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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

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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 Ax = b2**higher preconditioner quality Preconditioned iterative solver (IC,ILU,AMG...) Direct solver Iterative solver (CG,BiCG,MINRES,GMRES...) density power grid thermal analysis quadratic placement 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**adopt an iterative scheme 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 network of roads • 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**Solver using**the record What’s on the record? Row i**Recall that is**a solution to UL factorization**UL and LU**rev( ) :inverse ordering operator**Doolittle LU factorization is unique.**Therefore Convert to LU If symmetric**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**• Quadratic placement instances • 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**• Reasonable preconditioning overhead • 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 • Random-walk preconditioned Conjugate Gradient • Extendable to more general matrices • Promising results on placement matrices • Up to 7X speedup over ICCG • Favor large and dense matrices**Thank you**Solver package download http://www.ece.umn.edu/users/qianhf/hybridsolver**Backup: accuracy control**• Stopping criterion must be • Function of A • Independent ofb