., A. Mariano, Displacement Data Assimilation, J. Comp. Phys. 2016E. Chunikhina, J.M.R. Compressed Sensing and Optimal Sensor Placement in Data Assimilation, in preparation

.


DISPLACEMENT MAPS VIA CANONICAL

TRANSFORMATIONS

Find M such that

min ||q(M(x))− q0||22.

here (x, y) M→(X,Y).

In 2-Dimensions, the generating function isG(X, y) = Xy + f (X, y).

x =∂G∂y

= X + fy(X, y)

Y =∂G∂X

= y + fX(X, y).

invertible if fyX > −1.


DISPLACEMENT ASSIMILATION USING EKFTarget: min ||q(M(x))− q0||22.

Analysis

Displacement Map M


TAME DISTORTION:

Exploit strain tensor σ:

σ =

[x∆x y∆xx∆y y∆y

]=

11 + fyX

[−fyX −fyyfXX fXy − |H[f ]|

]Diagonals: normal strains, off-diagonals: shear strains.

Minimize instead:

J [f ] =

∫D

[q(f )− q0]2 dx dy

+

∫Dα[(x∆x)2 + (y∆y)2

]+ β

[(y∆x)2 + (x∆y)2

]dx dy

α and β adjustable weights.


AREA-PRESERVING MAPS

Map Regularized Map


DISPLACEMENT ASSIMILATION USING ENKF

Truth, EnKF Truth, EnKF+Displacement

yields up to 70% improvements for small ensembles...


THE IMPORTANCE OF DETERMINING A TREND

Eking out change means determining systematic variability, and thedetermination of a trend.

I Global temperature, CO2, greenhouse gases, oceanacidification, ...

I Mean sea level.I Climate interpretation (Variability of weather and climate).I Many applications in econometrics, geosciences,

engineering.

J.M.R., D. Comeau, H. Flaschka, Defining a Trend using the Intrinsic Time Decomposition, New J. Phys. 2014


MATHEMATICAL FACT ABOUT EXTREMES

1880 1900 1920 1940 1960 1980 2000year

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RANDOM DATA

An n random data set has about same number of extreme highs andlows. Their occurrence declines as 1/n, the number of datum.


NOT SEEN IN TEMPERATURE EXTREMES3

1880 1900 1920 1940 1960 1980 2000year

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RANDOM DATA

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MOSCOW JULY DATA

An n random data set has about same number of extreme highs andlows. Their occurrence declines as 1/n, the number of datum.

3S. Rahmstorf, D. Coumou, Increase of Extreme Events in a Warming World,PNAS, 2011.


IS HARVEY A RARE/EXTREME EVENT?

I Harvey is not a rare event,but a manifestation of weather change.

Extreme, but not rare.


IS HARVEY A RARE/EXTREME EVENT?

I Harvey is not a rare event, but a manifestation of weatherchange. Harvey is extreme, but not rare.

I

I Establishing the connection between climate change andweather outcomes is ongoing research.

I Basic thermodynamics can be used to establish theoutcomes of climate change.


HOW WARM IS IT GETTING?

The time rate of change of the temperature

CdTdt

=14

(1− α)S︸︷︷︸incoming radiation

− σT4︸︷︷︸outgoing radiation

Sun Radiation: S = 1361 W/m2. Earth’s Albedo: α ≈ 0.3Calculated global temperature: -18oC.

Doubling CO2 decreases outgoing IR by 4.2 W/m2 (Held &Soden, 2000).Using the CO2 data, the greenhouse adjusted globaltemperature is 14.5oC.


HOW WARM IS IT GETTING?4

4Figure, from K. Emanuel


A CLIMATE SIGNAL...

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eg C

Vostok Ice Core data, Temperature


Given a finite-time time series Y(i), i = 1, 2, ...,N,

The Tendency T(i) is an Executive Summary of Y(i)

I Captures essentials of histogram in the abscissa of Y(i); andI Most essential multi scale information, derived from

ordinate of Y(i).

The Empirical Uncertainty U(i) := Y(i)− T(i)

I is simple entropically,I The histogram of U(i), is easy to parametrize


THE INTRINSIC TIME DECOMPOSITION (ITD)Given a sequence of real numbers {Y(i)}N

i=1,

Y(i) = BD +

D∑j=1

Rj(i)

where

Bj(i) = Bj+1(i) + Rj+1(i), j = 0, ...,D,and

B0(i) : = Y(i).

Bj are called BASELINES, and Rj are called ROTATIONS.

ITD is related to EMD (Empirical Mode Decomposition)

Frei and Osorio, Proc. Roy. Soc. London, (2006).


THE INTRINSIC TIME DECOMPOSITION (ITD)

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B0(i)=B1(i)+



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B0(i)=B1(i)+R1(i)databaselinerotation


How does the ITD (and EMD) work?

E [Bj] := {Sj, bj}.

{Sj}nj1 be locations of extrema of baselines, with values bj.

ITD:{Sj+1, bj+1} = E [(I + Mj)bj].

What is E?

(I + Mj)bj is ForwardTime/CenterDifference approximation of

∂

∂tB =

14wj(x)

∂

∂x

[wj(x)

∂B∂x

],

wj(x) = exp[2∫ x

0 pj(t)dt].


EXAMPLE CALCULATION

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Vostok Ice Core data, Temperature


Y(i) = BD +

D∑k=1

Rk(i), Bj+1(t) + Rj+1(i) = Bj(i).

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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2Raw Signal

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x 105

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2Baseline B1

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x 105

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Baseline B2

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x 105

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2Baseline B3

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x 105

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2Baseline B4

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x 105

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Baseline B5

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x 105

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2Baseline B6

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x 105

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2Baseline B7

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x 105

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2Baseline B8

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x 105

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2Raw Signal

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R1

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R2

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R3

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R4

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R5

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R6

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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proper rotation R7


FIND TENDENCYFind ITD:

Y(i) = BD +

D∑j=1

Rj(i),

Bj(i) = Bj+1(i) + Rj+1(i)

Choosing j∗ among the baselines {Bj(i)}Dj=1:

T(i) := Bj∗(i)

The ABSISSA information:I For j = 1, ..,D compute Fj := histogram[Y(i)− Bj(i)]I Determine the Symmetry sj of Fj via percentiles:

sj :=Prj

75 − 2 Prj50 − Prj

25

(Prj75 − Prj

25)


Choosing j∗ among the baseline {Bj(i)}Dj=1:

T(i) := Bj∗(i)

The ORDINATE information:Compute the Complexity cj vector cj := corr(Bj,Rj)

0 0.5 1 1.5 2 2.5 3 3.5 4

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2Raw Signal

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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2Baseline B1

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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Baseline B2

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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2Baseline B3

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x 105

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2Baseline B4

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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Baseline B5

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

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2Baseline B6

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x 105

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2Baseline B7

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x 105

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2Baseline B8

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x 105

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2Raw Signal

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

02

proper rotation R1

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

02

proper rotation R2

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

02

proper rotation R3

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

02

proper rotation R4

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

02

proper rotation R5

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

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proper rotation R6

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−4−2

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proper rotation R7


The Tendency T(i), and the Vostok signal Y(i)

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Raw Signal

tendency


HISTOGRAMS

Vostok: Y(i). U(i) := Y(i)− T(i) Empirical Uncertainty.

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0.5pdf[Y-mean(Y)]pdf[Y-tendency]

0 0.5 1 1.5 2 2.5 3 3.5 4Age #105

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Raw Signal

tendency


FINDING TENDENCY: AN APPLICATION TO DATA

ASSIMILATION

I Modeling Tool with which to discern ImportantStructures in Data

I Generates a compact surrogate model of the form

dXt = f (Xt, t)dt + noiset.

I T(i) is the cummulant of the drift term f (·).I Empirical moments would be obtained from hist(Y− T).


NON-EQUILIBRIUM DYNAMICS AND ITS DATA

I Data Assimilation (Essential in Geosciences):I high dimensioned, good models, and very sparse data.

I Research challenges:I Focus on Variance/uncertainty estimation.I Better models means less uncertainty.I Beyond Least Squares...I Finding Trends (and bias in time dependent problems).I Better observation networks for better (sparse) data sets.I Efficient marginalization strategies for multiscale problems.I Integration of modeling, data, and computation.

With funding from


MY COLLABORATORSI S. Venkataramani (U. Arizona)I G. Eyink (Johns Hopkins)I F. Alexander (LANL)I H. Flaschka (U. Arizona)I J. Ramırez (U. Nacional de Colombia)I A. Mariano (U. of Miami)I C. Dawson (UT Austin)I R. Miller (Oregon State)I S. Rosenthal (PNNL), D. Comeau (LANL), A. Jensen

(OSU), K. Bergstrom (Intel), W. Mayfield (OSU)

Further Informationhttp://www.math.oregonstate.edu/∼restrepo


NON-EQUILIBRIUM DYNAMICS AND ITS DATAI Data Assimilation (Essential in Geosciences):

I high dimensioned, good models, and very sparse data.I Research challenges:

I Finding Trends.I Better observation networks for better (sparse) data sets.I Efficient marginalization strategies for multiscale problems.I Focus on Variance/uncertainty estimation.I Better models means less uncertainty.I Integration of modeling, data, and computation.I Beyond Least Squares.

I Current Work in data-driven dynamics and estimation:I Stochastic Parametrization: improving fidelity in models.I Dimension Reduction: glassy systemsI Fidelity Computing: development of an oil-spill model

With funding from


NEW: HOMOTOPY PATH INTEGRAL DATA

ASSIMILATIONFind

Z1 =∫

P(Y|X)P(X)dx =∫

M(x)dx,starting from Z0 =

∫P(Y|X)dx =

∫L(x)dx:

Known

Create an optimal schedule S(δs,N), for Zs =∫

MsL1−sdx.A. Jensen, J. M. R., R. Miller, Homotopy Path Integral Data Assimilation, Dynamics & Statistics of the Climate System,2017


NEW: DIMENSION REDUCTION,WHEN YOU JUST DON’T HAVE A CHOICE

I Motivation: An ocean oiltransport model:

I 104 chemicals,I 102 droplet sizes,I O(N6) spatial dofs,I O(108) time steps.

I Applications: Chemicalreactions, combustion,Glassy systems.

S. C. Venkataramani, R. Venkataramani, J. M. R. Dimension Reduction for Systems with Slow Relaxation In Memory of LeoP. Kadanoff, J. Stat. Phys. 2017


NEW: STOCHASTIC PARAMETRIZATIONMotivation: Model Fidelity

I Data:

I Imperfect Model:dv = F(x, t, α+ δα)dt

I Sensitive-dependence onparameters:var[v] = 1

F2 | dFdα |var[α][〈v〉]2

Historgram of Data

Data Assimilation, usingstochastic model and data

J. M. R., S. Venkatarmani, Stochastic Longshore Model Dynamics, J. Water Res. 2017J.M.R. Wave Breaking Dissipation in the Wave-Driven Ocean Circulation, J. Phys. Ocean. 2007J.M.R, J. Ramirez, Banner, J. C. McWilliams, J. Phys. Ocean. 2010J. Ramirez, J.M.R., Luc Deike, Ken Melville, Stochastic Progressive Wave Breaking, J. Phys. Ocean. in prep. 2017


NEW: HIGH FIDELITY COMPUTING

Application of ideas presented here to build an oil transportmodel:

I Dimension reduction and data assimilation.I Stochastic parametrization and filtering ideas.I Couples computational resolution and the physics at

relevant scales.I Modeling, computation, and data assimilation are tightly

coupled, leading to computational efficiency.

J.M. R., J. Ramirez, S. Venkataramani, An Oil Fate Model for Shallow Waters, J. Mar. Sci. Eng, 2015J.M.R., S. Venkataramani, C. Dawson, Nearshore Sticky Waters, Ocean Modelling 2016J. Ramirez, S. Moghimi, J.M.R. Mass Exchange Dynamics of Oceanic Surface and Subsurface Oil, submitted, Bull. Mar.Poll., 2017


MY COLLABORATORSI S. Venkataramani (U. Arizona)I G. Eyink (Johns Hopkins)I F. Alexander (LANL)I H. Flaschka (U. Arizona)I J. Ramırez (U. Nacional de Colombia)I A. Mariano (U. of Miami)I C. Dawson (UT Austin)I R. Miller (Oregon State)I S. Rosenthal (PNNL), D. Comeau (LANL), A. Jensen

(OSU), K. Bergstrom (Intel), W. Mayfield (OSU)

Further Informationhttp://www.math.oregonstate.edu/∼restrepo

Data Driven, Non-Equilibrium Dynamics

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