UT DA Power Grid Reduction by Sparse Convex Optimization Wei Ye 1 - - PowerPoint PPT Presentation

UT DA

Wei Ye1, Meng Li1, Kai Zhong2, Bei Yu3, David Z. Pan1

1ECE Department, University of Texas at Austin 2ICES, University of Texas at Austin 3CSE Department, Chinese University of Hong Kong

Power Grid Reduction by Sparse Convex Optimization

On-chip Power Delivery Network

t Power grid

› Multi-layer mesh structure › Supply power for on-chip devices

t Power grid verification

› Verify current density in metal wires (EM) › Verify voltage drop on the grids › More expensive due to increasing sizes of grids

» e.g., 10M nodes, >3 days

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[Yassine+, ICCAD’16]

Modeling Power Grid

t Circuit modeling

› Resistors to represent metal wires/vias › Current sources to represent current drawn by underlying devices › Voltage sources to represent external power supply › Transient: capacitors are attached from each node to ground

t Port node: node attached current/voltage sources t Non-port node: only has internal connection

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Port node Non-port node Current source Voltage source

Linear System of Power Grid

t Resistive grid model:

𝑀𝑤 = 𝑗 › 𝑀 is 𝑜×𝑜 Laplacian matrix (symmetric and diagonally- dominant): › 𝑕(,* denotes a physical conductance between two nodes 𝑗 and 𝑘

t A power grid is safe, if ∀𝑗:

𝑤( ≤ 𝑊

/0

t Long runtime to solve 𝑀𝑤 = 𝑗 for large linear systems

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Li,j = (P

k,k6=i g(i, k),

if i = j g(i, j), if i 6= j

Previous Work

t Power grid reduction

› Reduce the size of power grid while preserving input-

• utput behavior

› Trade-off between accuracy and reduction size

t Topological methods

› TICER [Sheehan+, ICCAD’99] › Multigrid [Su+, DAC’03] › Effective resistance [Yassine+, ICCAD’16]

t Numerical methods

› PRIMA [Odabasioglu+, ICCAD’97] › Random sampling [Zhao+, ICCAD’14] › Convex optimization [Wang+, DAC’15]

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t Input:

› Large power grid › Current source values

t Output: reduced power grid

› Small › Sparse (as input grid) › Keep all the port nodes › Preserve the accuracy in terms of voltage drop error

Problem Definition

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Overall Flow

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Node and edge set generation Store reduced nodes and edges Node elimination by Schur complement Edge sparsification by GCD Large graph partition

For each subgraph:

Node Elimination

t Linear system: 𝑀𝑤 = 𝑗 t 𝑀 can be represented as a 2×2 block-matrix:

𝑀 = 𝑀33 𝑀34 𝑀34

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𝑀44

t 𝑤 and 𝑗 can be represented as follows:

𝑤 = 𝑤3 𝑤4 and 𝑗 = 𝑗3

t Applying Schur complement on the DC system:

𝑀 7 = 𝑀33 − 𝑀34𝑀44

93𝑀34 5

which satisfies: 𝑀 7𝑤3 = 𝑗3

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L L11 L12 L>

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L22 b L

Node Elimination (cont’d)

t Output graph keeps all the nodes of interest t Output graph is dense t Edge sparsification: sparsify the reduced Laplacian without losing

accuracy

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a b d e c f g h i j a b d e c a b d e c

Node Elimination Edge Sparsification

Edge Sparsification

t Goal of edge sparsification

› Accuracy › Sparsity reduce the nonzero elements off-the-diagonal in L

t Formulation (1): t Formulation (2): [Wang+, DAC2014]

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min

X∈Rn×n

1 2m

m

X

k=1

k(X L)vkk2

2 + λ n

X

i=1

Xi,i, s.t. X is a Laplacian matrix

min

X∈Rn×n

1 2m

m

X

k=1

k(X L)vkk2

2 + λkXk0,

s.t. X is a Laplacian matrix

min

X∈Rn×n

1 2m

m

X

k=1

k(X L)vkk2

2 + λkXk1,

s.t. X is a Laplacian matrix

. . . .

L2 norm L1 norm

min

X∈Rn×n

1 2m

m

X

k=1

k(X L)vkk2

2 + λ n

X

i=1

Xi,i, s.t. X is a Laplacian matrix

Edge Sparsification

t Formulation (2): [DAC2014 Wang+] t Formulation (3):

› Weight vector: › Strongly convex and coordinate-wise Lipschitz smooth

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Problem: accuracy on the Vdd node does not guarantee accuracy on the current source nodes i0 ik, 8i = 1, · · · , 9

|∆i0|2 + |∆i1|2 + · · · |∆i9|2

min

X∈Rn×n

1 2m

m

X

k=1

k((X L)vk) wk2

2 + λ n

X

i=1

Xi,i, s.t. X is a Laplacian matrix

w0 = 1/n, wi = 1, ∀i = 1, · · · , n

Coordinate Descent (CD) Method

t Update one coordinate at each iteration t Coordinate descent:

Set 𝑢 = 1 and 𝑌3 = 0 For a fixed number of iterations (or convergence is reached):

Choose a coordinate (𝑗, 𝑘) Compute the step size 𝜀∗ by minimizing argmin

G

f(𝑌 + 𝜀𝑓(,*) Update 𝑌(,*

/K3 ← 𝑌(,*

/ + 𝜀∗

t How to decide the coordinate?

› Cyclic (CCD) › Random sampling (RCD) › Greedy coordinate descent (GCD)

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CD vs Gradient Descent

t Gradient descent (GD) algorithm:

𝑌/K3 ← 𝑌/ − 𝛽∇𝑔(𝑌)

t GD/SGD update 𝑃(𝑜4) elements in 𝑌 and gradient matrix

𝐻 at each iteration

t CD updates 𝑃 1 elements in 𝑌 (Laplacian property) t CD proves to update 𝑃 𝑜 elements in 𝐻 for Formulation

(2) and (3).

i? j? i? j? i? j? i? j?

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(4,4):6 … (1,2):0 (1,3):8 (1,4):4 … (1,2):0 (4,5):6

Greedy Coordinate Descent (GCD)

1 2 3 4 5

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(3,4):1 … (1,2):0 (2,3):2

Max-heap Input L Output X

GCD vs CCD

t GCD produces sparser results

› CCD (RCD) goes through all coordinates repeatedly › GCD selects the most significant coordinates to update

Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration T GCD: CCD: Input graph Add an edge Update an edge

5 10 15 20 100 101 102 103 104 105 Edge Weight Edge Count CCD GCD

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GCD Coordinate Selection

t General Gauss-Southwell Rule: t Observation: the objective function is quadratic w.r.t. the

chosen coordinate

t GCD is stuck for some corner cases: t A new coordinate selection rule:

(i∗, j∗) = arg max

(i,j)∈[n]×[n]

|Gi,j|

(i∗, j∗) = arg max

(i,j)∈[n]×[n]

|Gi,j| s.t. Gi,j > 0 or yi,j 6= 0

yt

i,j

yt+1

i,j

yt

i,j

yt+1

i,j

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GCD Speedup

t Time complexity is 𝑃(𝑜4) per iteration

› traverse 𝑃(𝑜4) elements to get the best index › As expensive as gradient descent

t Observation: each node has at most 𝑜 neighbors → heap t Heap to store 𝑃(𝑜4) elements in 𝐻:

› Pick the largest gradient, 𝑃(1) › Update 𝑃(𝑜) elements, 𝑃(𝑜 log 𝑜)

t Lookup table

› 𝑃(𝑜4) space; 𝑃 1 for each update

t Improved time complexity 𝑃(𝑜 log 𝑜)

i? j? i? j?

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Experimental Results

t Sparsity and accuracy trade-off t Accuracy and runtime trade-off

10−7 10−6 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 λ Voltage Error (mV) 10−7 10−6 10−5 10−4 10−3 10−2 10−1 500 1,000 1,500 2,000 2,500 3,000 3,500 #Edges Error Edge 0.2 0.4 0.6 0.8 1 ·104 10−4 10−3 10−2 #Iterations Voltage Error (mV) 0.2 0.4 0.6 0.8 1 ·104 500 1,000 1,500 2,000 2,500 3,000 #Edges Error Edge

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Gradient Descent Comparison

rand1 rand2 rand3 rand4 102 103 104 105 99 499 999 4,999 3,169 71,548 1.51 · 105 3.05 · 105 1,068 2,743 3,920 10,003 #Edges SGD CCD GCD rand1 rand2 rand3 rand4 10−2 10−1 100 101 102 5.09 5.33 5.23 7.52 7 · 10−2 5 · 10−2 3 · 10−2 2.18 2 · 10−2 1 · 10−2 1 · 10−2 1.3 Voltage error (mV) SGD CCD GCD rand1 rand2 rand3 rand4 10−2 10−1 100 101 102 0.12 2.93 29.82 144.43 1 · 10−2 1.92 19.09 230.52 6 · 10−2 0.74 2.75 32.6 Runtime (s) SGD CCD GCD

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Sparsity Accuracy Runtime

Experimental Results

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CKT ibmpg2 ibmpg3 ibmpg4 ibmpg5 ibmpg6 #Port Nodes Before 19,173 100,988 133,622 270,577 380,991 After 19,173 100,988 133,622 270,577 380,991 #Non-port Nodes Before 46,265 340,088 345,122 311,072 481,675 After #Edges Before 106,607 724,184 779,946 871,182 1283,371 After 48,367 243,011 284,187 717,026 935,322 Error 1.2% 0.7% 4.8% 2.2% 2.0% Runtime 38s 106s 132s 123s 281s

Conclusion

t Main Contributions:

› An iterative power grid reduction framework › Weighted convex optimization-based formulation › A GCD algorithm with optimality guarantee and runtime efficiency for edge sparsification

t Future Work:

› Extension to RC grid reduction

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Thanks