Split and Merge Strategies for Solving Uncertain Equations Using Affine Arithmetic

Transcription

1 Split and Merge Strategies for Solving Uncertain Equations Using Affine Arithmetic Oliver Scharf, Markus Olbrich, Erich Barke SIMUTOOLS 2015

2 Outline Introduction Uncertain parameters Affine arithmetic Circuit simulation EPD algorithm Split strategies Transient simulation Merge strategies Results Seite 2

3 Simulation of parameter uncertainties Parameter uncertainties Manufacturing tolerances Ageing Environment, i.e. temperature Uncertainties have influence on system behaviour Simulation using Monte Carlo method Corner Case method Range arithmetic methods (i.e. interval arithmetic, affine arithmetic) Seite 3

4 Affine forms Central value Linear partial deviations Deviation symbols Exact affine operations Other non exact operations Approximation One symbol per operation is generated Linear correlation is maintained Inclusion guaranteed Seite 4

5 Types of deviations Equation: r =f( x, p ) 1D example: PPDs (Parameter Partial Deviations) contain linear correlation with parameters EPDs (Enhanced PDs) added to maintain inclusion NLPDs (Nonlinear PDs) generated during computation of non-exact functions Seite 5

6 Over-approximation No optimal result (minimal inclusion) can be calculated Affine arithmetic safely includes optimal result Possible approximation: Min/Max from Monte Carlo Over-approximation: difference between optimal and affine result d A d MC Over-approximation Measure: Mean relative over-approximation m OA n affine n Monte Carlo Seite 6

7 Circuit simulation R 1 C 1 f(x,p,t) Solver schematic equations output f(x,p,t) is implicit, nonlinear and time-dependent Seite 7

8 Circuit simulation with affine arithmetic R 1 C 1 f( x, p,t) EPD Solver schematic equations output f( x, p,t) is parametric, implicit, nonlinear and timedependent Seite 8

9 EPD algorithm without splitting Central value = nominal PPDs from linearisation Add EPDs Calculation of residuum Convergence limited Larger parameter deviations not solvable Scaling of the EPDs Check abort criterion Seite 9

10 Splitting Split parameter f(x,p) Solve parts separately x Merge parts Seite 10

11 EPD algorithm with splitting Split parameter Central value = nominal solution PPDs from linearisation Add EPDs Calculation of residuum Split required Check splitting criterion Scaling of EPDs Check abort criterion Merge split parts Seite 11

12 Parameter tree/solution tree p p p p, p, x x x x, x, parameter tree solution tree Split parameter are stored in a tree Solution is stored in corresponding leave of another tree Seite 12

13 Split of single parameters Split all parameters: 2 n parts large runtime Idea: Select one parameter to split Relation between parameters and solution space is nonlinear and implicit Which parameter should be split? Seite 13

14 Split strategies Select by deviation : Parameter with largest deviation Select by sensitivity : Parameter with largest entry in df/ dp Select by sensitivity and deviation : Parameter deviation scaled with sensitivity Select by EPD and PPD : Equation with largest EPD, from this the parameter with largest deviation Select by EPD and sensitivity : Equation with largest EPD, from this the parameter with largest sensitivity Seite 14

15 Transient simulation (TR) U/I f(x, x,p,t) t = 0 Integration TR is the most common simulation type DAE is solved by numerical integration x = x( t n+1 ) x( t n ) / t n+1 t n t Solve t = t + tstep t tend Output Seite 15

16 Derivations x x x x x x x x x time step t n-1 time step t n time step t n+1 x = x( t n+1 ) x( t n ) / t n+1 t n Seite 16

17 Derivations x x x x, x, x x x x, x, x x x x, x, time step t n-1 time step t n time step t n+1 x = x( t n+1 ) x( t n ) / t n+1 t n Seite 17

18 Transient simulation merge strategies t = 0 Integration Split EPD algorithm Merge t = t + tstep t tend t = 0 Integration EPD algorithm Split t = t + tstep t tend Merge Merge increases over-approximation Recreation of split increases runtime Seite 18

19 Merge strategies Visited not solvable Visited - solvable Unvisited Merge each Merge after each step t Don t merge Keep all splits Merge every n th step Merge after n steps Try root first Test without split, use splits if necessary Seite 19

20 Example circuits Inverter R B R L V out V dd Symbol Central value Deviation R B 5KΩ ±450Ω R L 10KΩ ±50Ω Bandpass Symbol Central value Deviation R 2 R KΩ ±1.125kΩ V in R 1 C 2 C 1 R 3 R 4 V out R KΩ ±1.11kΩ R KΩ ±3.01kΩ R 4 20KΩ ±3kΩ R 5 R 5 20KΩ ±3kΩ C 1 10nF ±0.2nF C 2 10nF ±0.2nF Seite 20

21 Runtime split strategies - forced splits Split strategy # forced splits Select all! by deviation! by sensitivity! by sensitivity and deviation! by EPD and PPD! by EPD and sensitivity! random! Circuit: Inverter Circuit not solvable without splitting Split strategies decrease runtime compared to select all Seite 21

22 Runtime split strategies (normalized to select all ) "select by deviation" 0.8 "select by sensitivity" "select by sensitivity and deviation" "select random" Inverter Bandpass "select by EPD and PPD" è Select by deviation is the fastest strategy Seite 22

23 Runtime merge strategies (normalized to Don t Merge ) "Don't merge" "Merge each" "Merge every nᵗʰ step" (n=5) "Try root first" 0 Inverter Bandpass è Merge every n th step is the fastest for inverter è Merge each is the fastest for bandpass Seite 23

24 Merge strategies: over-approximation "Don't merge" "Merge each" "Merge every nᵗʰ step" (n=5) "Try root first" 0 Inverter Bandpass è Don t merge causes least over-approximation Seite 24

25 Conclusion Split & merge allow to solve circuits with larger deviations Proposed split strategies Useful for larger circuits (~3% of the runtime) Smallest runtime for select by deviation Proposed merge strategies Smallest runtime for merge each (bandpass) Smallest over-approximation for don t merge Over-approximation can be traded for runtime Seite 25

26 Thanks for your attention! Questions? Seite 26