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MULTISCALE COMPUTATIONAL METHODS

MULTISCALE COMPUTATIONAL METHODS. Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi. Computational bottlenecks:. Elementary particles Physics standard model. Chemistry, materials science. Schrödinger equation. Molecular dynamics forces.

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MULTISCALE COMPUTATIONAL METHODS

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  1. MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi

  2. Computational bottlenecks: • Elementary particles Physics standard model • Chemistry, materials science Schrödinger equation Molecular dynamics forces • (Turbulent) flows Partial differential equations • Vision: recognition • Seismology • Tomography (medical imaging) • Graphs: data mining,… • VLSI design

  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-particle Lennard-Jones potential Particle distance r0 00 + external forces…

  5. small step Moving one particle at a timefast local ordering r0 slow global move

  6. Numerical solution of a partial differential equation (PDE) e.g., approximating Laplace eq. on a fine grid

  7. fine grid h u= average ofu's approximating Laplace eq.

  8. u given on the boundary h e.g., u= average ofu's Solution algorithm: approximating Laplace eq. Point-by-point RELAXATION

  9. 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

  10. 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

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

  12. Macromolecule ~ 10-15 second steps

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

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

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

  16. Scale-born obstacles: • Many variables n gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … 1. Localness of processing 2. Attraction basins • Multiple solutions Inverse problems / Optimization Many eigenfunctions Statistical sampling Removed by multiscale algorithms

  17. 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

  18. , error Approximation Residual equation: Relaxation of linear systems Ax=b Eigenvectors: Relaxation:Fast convergence of high modes

  19. When relaxation slows down: the error is a sum of low eigen-vectors ELLIPTIC PDE'S (e.g., Poisson equation) the error is smooth

  20. 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

  21. When relaxation slows down: the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth DISCRETIZED PDE'S the error is smooth Along characteristics

  22. When relaxation slows down: the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth DISCRETIZED PDE'S the error is smooth Along characteristics GENERAL SYSTEMS OF LOCAL EQUATIONS The error can be approximated by a far fewer degrees of freedom (coarser grid)

  23. When relaxation slows down: the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth The error can be approximated on a coarser grid

  24. h LhUh=Fh LU=F 2h L2hU2h=F2h 4h L4hU4h=F4h

  25. Local relaxation approximation smooth h 2h LhUh=Fh L2hU2h=F2h

  26. ~ ~ ~ ~ = + h h 2 2 h h u u v v new old TWO GRID CYCLE Fine grid equation: 1. Relaxation Approximate solution: Smooth error: Residual equation: residual: 2. Coarse grid equation: Approximate solution: 3. Coarse grid correction: 4. Relaxation

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