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MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling

MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling. A. Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi. Major scaling bottlenecks: computing. Elementary particles (QCD) Schrödinger equation molecules condensed matter

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MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling

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  1. MULTISCALE COMPUTATION:From Fast SolversTo Systematic Upscaling A. Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi

  2. Major scaling bottlenecks:computing Elementary particles (QCD) Schrödinger equationmoleculescondensed matter Molecular dynamicsprotein folding, fluids, materials Turbulence, weather, combustion,… Inverse problemsda, control, medical imaging Vision, recognition

  3. Scale-born obstacles: • Many variablesn gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … • due to • Localness of processing

  4. Two-atom Lennard-Jones small potential step distance r0 00 r0

  5. small step Moving one particle at a timefast local ordering slow global move

  6. 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 smoothingslow solution

  7. Scale-born obstacles: • Many variablesn gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … • due to • Localness of processing 2. Attraction basins

  8. FluidsGas/Liquid r0 • Positional clustering • Lennard-Jones • Electrostatic clustering • Dipoles Water:1& 2

  9. E(r) r Optimizationmin E(r) multi-scale attraction basins

  10. t Macromolecule Dihedral potential G2 G1 T t 0 -p p + Lennard-Jones + Electrostatic ~104Monte Carlo passes for one T Gi transition

  11. E ri tijkl rij rl rj Potential Energy Lennard-Jones Electrostatic Bond length strain Bond angle strain torsion hydrogen bond rk

  12. Scale-born obstacles: • Many variablesn gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … • due to • Localness of processing 2. Attraction basins Removed by multiscale processing

  13. 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 smoothingslow solution

  14. h LU = F LhUh = Fh 2h L2hU2h = F2h L2hV2h = R2h 4h L4hV4h = R4h

  15. h 2h . . . h0/4 h0/2 h0 * * * * interpolation (order l+p) to a new grid interpolation (order m) of corrections residual transfer enough sweeps or direct solver relaxation sweeps * algebraic error < truncation error Full MultiGrid (FMG) algorithm

  16. Multigrid solversCost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)

  17. Multigrid solversCost: 25-100 operations per unknown Linear scalar elliptic equation (~1971) Nonlinear FAS (1975)

  18. h LhUh = Fh LU = F 2h U2h = Uh,approximate +V2h L2hV2h = R2h L2hU2h = F2h Fine-to-coarse defect correction 4h L4hU4h = F4h

  19. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  20. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  21. Same fast solver Local patches of finer grids • Each level correct the equations of the next coarser level • Each patch may use different coordinate system and anisotropic grid “Quasicontiuum” method [B., 1992] and differet physics; e.g.atomistic • Each patch may use different coordinate system and anisotropic grid anddifferent physics; e.g. Atomistic

  22. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  23. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  24. ALGEBRAIC MULTIGRID (AMG) 1982

  25. ALGEBRAIC MULTIGRID (AMG) 1982 Coarse variables - a subset 1. “General” linear systems 2. Variety of graph problems

  26. Graph problems Partition: min cut Clustering bioinformatics Image segmentation VLSI placement Routing Linear arrangement: bandwidth, cutwidth Graph drawing low dimension embedding Coarsening: weighted aggregation Recursion: inherited couplings (like AMG) Modified by properties of coarse aggregates General principle: Multilevel objectives

  27. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  28. 1D Wave Equation: u”+k2u=f 2p/w wavelength 2h h Non-local components: eiwx,w ≈ ±k Slow to converge in local processing The error afterrelaxation v(x) = A1(x)eikx + A2(x) e-ikx A1(x), A2(x) smooth Ar(x) are represented on coarser grids: A1 + 2 i k A1′ = f1 = rh(x)e-ikx

  29. 2D Wave Equation: Du+k2u=f Non-local: w2 (a3,b3) (a2,b2) (a1,b1) (a4,b4) k w1 (a5,b5) (a8,b8) (a7,b7) O(H) (a6,b6) ei(w1x + w2y) w12+ w22≈ k2 On coarser grid (meshsize H): • Fully efficient multigrid solver • Tends to Geometrical Optics • Radiation Boundary Conditions: directly on coarsest level

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

  31. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  32. N eigenfunctions Electronic structures (Kohn-Sham eq): i = 1, …, N= # electrons O(N) gridpoints per yi O(N2 ) storage Orthogonalization O(N3) operations Multiscale eigenbase 1D: Livne O(NlogN) storage & operations V = Vnuclear+ V(y) One shot solver

  33. Multigrid solversCost: 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 equationsFull matrix Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) (1977,1982) FAS (1975) Withinonesolver

  34. Integro-differential Equation differential , dense Multigrid solver Distributive relaxation: 1st order 2nd order Solution cost ≈ one fast transform(one fast evaluation of the discretized integral transform)

  35. Integral Transforms G(x,x) Transform Complexity O(n logn) Fourier Laplace O(n logn) O(n) Gauss Potential O(n) G(x,x) * Exp(ikx) O(n logn) Waves

  36. G(x,y) Glocal Gsmooth s |x-y| ~ 1 / | x – y | G(x,y) = Gsmooth(x,y) + Glocal(x,y) s ~ next coarser scale O(n) not static!

  37. Multigrid solversCost: 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 Monte-Carlo Massive parallel processing *Rigorous quantitative analysis (1986) (1977,1982) FAS (1975) Withinonesolver

  38. Multiscale ~ DiscretizationLattice for accuracy Monte Carlocost ~ “volume factor” “critical slowing down” Multigrid moves Many sampling cycles at coarse levels

  39. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  40. Same fast solver Local patches of finer grids • Each level correct the equations of the next coarser level • Each patch may use different coordinate system and anisotropic grid “Quasicontiuum” method [B., 1992] and differet physics; e.g.atomistic • Each patch may use different coordinate system and anisotropic grid anddifferent physics; e.g. Atomistic

  41. Repetitive systemse.g., same equations everywhere UPSCALING: Derivation of coarse equationsin small windows

  42. Scale-born obstacles: • Many variablesn gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … • due to • Localness of processing • Attraction basins Removed by multiscale processing

  43. Systematic Upscaling • Choosing coarse variables • Constructing coarse-leveloperational rules equations Hamiltonian

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