multiscale methods of data assimilation achi brandt the weizmann institute of science ucla...

Multiscale Methods of Data Assimilation

Achi BrandtThe Weizmann Institute of ScienceUCLA

[email protected]

INRODUCTION

EXAMPLE FOR INVERSE PROBLEMS-----------

Data Assimilation4D

PDEs: , ,u

x t Lu x tt

Observation Projection y Pu y

Nonlinear Multigrid PDE solver

• Nonlinear implicit time steps

• Adaptable discretization

• Space + time parallel processing

One-Shot Solver + Assimilator

• Not just initial-condition control

• Multiscale observational data

• Multiscale covariance matrices

• Improved regularization

• Continual assimilation

Nonlinear Multigrid PDE solver• Nonlinear implicit time steps


Cost per time step comparable to explicit step ??

Avoid forward extrapolation of nonlinear terms

Unconditional stability of long Rossby waves

Irad Yavneh and Jim Mcwilliams, 1994, 1995

Shallow water balance equations

Ray Bates, Yong Li, Steve McCormick and Achi Brandt, 1995, 2000, 2000

Global shallow water (&3D), semi-Lagrangian advection of potential vorticity

Solving PDE: Influence of pointwiserelaxation on the error

Error of initial guess Error after 5 relaxation sweeps

Error after 10 relaxations Error after 15 relaxations

Fast error smoothingslow solution

LU = F

h

2h

4h

LhUh = Fh

L4hU4h = F4h

h2

h4

Fine-to-coarse defect correction

L2hU2h = F2h

4

3

2

1

correctionTruncation error estimator

interpolation of changes

interpolation (order l+p)to a new grid

interpolation (order m)of corrections

relaxation sweeps

algebraic error< truncation error

residual transfer

ν νenough sweepsor direct solver*

**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν

Full MultiGrid (FMG) algorithm

..

.

*

1ν

1ν

1ν

2ν

2ν

2ν

Vcyclemultigrid

h0

h0/2

h0/4

2h

h

F cycle

h0

h0/2

h0/4

2h

h

**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν

...

*

1ν

1ν

1ν

2ν

2ν

2ν



relaxation sweeps


residual transfer


residual transferno relaxation

Multigrid solversCost: 25-100 operations per unknown

• Linear scalar elliptic equation (~1971)


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs* (1980)

• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics

Massive parallel processing*Rigorous quantitative analysis

(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

Σr = 1

m

Ar(x) φr(x)

Generally: LU=F

Non-local part of U has the form

L φr ≈ 0

Ar(x) smooth

{φr } found by local processing

Ar represented on a coarser grid

m coarser grids


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs* (1980)

• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations Full matrix• Statistical mechanics


(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs*

(1980)

• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics


(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

• Same fast solver

Local patches of finer grids

• Each patch may use different coordinate system and anisotropic grid

• Each patch may use different coordinate system and anisotropic grid and different

physics; e.g. Atomistic

and differet physics

• Possibly once for all corrections

• Each level correct the equations of the next coarser level ( ) correction



• Parallel processing across space + time




Local refinements in space + time

Uniform discretization stencils

enabling efficient high order

Coarser levels extending farther

• Parallel processing across space + time

Natural self adaptation criteria

based on local size of


4D Multigrid Solver + Data Assimilation

• Correlations extending far in time

Full-Control Data Assimilation4D

PDEs: , ,u

x t Lu x tt

Observation projection y Pu y

Residuals: , , ,u

x t x t Lu x tt

Derivations: Observation d y Pu y y

Discretized: vectors ,dControl to minimize

T TE S d Wd

positive definite covariance matrices ,S W

: discretization errors + modeling error estimates

1S

: measurement + representativeness errors 1W


4D Full-Control Data Assimilation


• COST = ??

Multiscale organization of observational data

Oct trees

Level-by-level coarsening (FAS interpolation)

Replacing dense observation by weighted averages (reduced errors)

Multiscale representation of covariance

1, ', ' , , ', ' ', 'W x t x t w x t G x t x t w x t

, ,

, ', 'G x t x t , is asymptotically smooth

kG G

kG is local on scale 02kkh h

Fast multigrid inversion of 1W

One-Shot Multigrid solution + assimilation

Multigrid solver for the PDEs

incorporating adjustments for the control

F cycle

h0

h0/2

h0/4

2h

h

**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν

...

*

1ν

1ν

1ν

2ν

2ν

2ν



relaxation sweeps


residual transfer


residual transferno relaxation

Relaxation of the Control

k-th step: , , ,k kx t x t x t

0 0 0 0 0 0 0

0 0 1 1 0 0 0

0 0 1 1 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

k x

-

-

Relaxation of the Control

k-th step: , , ,k kx t x t x t

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 1 1 0 0 0

0 0 1 1 0 0 0

0 0 0 0 0 0 0

k x

-

-

, , ,k ku x t u x t x t

k approximately local, calculated locally

k calculated to best reduce E

Complemented by changes at coarser levels

Not done at the finest PDE levels

One-Shot 4D solution + assimilation

• Large-scale assimilation - at coarse levels

• Local deviations processed locally • Multiscale windows advanced in time



One-Shot 4D solution + assimilation

• Large-scale assimilation - at coarse levels

• Local deviations processed locally • Multiscale windows advanced in time • Coarse levels extending far in time • Further grid adaptation in space+time

• COST = O( # discrete variables )



For fronts, orography, human interest…

As far as extend on that coarse scale,

hence extending only locally on that scale

Still fully accurate (frozen )

k



4D Full-Control Data Assimilation

• COST = ??• COST comparable to the direct solver

At the finest flow levels:

No relaxation of the control

is interpolated from coarser levels

SIMPLE MODEL: 1D + time WAVE EQUATION

],0[ Tucutt )(tpu

)(00 xuu t

)(0 xgu tt

The model (c, p, u0, g) only partly and approximately

known, but instead given .kjkj dtxu ),(

(1)(2)

(3)

(4)

in

For example, let u0 be unknown and c known only approximately. .

],0[ T

Comparing full-flow control with initial-value control

Rima Gandlin:

Find u in

measurements .

Initial control vs. Residual control

Discretization and algebraic errors (phase errors)

Noised data

Ak = cos(kω δt), ω = 10, h = 0.1, δt = 0.01

Ak = cos(kωδt) + r, r(-0.5,0.5) × 0.1 or 0.3ω=10, h=0.1, δt=0.1

Exact solution: u(x,t) = eiωxcos(ωt)

Modeling error

phase error + noised data + modeling error:

c = 1.1 or 1.2, ω = 10, h = 0.1, δt = 0.01

ω = 10, h = 0.1, δt = 0.1

Exact solution: u(x,t) = eiωxcos(ωt)

Initial control vs. Residual control

Improved Regularization

• Natural to multiscale solvers

• Scale-dependent regularization

Reduced noise at coarse levels

Scale-dependent statistical theories of the atmosphere

Scale dependent data types

• Fine scale fluctuations -- Coarse scale amplitudes

• Just local re-processing at each level

Fast continual assimilation of new data

Multiscale statistical ensembles

• Few local ensemble Ensembles of fine-to-coarseFine-to-Coarse correction to covariance W

Multiscale attractors

correction

Data Assimilation4D

PDEs: , ,u

x t Lu x tt

Observation Projection y Pu y




• Space + time parallel processing

One-Shot Solver + Assimilator


• Multiscale observational data

• Multiscale covariance matrices

• Improved regularization

• Continuous assimilation

THANK YOU!

multiscale methods of data assimilation achi brandt the weizmann institute of science ucla...

Documents

h h slide

h u h slide

h fine

f cycle h0h0 h

relaxation slide

relaxation sweeps error

relaxations error

multigrid solvers cost