A Hybrid Linear Equation Solver and its Application in Quadratic Placement

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A Hybrid Linear Equation Solver and its Application in Quadratic Placement Haifeng Qian, Univ of Minnesota Sachin S. Sapatnekar, Univ of Minnesota Quadratic placement Variables Cost function Computation My topic A x = b where | A | ≠ 0 # re-solve A x = b 2

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### A Hybrid Linear Equation Solver and its Application in Quadratic Placement

Haifeng Qian, Univ of Minnesota

Sachin S. Sapatnekar, Univ of Minnesota

• Variables
• Cost function
• Computation

My topic

A x = b

where |A| ≠ 0

# re-solve Ax = b2

higher preconditioner quality

Preconditioned iterative solver (IC,ILU,AMG...)

Direct solver

Iterative solver (CG,BiCG,MINRES,GMRES...)

density

power grid

thermal analysis

The big picture
Status quo
• Preconditioning
• Popular choice: Incomplete LU
• Placement matrices
• Symmetric positive definite
• Popular choice: ICCG with
• Different ordering
• Different dropping rules
Status quo

Is this the best we can do?

done

done

• Rules
• Pattern
• Min value
• Size limit

stochastic precondi-tioning

Iterative

Solver

The Hybrid

Solver

Stochastic

Solver

We present …
• Limited topic today
• Symmetric
• Positive diagonal entries
• Negative off-diagonal entries
• Irreducibly diagonally dominant
• These are sufficient, NOT necessary, conditions

is sufficient not necessary

Equations

Random event(s)

Variables

Expectation of random variables

Approximate solution

Average of random samples

The third and forgotten category

Stochastic solver methodology

\$

Home

Home

Home

Random walk overview

• Given:
• A motel at eachintersection
• A set of homes
• Random walk
• Walk one (randomlychosen) road every day
• Stay the night at a motel

(pay for it!)

• Keep going until reaching home
• Win a reward for reaching home!
• Problem: find the expected amount of money in the end as a function of the starting node x

4

px,4

x

1

3

px,1

px,3

px,2

2

Random walk overview

• For every node

Linear equation set

Random walk game

M walks from i-th node

Take average

i-th entry of solution

Random walk overview

Weakness

Error ~ M-0.5

3% error to be faster than direct/iterative solvers

Stochastic

Solver

New solution:

Sequential Monte Carlo

Stochastic

Solver

approx. solve

error & residual

approx. solve

Benefit: ||r||2<<||b||2 ||y||2<<||x||2

same relative error = lower absolute error

A. W. Marshall 1956, J. H. Halton 1962

Start

Start

Trick #1: new homes

New home

Previously calculated node

Benefit: more and more homes

shorter and shorter walks

one walk = average of multiple walks

Qian et al., DAC2003

Trick #2: journey record

Keep a record: motel/award list

New RHS: Ax = b2

Update motel prices, award values

Use the record: pay motels, receive awards

New solution

Benefit: no more walks

only feasible after trick#1

Qian et al., DAC2003

Keep a record: motel/award list

Stochastic

Solver

New RHS: Ax = b2

Update motel prices, award values

Stochastic

Solver

Use the record: pay motels, receive awards

New solution

New solution:

Ring a bell?

Random walks:Initial solution

& Keep a record

New problem:

Update motel prices, award values

Use the record: pay motels, receive awards

produce

New solution:

Prototype
A second look

Solver using

the record

kth iteration

by linear operations

By definition

Substitution

By definition

This is preconditioned Gauss-Jacobi !

Why Gauss-Jacobi

Why not CG/BiCG/MINRES/GMRES

UL and LU

rev( ) :inverse ordering operator

Recall

LDL factorization
• What we need for symmetric A
• What we have
• How to find ?
• Please refer to the paper

A path exists from i to j

through node set {i+1,…N}

Incomplete factorization
• Non-zero pattern proof
• Accuracy control
• Please refer to the paper

Conclusion: Incomplete LDL factorization to

precondition any iterative solver

Compare to existing ILU
• Existing ILU
• Gaussian elimination
• Drop edges by pattern, value, size
• Error propagation
• A missing edge affects subsequent computation
• Exacerbated for larger and denser matrices

b2

b2

b1

b1

a

b3

b3

b5

b5

b4

b4

Superior because …
• Each row of L is independently calculated
• No knowledge of other rows
• Responsible for its own accuracy
• No debt from other steps
Test setup
• Set #1: matrices and rhs’s by an industrial placer
• Set #2: matrices by UWaterloo placer on ISPD02 benchmarks, unit rhs’s
• LASPACK: ICCG with ILU(0)
• MATLAB: ICCG with ILUT
• Approx. Min. Degree ordering
• Tuned to similar factorization size
• Same accuracy:
• Complexity metric: # double-precision multiplications
• Solving stage only
Physical runtimes on P4-2.8GHz
• Less than 3X solving time
• One-time cost, amortized over multiple solves
Conclusion
• Hybrid linear solver
• Combining stochastic and iterative techniques
• Special case: symmetric diagonally dominant
• Proven incomplete LDL factorization
• Extendable to more general matrices
• Promising results on placement matrices
• Up to 7X speedup over ICCG
• Favor large and dense matrices