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 process Not optimization !

Multigrid solvers Cost: operations per unknown Linear scalar elliptic equation (~1971)* Nonlinear Grid adaptation General boundaries, BCs* Discontinuous coefficients Disordered: coefficients, grid (FE) Several coupled PDEs* 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)

u given on the boundary h u = function of u's and f Probability distribution of u = function of u's and f Point-by-point RELAXATIONPoint-by-point MONTE CARLO

Discretization Lattice for accuracy “volume factor” Multiscale cost ~ Multigrid cycles Many sampling cycles at coarse levels “critical slowing down” Monte Carlo cost ~ 2z L   Statistical samples

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

Repetitive systems e.g., same equations everywhere UPSCALING: Derivation of coarse equations in small windows Vs. COARSENING: For acceleration Or surrogate problems Etc.

A solution value is NOT generally determined just by few local equations N unknowns  O (N) solution operations UPSCALING: The coarse equation can be derived ONCE for all similar neighborhoods  # operations << N A coarse equation IS generally determined just by few local equations

Systematic Upscaling 1.Choosing coarse variables 2.Constructing coarse-level operational rules equations Hamiltonian

Macromolecule ~ second steps

ALGEBRAIC MULTIGRID (AMG) 1982 Coarse variables - a subset Criterion: Fast convergence of “compatible relaxation” Ax = b Relax Ax = 0 Keeping coarse variables = 0

Systematic Upscaling 1.Choosing coarse variables Criterion: Fast convergence of “compatible relaxation” 2. Constructing coarse-level operational rules (equations / Hamiltonian) Done locally Local dependence on coarse variables OR: Fast equilibration of In representative “windows” “compatible Monte Carlo”

Macromolecule ~ second steps

Macromolecule Two orders of magnitude faster simulation

Macromolecule      + Lennard-Jones ~10 4 Monte Carlo passes for one T G i transition G1G1 G2G2 T Dihedral potential + Electrostatic

Fluids £ Total mass £ Total momentum £ Total dipole moment £ average location

1 1 2

Windows Coarser level Larger density fluctuations Still coarser level

Fluids Total mass: Summing

Lower Temperature T Summing also 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 Applicable to fluids, solids, macromolecules, electronic structures, elementary particles, turbulence, … Efficient transitions between meta-stable configurations.

Upscaling Projects QCD (elementary particles): Renormalization multigrid Ron BAMG solver of Dirac eqs. Livne, Livshits Fast update of, det 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 ??? defects, dislocations, grains Navier Stokes Turbulence McWilliams Dinar, Diskin

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