Stochastic Integral Equation Solver for Efficient Variation-Aware Interconnect Extraction

1. Stochastic Integral Equation Solver for Efficient Variation-Aware Interconnect Extraction Tarek A. El-Moselhy and Luca Daniel

2. Motivation: On/Off-chip Variations • Rough-surfaces: On-package and on-board • Irregular geometries: On-chip [Courtesy of IBM and Cadence] [Braunisch06] • Irregularities change in impedance • Irregularities are random but current extraction tools are deterministic

3. Definition of Stochastic Solver Geometry of interconnect structure Stochastic Field Solver Statistics of interconnect input impedance Distribution describing the geometrical variations PDF PDF input impedance width

4. J J J J Magneto-Quasistatic MPIE Vm Vk Current conservation Piecewise constant basis functions + Galerkin testing Mesh matrix M Stochastic

5. Linear System Abstraction • The system matrix elements are functions of the random variables describing the geometry • Vector represents n (Gaussian) correlated random variables • Single matrix element depends on a small subset of the physical parameters • The objective is to find the distribution of the unknown vector

6. Outline • Motivation and Problem Definition • Previous Work and Standard Techniques • Contribution • New Theorem for orthogonal projection • New simulation technique • Results

7. Sampling-Based Techniques • Monte Carlo, stochastic collocation method [H.Zhu06] • Solve the system Mc times for Mc different realizations of • Compute required statistics from the ensemble • Advantages: Highly parallelizable, very simple • Disadvantages: requires solving the system Mc times which means complexity is

8. Neumann Expansion convergence criterion • Computing the statistics is very expensive =0 Complexity: O(N4) • 2D capacitance [Z.Zhu04], 3D inductance [Moselhy07], on-chip capacitance [Jiang05]

9. Stochastic Galerkin Method [Ghanem91] Step 1 need to decouple random variables Step 2 Step 3 write as a summation of same polynomials expand in terms of orthogonal polynomials Step 4 substitute and assemble linear system to compute the unknowns

10. Stochastic Galerkin Method (Con’t) Step 2. Polynomial Chaos Expansion • Use multivariate Hermite polynomials M-dimensional • For a typical interconnect structure M > 100 Problem 1: Very expensive multi-dimensional integral

11. Stochastic Galerkin Method (Con’t) K+1 unknowns each of length N Step 4. System Assembly • Use Galerkin Testing to obtain a deterministic linear system of equations Problem 2: Very large linear system O(KN)

12. Outline • Motivation and Problem Definition • Previous Work and Standard Techniques • Contribution • New Theorem for orthogonal projection • New simulation technique • Results

13. Solution of Problem 1: Efficient Multi-Dimensional Projection • Current techniques include: • Monte Carlo integration • Quasi-Monte Carlo integration • Sparse grid integration • We propose to solve the problem by reducing the dimension of the integral.

14. Solution of Problem 1: Efficient Multi-Dimensional Projection (con’t) is a small subset of the vector containing the physical parameters for a second order expansion

15. Corollary 100-D Integral Original Polynomial Chaos Expansion 8-D Integral New Theorem Matrix elements depend on a small subset of the physical random variables Second order expansion

16. Theorem • Given the matrix elements the coefficients of the Hermite expansion ( ) are given by: where is the subset of parameters on which the matrix element depends and is the subset of random variables on which the polynomial depends • If dimension of then the above formula is more efficient than the traditional approach

17. Solution of Problem 2: Efficient Stochastic Solver • Use Neumann expansion to reduce system size • Use Polynomial Chaos expansion to simplify computation of the statistics: • Rearranging above expansion we obtain the required expansion of the output:

18. Efficient Stochastic Solver • Obtain directly an expansion of the output in terms of some orthogonal polynomials • Complexity is transformed into a large number of vector matrix products • Highly parallelizable • Requires independent system solves (same system matrix), currently implemented using direct system solvers and re-using the LU factorization • Efficiency can be even further enhanced using block iterative solvers

19. Outline • Motivation and Problem Definition • Previous Work and Standard Techniques • Contribution • New Theorem for orthogonal projection • New simulation technique • Results

20. Definition of Stochastic Solver Geometry of interconnect structure Stochastic Field Solver Statistics of interconnect input impedance Distribution describing the geometrical variations Rough surface with Gaussian profile and correlation

21. Results: Accuracy Validation • Microstrip line W=50um, L=0.5mm, H=15um • sigma=3um, correlation length=50um • mean: 0.0122, std (MC, SGM) = 0.001, std (New algorithm)= 0.00097 SGM +

22. Results: Complexity Validation

23. Results: Large Example correlation length = 5um • Two-turn inductor • Simulation at 1GHz for different rough surface profiles • Input resistance is 9.8%, 11.3% larger than that of smooth surface for correlation lengths 5um, 50um, respectively • Variance increases proportional to the correlation length • Inductance is decreased by about 5% • Quality factor decreases correlation length = 50um

24. Conclusion • Developed a new theorem: • efficient Hermite polynomial expansion • new inner product • many orders of magnitude reduction in computation time • suitable for any algorithm that relies on polynomial expansion • Developed new simulation algorithm: • merged both Neumann and polynomial expansion • does not require the solution of a large linear system • easy to compute the statistics • parallelizable. • Verified our algorithm on a variety of large examples that were not solvable before.

25. Thank You

26. Inductor Example

27. Proof • The main step is to prove the orthogonality of the polynomial using the modified inner product definition • Consequently,

28. Alternative Point of View • The same theorem can be proved by doing a variable transformation and making use of Mercer Theorem: Remember from Mercer Theorem: