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

2. 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

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. Magneto-Quasistatic MPIEJ 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 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]

9. 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

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
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
SGM +
Microstrip line W=50um, L=0.5mm, H=15um
sigma=3um, correlation length=50um
mean: , std (MC, SGM) = 0.001, std (New algorithm)=

22. 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

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: