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

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

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

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

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

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

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

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

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

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

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)

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

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.

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

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

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

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:

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

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

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

Results: Accuracy Validation SGM + 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

Results: Complexity Validation Example Technique Properties for 5% accuracy Memory Time Long Microstrip line DC Only 400 unknowns Monte Carlo Neumann* SGM New Algorithm 10, 000 2nd order 96 iid, 4753 o.p. 1.2 MB (72 GB) 2.4 hours 0.25 hours - 0.5 hours Transmission Line 10 freq. points 800 unknowns 105 iid, 5671 o.p. 10 MB (300 TB) 16 hours 24 hours 7 hours Two-turn Inductor 2750 unknowns 400 iid, 20604* 121 MB (800 PB) (150 hours) X 4p (828 hours) X 4p 8 hours X 4p

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

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.

Thank You

Inductor Example

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

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: