Document Sample

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 Molecular dynamics protein folding, fluids, materials Turbulence, weather, combustion,… Inverse problems da, control, medical imaging Vision, recognition Scale-born obstacles: • Many variables n gridpoints / particles / pixels / … • Interacting with each other O(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … due to 1. Localness of processing Moving one particle at a time fast local ordering small step slow global move 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 Scale-born obstacles: • Many variables n gridpoints / particles / pixels / … • Interacting with each other O(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … due to 1. Localness of processing 2. Attraction basins Optimization min E(r) E(r) r multi-scale attraction basins Scale-born obstacles: • Many variables n gridpoints / particles / pixels / … • Interacting with each other O(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … due to 1. Localness of processing 2. Attraction basins Removed by multiscale processing 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 h LU = F LhUh = Fh 2h L2hV2h = R2h L2hU2h = F2h 4h L4hV4h = R4h Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971) Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971) • Nonlinear FAS (1975) h LU = F LhUh = Fh 2h 2h 2h 2h h,approximate 2h 2h 2h 2h U =U +V U F L V =R Fine-to-coarse defect correction 4h 4h 4h 4h 4h L U =F Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) Local patches of finer grids • Same fast solver • Each level correct the equations of the next coarser level • Each patch may use different coordinate system and anisotropic grid and differet physics; e.g. atomistic “Quasicontiuum” method [B., 1992] Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) ALGEBRAIC MULTIGRID (AMG) 1982 ALGEBRAIC MULTIGRID (AMG) 1982 Coarse variables - a subset 1. “General” linear systems 2. Variety of graph problems 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 Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) 1D Wave Equation: u”+k2u=f Non-local components: e iwx, w ≈ ±k Slow to converge in local processing 2p/w wavelength The error after relaxation v(x) = A1(x) eikx + A2(x) e-ikx A1(x), A2(x) smooth Ar(x) are represented on coarser grids: 2h A1 + 2 i k A1′ = f1 = rh(x) e-ikx h 2D Wave Equation: Du+k2u=f w2 Non-local: (a3,b3) (a2,b2) (a4,b4) (a1,b1) ei(w1 x + w2 y) k w1 w12 + w22 ≈ k2 (a5,b5) (a8,b8) O(H) (a7,b7) (a6,b6) On coarser grid (meshsize H): cH v(x) A r (x, y) ei (a r x b r y) r 1 Fully efficient multigrid solver Tends to Geometrical Optics Radiation Boundary Conditions: directly on coarsest level Generally: LU=F Non-local part of U has the form m Σ r=1 Ar(x) φr(x) L φr ≈ 0 Ar(x) smooth {φr } found by local processing Ar represented on a coarser grid Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) N eigenfunctions Electronic structures (Kohn-Sham eq): V( x)ψi ( x) i ψi ( x) i = 1, …, N = # electrons O (N) gridpoints per yi O(N2 ) storage Orthogonalization O(N3 ) operations Multiscale eigenbase 1D: Livne O(N log N) storage & operations V = Vnuclear + V(y) One shot solver Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations Full matrix • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) Integro-differential Equation Lu ( x) G ( x, y ) u ( y ) dy differential nn Au f , dense A Multigrid solver Distributive relaxation: 1st order 2nd order 2 Solution cost ≈ one fast transform (one fast evaluation of the discretized integral transform) Integral Transforms V(x) Ω G(x, u( ) d x ' G(x, Transform Complexity ix O(n logn) e Fourier - x e Laplace O(n logn) - | x - |2 / 2 O(n) e Gauss 1 Potential O(n) | x - | G(x, *Exp(ik Waves O(n logn) G(x,y) ~ 1 / | x – y | Glocal Gsmooth s |x-y| G(x,y) = Gsmooth(x,y) + Glocal(x,y) s ~ next coarser scale O(n) not static! Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Monte-Carlo Massive parallel processing *Rigorous quantitative analysis (1986) Discretization Lattice L L for accuracy ε : L ~ εq Monte Carlo cost ~ L Ld z 2 “volume factor” “critical slowing down” Multiscale ~ ε 2 Multigrid moves Many sampling cycles at coarse levels Multigrid solvers Cost: 25-100 operations per unknown • Linear scalar elliptic equation (~1971)* • Nonlinear FAS (1975) • Grid adaptation (1977,1982) • General boundaries, BCs* • Discontinuous coefficients • Disordered: coefficients, grid (FE) AMG • Several coupled PDEs* (1980) • Non-elliptic: high-Reynolds flow • Highly indefinite: waves u k 2u f • Many eigenfunctions (N) O( N log N ) • Near zero modes • Gauge topology: Dirac eq. • Inverse problems • Optimal design Within one solver • Integral equations • Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) Local patches of finer grids • Same fast solver • Each level correct the equations of the next coarser level • Each patch may use different coordinate system and anisotropic grid and differet physics; e.g. atomistic “Quasicontiuum” method [B., 1992] UPSCALING: Derivation of coarse equations in small windows Repetitive systems e.g., same equations everywhere Scale-born obstacles: • Many variables n gridpoints / particles / pixels / … • Interacting with each other O(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … due to 1. Localness of processing 2. Attraction basins Removed by multiscale processing Systematic Upscaling 1. Choosing coarse variables 2. Constructing coarse-level operational rules equations Hamiltonian Macromolecule ~ 10-15 second steps Systematic Upscaling 1. Choosing coarse variables Criterion: Fast convergence of “compatible relaxation” Systematic Upscaling 1. Choosing coarse variables Criterion: Fast equilibration of “compatible Monte Carlo” OR: Fast convergence of “compatible relaxation” Local dependence on coarse variables 2. Constructing coarse-level operational rules Done locally In representative “windows” fast Macromolecule Macromolecule Two orders of magnitude faster simulation Fluids £ Total mass £ Total momentum £ Total dipole moment £ average location 1 1 2 Windows Fine level : density ~ 1 Coarser level Larger density fluctuations Fine level : density ~ 2 Fine level density : 3 Still coarser level Fluids Total mass: Summing m(x) Lower Temperature T Summing also i w x e m(x) v u w u 2p , w v 0 Still lower T: More precise crystal direction and periods determined at coarser spatial levels Heisenberg uncertainty principle: Better orientational resolution at larger spatial scales Optimization by Multiscale annealing Identifying increasingly larger-scale degrees of freedom at progressively lower temperatures Handling multiscale attraction basins E(r) r Systematic Upscaling Rigorous computational methodology to derive from physical laws at microscopic (e.g., atomistic) level governing equations at increasingly larger scales. Scales are increased gradually (e.g., doubled at each level) with interscale feedbacks, yielding: • Inexpensive computation : needed only in some small “windows” at each scale. • No need to sum long-range interactions • Efficient transitions between meta-stable configurations. Applicable to fluids, solids, macromolecules, electronic structures, elementary particles, turbulence, … Upscaling Projects • QCD (elementary particles): Renormalization multigrid Ron BAMG solver of Dirac eqs. M x f Livne, Livshits Fast update of M 1, det M Rozantsev • (3n +1) dimensional Schrödinger eq. Filinov Real-time Feynmann path integrals Zlochin multiscale electronic-density functional • DFT electronic structures Livne, Livshits molecular dynamics • Molecular dynamics: Fluids Ilyin, Suwain, Makedonska Polymers, proteins Bai, Klug Micromechanical structures Ghoniem defects, dislocations, grains • Navier Stokes Turbulence McWilliams Dinar, Diskin THANK YOU

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 1 |

posted: | 10/5/2011 |

language: | English |

pages: | 58 |

OTHER DOCS BY liuqingyan

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.