1. Multiscale Methods of Data Assimilation Achi Brandt The Weizmann Institute of Science UCLA abrandt@math.ucla.edu INRODUCTION EXAMPLE FOR INVERSE PROBLEMS -----------

2. Data Assimilation PDEs: Observation Projection 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

3. Nonlinear Multigrid PDE solver Nonlinear implicit time steps Nonlinear Multigrid PDE solver 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

5. Solving PDE : Influence of pointwise relaxation on the error Error of initial guess Error after 5 relaxation sweeps Error after 10 relaxations Error after 15 relaxations Fast error smoothing slow solution

6. LU = F h 2h 4h L h U h = F h L 2h U 2h = F 2h L 4h V 4h = R 4h L 2h V 2h = R 2h R 2h =: F h -L h U h

7. LU = F h 2h 4h L h U h = F h L 4h U 4h = F 4h Fine-to-coarse defect correction L 2h U 2h = F 2h 4 3 2 1 correction Truncation error estimator interpolation of changes

8. interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * *** Full MultiGrid (FMG) algorithm... * h0h0 h 0 /2 h 0 /4 2h h

9. F cycle h0h0 h 0 /2 h 0 /4 2h h ***... * interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * residual transfer no relaxation

10. Multigrid solvers Cost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)

11. Multigrid solvers Cost: 25-100 operations per unknown 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 (1977,1982)

12. Multigrid solvers Cost: 25-100 operations per unknown 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 (1977,1982)

13. Multigrid solvers Cost: 25-100 operations per unknown 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 (1977,1982)

15. Multigrid solvers Cost: 25-100 operations per unknown 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 (1977,1982)

16. 2h h 2  wavelength Non-local components: e i  x,  ≈ ±k Slow to converge in local processing The error after relaxation v(x) = A 1 (x) e ikx + A 2 (x) e -ikx A 1 (x), A 2 (x) smooth A r (x) are represented on coarser grids: A 1  + 2 i k A 1 ′ = f 1 = r h (x) e -ikx 1D Wave Equation: u”+k 2 u=f

17.  k   8,  8 )  1,  1 )  2,  2 )  3,  3 )  4,  4 )  5,  5 )  6,  6 )  7,  7 ) O(H) 2D Wave Equation: Du+k 2 u=f Non-local: e i(   x +  2 y)    +    ≈ k 2 On coarser grid (meshsize H):  Fully efficient multigrid solver  Tends to Geometrical Optics  Radiation Boundary Conditions: directly on coarsest level

18. Σ r = 1 m A r (x) φ r (x) Generally: LU=F Non-local part of U has the form L φ r ≈ 0 A r (x) smooth {φ r } found by local processing A r represented on a coarser grid m coarser grids

19. Multigrid solvers Cost: 25-100 operations per unknown 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 Massive parallel processing *Rigorous quantitative analysis (1986) FAS (1975) Within one solver (1977,1982)

20. Multigrid solvers Cost: 25-100 operations per unknown 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 (1977,1982)

21. 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

23. Nonlinear implicit time steps Nonlinear Multigrid PDE solver Parallel processing across space + time

25. Nonlinear implicit time steps Nonlinear Multigrid PDE solver Adaptable discretization 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

26. Not just initial-condition control 4D Multigrid Solver + Data Assimilation Correlations extending far in time

27. Full-Control Data Assimilation PDEs: Observation projection Residuals: Derivations: Observation Discretized: vectors Control to minimize positive definite covariance matrices : discretization errors + modeling error estimates : measurement + representativeness errors

28. Not just initial-condition control 4D Full-Control Data Assimilation Correlations extending far in time COST = ??

29. 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 is asymptotically smooth is local on scale Fast multigrid inversion of

30. One-Shot Multigrid solution + assimilation Multigrid solver for the PDEs incorporating adjustments for the control

31. F cycle h0h0 h 0 /2 h 0 /4 2h h ***... * interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * residual transfer no relaxation

32. Relaxation of the Control k-th step:

33. Relaxation of the Control k-th step: approximately local, calculated locally calculated to best reduce E Complemented by changes at coarser levels Not done at the finest PDE levels

34. One-Shot 4D solution + assimilation Large-scale assimilation - at coarse levels Local deviations processed locally Multiscale windows advanced in time Multigrid solver for the PDEs incorporating adjustments for the control

36. 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 ) Multigrid solver for the PDEs incorporating adjustments for the control For fronts, orography, human interest… As far as extend on that coarse scale, hence extending only locally on that scale Still fully accurate (frozen )

37. Correlations extending far in time Not just initial-condition control 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

38. SIMPLE MODEL : 1D + time WAVE EQUATION The model (c, p, u 0, g) only partly and approximately known, but instead given. (1) (2) (3) (4) in For example, let u 0 be unknown and c known only approximately.. Comparing full-flow control with initial- value control Rima Gandlin: Find u in measurements.

39. Initial control vs. Residual control Discretization and algebraic errors (phase errors) Noised data A k = cos(kω δt), ω = 10, h = 0.1, δt = 0.01 Ak = cos(kωδt) + r, r  (-0.5,0.5) × 0.1 or 0.3 ω=10, h=0.1, δt=0.1 Exact solution: u(x,t) = e iωx cos(ωt)

40. 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) = e iωx cos(ωt) Initial control vs. Residual control

41. 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

42. Just local re-processing at each level Fast continual assimilation of new data Multiscale statistical ensembles Few local ensemble  Ensembles of fine-to- coarse  Fine-to-Coarse correction to covariance W Multiscale attractors correction

43. Data Assimilation PDEs: Observation Projection 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 Continuous assimilation

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