28 Sep 2015

Big O Theory Club , Georgia Tech

FAST MULTIPOLE

METHODS (FMM)

SKANDA PRASAD

skanda@gatech.edu

In Gravitation or Electrostatics

Φ 𝒚𝑗 = 𝑘

𝑖=1

𝑁𝑞𝑖

𝒙𝑖 − 𝒚𝑗

OR

𝜱 𝒚 = 𝑘. 𝑅𝒒(Matrix-Vector product form)

Similar problems in acoustics, scattering, magnetostatics, illumination, thermodynamics etc.

Essentially, problems of the form

𝑢 𝒚 = 𝐾 𝒙, 𝒚 𝑤 𝒙 𝑑𝒙

Or

𝑢 𝒚 = 𝑖=1𝑁 𝑤𝑖𝐾 𝒚, 𝒙𝑖

Can approximate integrals as sums.

THE CLASSIC N-BODY PROBLEM

Highly Complex

Matrix-Vector Product

O(MN) for dense matrices

O(N2) for N-body problem

Complex in time AND memory!

Fast Summation Methods

Achieve in O(Nα), with 1 ≤ α < 2

Eg: FFT

Sequential Algorithms

○ Appel (1985), Barnes and Hut (1986)

THE PROBLEM??

Wickedly Fast ( O(N log N) )…

Exact

… BUT:

Restrictive Problem Structures.

Only uniform sample grids.

Very high sample rates required

Not great for sparse or uneven

structures.

Not robust

Can’t work for 2D manifolds in

3D space.

WHAT ABOUT THE FFT?

FMMS TO THE RESCUE!

• Greengard and Rokhlin, 1985.

• Original Aim: Calculate N-body / MoM problems

faster.

• One of the Top 10 Algorithms of the 20th Century

(SIAM)

• Monte Carlo Method

• Simplex Method for LP

• Krylov Subspace iteration

• Matrix Decompositions

• Integer relation detection algo.

• Fortran

• QR Algorithm

• Quicksort

• FFT

• The FMM

Reduced computational cost From O(N2) to O(N log N) – or even O(N).

Reduced Memory requirements O(N2) to ~ O(N log N).

Just-in-time storage possible

Numerically exact solution. Can achieve machine precision.

Pre-computation to achieve speedup.

FMMS TO THE RESCUE!

FMM - CHARACTERISTICS

• Common uses:

• Method of Moments (MoM) and N-Body problems.

• Computational EMag, Large scale scattering problems.

• Tree Code:

• Specified accuracy, є.

• Hierarchical subdivision of space.

• Far field expansion of ‘kernel’ K(x, y), separating source and

target influences.

• (Optional) Converting Far field expansions into local

expansions.

FMM – THE BASIC INGREDIENTS

• SUGAR: Finite Rank / Degenerate

Kernels

• SPICE: Error Bounding through

truncated series expansion.

• EVERYTHING NICE: Hierarchical

Partitioning / Recursive computation

• For MLFMA

• … And CHEMICAL X:

Translation + Precomputation Takes the FMM from O(N logN) to O(N)

Finite Rank Approximation.

𝐾 𝒙, 𝒚 = 𝑘=0𝑃−1𝜙𝑘 𝒙 𝜓𝑘 𝒚

• Decouples source and target influences

• Is a low rank approximation of the field.

Example:

1

𝑦− 𝑥𝑖=1

𝑦− 𝑥∗. 𝑚=0∞ 𝑥𝑖− 𝑥∗

𝑦− 𝑥∗

𝑚

“Multipole Expansion”

Regular (R) & Singular (S) Expansions

SUGAR – DEGENERATE KERNELS

Multipole Expansions: Guaranteed far field convergence Far Field: Target box is ‘well separated’ from Source box.

How far is ‘well separated’? Ask MAC – Multipole Acceptance Criterion

Rigorous Error bounds Error bounding is NOT a matter of trial and error - Can be ANALYTICALLY

bounded

Can a priori choose truncation number.

SPICE – ERROR BOUNDING

Single Evaluation

Centre.

THE MIDDLEMAN APPROACH

MULTIPLE EVALUATION CENTERS

“Approximation” can be as exact as we’d like

Given a tolerance ‘ε’, choose

expansion rank, P.

SINGLE LAYER FMM / BARNES-HUT

Given a tolerance ‘ε’, choose expansion rank, P.

Partition the source space into cells

Each cell has O(1) no. of sources.

IMPORTANT: Assumed uniform distribution

SINGLE LAYER FMM / BARNES-HUT

Given a tolerance ‘ε’, choose expansion rank, P.

Partition the source space into cells

Each cell has O(1) no. of sources.

IMPORTANT: Assumed uniform distribution

Form multipole expansions about each cell centre.

SINGLE LAYER FMM / BARNES-HUT

Given a tolerance ‘ε’, choose expansion rank, P.

Partition the source space into cells

Each cell has O(1) no. of sources.

IMPORTANT: Assumed uniform distribution

Form multipole expansions about each cell centre.

For each target, do:

Aggregate contributions from each far-field cell

SINGLE LAYER FMM / BARNES-HUT

Given a tolerance ‘ε’, choose expansion rank, P.

Partition the source space into cells

Each cell has O(1) no. of sources.

IMPORTANT: Assumed uniform distribution

Form multipole expansions about each cell centre.

For each target, do:

Aggregate contributions from each far-field cell

Individually add the contributions from each near field source point.

SINGLE LAYER FMM / BARNES-HUT

SINGLE LAYER FMM

EVERYTHING NICE – HIERARCHICAL PARTITIONING

EVERYTHING NICE – HIERARCHICAL PARTITIONING

L = logb (N) levels.

Near Field

Interaction List

Far Field

AND NOW…

… FOR

CHEMICAL X

Multipole to Multipole Translation (M2M): Or “Outgoing-to-Outgoing”

Transforms the multipole expansion of a node’s children to its own multipole expansion.

Multipole to Local Translation (M2L): Transforms the multipole expansion of a node into

the local expansion of a node in its interaction list

Local to Local Translation (L2L): Or “Incoming / local expansion”

Transforms the local expansion of a node’s parent to its own local expansion

TRANSLATION OPERATORS

1. Aggregation (Multipole

Expansion)

2. Translation

1. Outgoing-to-outgoing

2. Outgoing-to-local/incoming

3. Disaggregation

TRANSLATION OPERATORS

12 3

PREPARATION:

1. Choose 𝑝 ≈ 𝑙𝑜𝑔𝑏 𝜖𝒅 1

𝑜𝑟 𝑝 ≈ log2 1𝜖

‘b’ varies with problem.

‘d’ – leaf cell size

2. Choose number of levels, 𝐿 ≈ log2𝑁 .

3. Partition the space so that each leaf cell has O(1) sources.

PRECOMPUTATION:

1. Form multipole expansions (moments) at the tree’s finest scale.

2. UPWARD PASS: Translate expansions from lower levels to next coarser level.

Multipole expansions of each leaf cell translated to parent cell.

THE MULTI-LAYER FMM ALGORITHM (MLFMA)

ALGORITHM:

3. DOWNWARD PASS: For each (target) region, Transform multipole expansions of all far-

field cells in the interaction list (at current level) into a local expansions about current

cell centre.

Algorithm bottleneck, but still O(N).

3. DOWNWARD PASS: For each region, translate the local expansion down the tree to the

centre of each of its 2d children.

4. Repeat 3 and 4 till finest level;

5. Evaluate series expansions at each target location and aggregate near neighbour

interactions directly.

THE MULTI-LAYER FMM ALGORITHM (MLFMA)

Asymptotic Cost Getting to O(N)

Stage 1: Compute all outgoing expansions O ( N log N ) Easy

Stage 2: Compute Far Field Potentials O ( N log N ) ??

Stage 3: Compute Near Field Potentials O ( N ) No need to fix

COMPUTATIONAL COMPLEXITY

… BECAUSE IT WON’T BE BIG-O WITHOUT IT

Single Layer FMM

Need to compute all interactions

Too much “communication” between boxes

Multi-level FMM

O(N log N) - Each particle (N) communicates with all levels above it.

Tradeoff: decimate for smaller cells vs increasing # of levels

LevelP = expansion size,

L ~ log N, d = dimensions# (Opers)

L (leaf) “outgoing from sources” N.P

L - 1 “outgoing from outgoing” Nboxes.P2

L – 2 - do -(Nboxes/4).P2

L – 3 - do - (Nboxes/16).P2

… … …

COMPUTATIONAL COMPLEXITY

Single Layer FMM

Need to compute all interactions

Too much “communication”

between boxes

Multi-level FMM

O(N log N) - Each particle (N)

communicates with all levels

above it.

Tradeoff: decimate for smaller

cells vs increasing # of levels Cost of computing all outgoing expansions = O(N)

COMPUTATIONAL COMPLEXITY

PUTTING IT ALL TOGETHER …

Standard M-V

Product

Middle approach

Middleman with Target

Clusters

Middleman with Source

Clusters

Single Layer FMM

PUTTING IT ALL TOGETHER

ADVANTAGES OF FMM

• O( N log N ) complexity, Going down to O(N)

• Non-uniform distributions supported, (slightly ↑sed complexity).

• (Rigourous) Error Bounding to arbitrary precision.

• Massive Memory saving.

• FMM parallelizes very well.

• Compare with FFT

• FFT – Exact, but Brittle.

• FMM – Approximate but robust. Rigorous a priori error bound possible.

Adaptive FMMs – 2 extra steps1. Partition carefully

2. Some exceptional boxes.

Fast Elliptic Solvers for PDEs PDE Integral Equation Sum Approx. FMM

SOME INTERESTING EXTENSIONS

Different Kernels Cauchy Kernel (Electrostatics)

Laplace’s Equation and Green’s Function

Helmholtz Equation

3D Laplace’s Equation

Solving certain PDEs Recast as Integral Equations

Stealth / low-RCS testing

CFD and Thermodynamics?

…What’s next?

EXPANDING HORIZONS

… Want to learn someone who isn’t a dummy?

Beatson & Greengard, A short course on FMMs

Raykar, A Short Primer on the FMM

LeXing, A pedestrian introduction to FMMs

Darve, The FMM: Numerical Implementation

… Want some CS / implementation specific stuff?

Chandramowlishwaran, The FMM at Exascale. (PhD Dissertation, Georgia Tech, 2013)

… feeling especially mathochistic?

Greengard & Rokhlin, A new version of the FMM for the 3D Laplace Equation.

Cheng, Greengard & Rokhlin, A fast adaptive multipole algorithm in 3 dimensions.

… not feeling punished enough?

FINE! Greengard’s original thesis it is then.

IF I HAVEN’T BORED YOU YET…

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