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Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments Miloslav Capek 1 , Luk nek 1 , Mats Gustafsson 2 a s Jel 1 Department of Electromagnetic Field Czech Technical University in Prague Czech Republic

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SLIDE 1

Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments

Miloslav ˇ Capek1, Luk´ aˇ s Jel´ ınek1, Mats Gustafsson2

1Department of Electromagnetic Field

Czech Technical University in Prague Czech Republic miloslav.capek@fel.cvut.cz

2Department of Electrical and Information Technology

Lund University Sweden

July 10, 2019 AP-S/URSI 2019 Atlanta, GA, US

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 1 / 19

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SLIDE 2

Outline

  • 1. Problem Parametrization
  • 2. Inversion-free Solution of Linear System
  • 3. Graph Representation
  • 4. Monte Carlo Analysis (Q-factor Optimization)
  • 5. Heuristically Restarted Topology Sensitivity
  • 6. Concluding Remarks

This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e., A (r, t) = Re {A (r) exp (jωt)}.

(Sub-)optimal solution of Q-factor minimization

  • ver triangularized grid, 753 DOF.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 2 / 19

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SLIDE 3

Problem Parametrization

Degrees of Freedom

Ω

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19

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SLIDE 4

Problem Parametrization

Degrees of Freedom

Ω → {Tt}

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19

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SLIDE 5

Problem Parametrization

Degrees of Freedom

Ω → {Tt} → {ψn (r)}

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19

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SLIDE 6

Problem Parametrization

Degrees of Freedom

Ω → {Tt} → {ψn (r)} → g ◮ g ∈ {0, 1}N×1 is characteristic vector (discretized characteristic function)

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19

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SLIDE 7

Problem Parametrization

Shape Optimization

Capability to effectively remove/add a degree of freedom.1

  • 1M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans.

Antennas Propag., vol. 67, no. 6, pp. 3889 –3901, 2019. doi: 10.1109/TAP.2019.2902749

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 4 / 19

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SLIDE 8

Problem Parametrization

Shape Optimization

Capability to effectively remove/add a degree of freedom.1 ◮ Perfectly compatible with method of moments;

◮ basis functions used as DOF.

4 4 14 1 19 5 8 9 26 1 1 13 14 37 1 1 17 18 43 1 18 17 37 1 14 13 26 1 1 9 8 14 1 1 4 4 5 19 1 9 1 30 10 −1 3 −10 −4 2 −1 −1 −5 −1 −1 −3 −1 −1 −10 −3 −1 −1 −1 2 −4 3 9 10 30 1 1 11 26 11 1 −1 −1 3 2 5 −4 9 −1 1 5 −13 −8 −1164 −854 −937 11 −8 −13 9 11 −937 −854 −4 5 −1 5 1 −1 1 2 3 −1 11 1 11 26 2 7 27 6 7 5 −1 85 22 23 5 −4 186 73 74 −478 −18 192 198 −18 −478 186 198 192 −4 5 85 74 73 −1 5 27 23 22 7 2 7 6

Example of topology sensitivity, ka = 1/2, plate fed in the middle.

  • 1M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans.

Antennas Propag., vol. 67, no. 6, pp. 3889 –3901, 2019. doi: 10.1109/TAP.2019.2902749

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 4 / 19

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SLIDE 9

Problem Parametrization

Shape Optimization

Capability to effectively remove/add a degree of freedom.1 ◮ Perfectly compatible with method of moments;

◮ basis functions used as DOF.

◮ Inversion-free for the smallest perturbations;

◮ gradient-based shape

  • ptimization possible

deterministically.

4 4 14 1 19 5 8 9 26 1 1 13 14 37 1 1 17 18 43 1 18 17 37 1 14 13 26 1 1 9 8 14 1 1 4 4 5 19 1 9 1 30 10 −1 3 −10 −4 2 −1 −1 −5 −1 −1 −3 −1 −1 −10 −3 −1 −1 −1 2 −4 3 9 10 30 1 1 11 26 11 1 −1 −1 3 2 5 −4 9 −1 1 5 −13 −8 −1164 −854 −937 11 −8 −13 9 11 −937 −854 −4 5 −1 5 1 −1 1 2 3 −1 11 1 11 26 2 7 27 6 7 5 −1 85 22 23 5 −4 186 73 74 −478 −18 192 198 −18 −478 186 198 192 −4 5 85 74 73 −1 5 27 23 22 7 2 7 6

Example of topology sensitivity, ka = 1/2, plate fed in the middle.

  • 1M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans.

Antennas Propag., vol. 67, no. 6, pp. 3889 –3901, 2019. doi: 10.1109/TAP.2019.2902749

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 4 / 19

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SLIDE 10

Inversion-free Solution of Linear System

Removing and Adding DOF2

DOF removal:

  • I =
  • yf − Yfb

Ybb yb

  • lfV0,

Admittance matrix update:

  • Y = CT
  • Y − 1

Ybb ybyT

b

  • C,

DOF addition:

  • I = CT
  • yf
  • + xfb

zb

  • xb

−1

  • lfV0,

Admittance matrix update:

  • Y = 1

zb CT zbY + xbxT

b

−xb −xT

b

1

  • C,

Cnn = ⇔ gn = b 1 ⇔

  • therwise

xb = Y zb, zb = Zbb − zT

b xb

Cmn = 1 ⇔ gn = S (m) ⇔

  • therwise

◮ All columns of C matrix containing solely zeros are eliminated before use.

  • 2M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19

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SLIDE 11

Inversion-free Solution of Linear System

Removing and Adding DOF2

DOF removal:

  • I =
  • yf − Yfb

Ybb yb

  • lfV0,

Admittance matrix update:

  • Y = CT
  • Y − 1

Ybb ybyT

b

  • C,

DOF addition:

  • I = CT
  • yf
  • + xfb

zb

  • xb

−1

  • lfV0,

Admittance matrix update:

  • Y = 1

zb CT zbY + xbxT

b

−xb −xT

b

1

  • C,

Cnn = ⇔ gn = b 1 ⇔

  • therwise

xb = Y zb, zb = Zbb − zT

b xb

Cmn = 1 ⇔ gn = S (m) ⇔

  • therwise

◮ All columns of C matrix containing solely zeros are eliminated before use.

  • 2M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19

slide-12
SLIDE 12

Inversion-free Solution of Linear System

Removing and Adding DOF2

DOF removal:

  • I =
  • yf − Yfb

Ybb yb

  • lfV0,

Admittance matrix update:

  • Y = CT
  • Y − 1

Ybb ybyT

b

  • C,

DOF addition:

  • I = CT
  • yf
  • + xfb

zb

  • xb

−1

  • lfV0,

Admittance matrix update:

  • Y = 1

zb CT zbY + xbxT

b

−xb −xT

b

1

  • C,

Cnn = ⇔ gn = b 1 ⇔

  • therwise

xb = Y zb, zb = Zbb − zT

b xb

Cmn = 1 ⇔ gn = S (m) ⇔

  • therwise

◮ All columns of C matrix containing solely zeros are eliminated before use.

  • 2M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19

slide-13
SLIDE 13

Inversion-free Solution of Linear System

Removing and Adding DOF2

DOF removal:

  • I =
  • yf − Yfb

Ybb yb

  • lfV0,

Admittance matrix update:

  • Y = CT
  • Y − 1

Ybb ybyT

b

  • C,

DOF addition:

  • I = CT

yf

  • + xfb

zb

  • xb

−1

  • lfV0,

Admittance matrix update:

  • Y = 1

zb CT zbY + xbxT

b

−xb −xT

b

1

  • C,

Cnn = ⇔ gn = b 1 ⇔

  • therwise

xb = Y zb, zb = Zbb − zT

b xb

Cmn = 1 ⇔ gn = S (m) ⇔

  • therwise

◮ All columns of C matrix containing solely zeros are eliminated before use.

  • 2M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19

slide-14
SLIDE 14

Inversion-free Solution of Linear System

Removing and Adding DOF2

DOF removal:

  • I =
  • yf − Yfb

Ybb yb

  • lfV0,

Admittance matrix update:

  • Y = CT
  • Y − 1

Ybb ybyT

b

  • C,

DOF addition:

  • I = CT

yf

  • + xfb

zb

  • xb

−1

  • lfV0,

Admittance matrix update:

  • Y = 1

zb CT zbY + xbxT

b

−xb −xT

b

1

  • C,

Cnn = ⇔ gn = b 1 ⇔

  • therwise

xb = Y zb, zb = Zbb − zT

b xb

Cmn = 1 ⇔ gn = S (m) ⇔

  • therwise

◮ All columns of C matrix containing solely zeros are eliminated before use.

  • 2M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19

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SLIDE 15

Inversion-free Solution of Linear System

Topology Sensitivity

Topology sensitivity is defined as: τ (P, Ω) = −

  • P (I) − P
  • I

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 6 / 19

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SLIDE 16

Inversion-free Solution of Linear System

Topology Sensitivity

Topology sensitivity is defined as: τ (P, Ω) = −

  • P (I) − P
  • I

For example, Q-factor is evaluated as P (I) ≡ Q = IHWI +

  • IHXI
  • IHRI

, W = ω∂X/∂ω, Z = R + jX ∈ CN×N. Optimization variable: binary vector g; ◮ analogy to characteristic function, ◮ determines which DOF are enabled/disabled.

Solid – enabled, dashed – disabled, grayed – fed edge.

g =     1 1 1    

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 6 / 19

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SLIDE 17

Graph Representation

Nearest Neighbors: Hamming graph H(N, 2) of Vectors g

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 Solution space for N = 4 represented as a hierarchic Hamming graph with nearest neighbors highlighted.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 7 / 19

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SLIDE 18

Graph Representation

Nearest Neighbors: Hamming graph H(N, 2) of Vectors g

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 Solution space for N = 4 represented as a hierarchic Hamming graph with nearest neighbors highlighted.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 7 / 19

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SLIDE 19

Graph Representation

Nearest Neighbors: Hamming graph H(N, 2) of Vectors g

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 Solution space for N = 4 represented as a hierarchic Hamming graph with nearest neighbors highlighted.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 7 / 19

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SLIDE 20

Graph Representation

Nearest Neighbors: Hamming graph H(N, 2) of Vectors g

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 Solution space for N = 4 represented as a hierarchic Hamming graph with nearest neighbors highlighted.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 7 / 19

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SLIDE 21

Graph Representation Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 8 / 19

i = 0, Q/QTM

lb

= 119.35

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SLIDE 22

Monte Carlo Analysis (Q-factor Optimization)

Shape Optimization: Monte Carlo Analysis

◮ Topology sensitivity over nearest neighbors, fitness function: Q-factor. ◮ Starting seeds g selected randomly, updated till local minimum reached. ◮ Number of restarts N = 5 · 104, i.e., statistics doable. . .

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 9 / 19

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SLIDE 23

Monte Carlo Analysis (Q-factor Optimization)

Relative Improvement of Q-factor (PDF)

−3 −2 −1 0.5 1 1.5 2 grid 4 × 8 log10 (Qfinal/Qinit) PDF

grid 4 × 8 grid 6 × 12

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 10 / 19

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SLIDE 24

Monte Carlo Analysis (Q-factor Optimization)

Effect of Removals × Additions Only (PDF)

−2 −1 2 4 6 log10 (Qfinal/Qinit) PDF

grid 4 × 8, removals grid 4 × 8, additions

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 11 / 19

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SLIDE 25

Monte Carlo Analysis (Q-factor Optimization)

Relative Number of Required Improvements (PDF)

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 2 4 6 8 10 12 grid 6 × 12 κ = gfinal ⊕ ginit/(N − 1) PDF

grid 4 × 8 grid 6 × 12

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 12 / 19

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SLIDE 26

Monte Carlo Analysis (Q-factor Optimization)

Performance of Found Structures (CDF, PDF)

1.1 1.2 1.3 1.4 1.5 10 20 Qfinal/QTM

lb

PDF

grid 4 × 8 grid 6 × 12

0.25 0.5 0.75 1 CDF

grid 4 × 8 grid 6 × 12

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 13 / 19

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SLIDE 27

Monte Carlo Analysis (Q-factor Optimization)

Shape Optimization: Monte Carlo Analysis

◮ Topology sensitivity over nearest neighbors, fitness function: Q-factor. ◮ Starting seeds g selected randomly, updated till local minimum reached. plate 4 × 8 6 × 12 8 × 16 DOF, N 180 414 744 runs, I 5 · 104 5 · 104 1 · 103

  • comp. time, T [s]

2.4 · 103 5.8 · 104 1.2 · 104 evaluated shapes 7.2 · 108 3.9 · 109 2.6 · 108 shapes per second 3 · 105 7 · 104 2 · 104

  • comp. time per run, T/I [s]

4.8 · 10−2 1.2 · 100 1.2 · 101 evaluated shapes per run 1.4 · 104 7.8 · 104 2.6 · 105 Qmin/QTM

lb

1.18 1.12 1.11

Computer: CPU Threadripper 1950X (3.4 GHz), 128 GB RAM.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 14 / 19

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SLIDE 28

Heuristically Restarted Topology Sensitivity

Ω, Einc, ka, P Z, W, . . . generate gi

  • Y = Z−1 (gi)

τ

  • I, P
  • min

n τn < 0

update Y p∗, Ω∗ yes no Ω

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 15 / 19

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SLIDE 29

Heuristically Restarted Topology Sensitivity

Ω, Einc, ka, P Z, W, . . . i ≤ I generate gi

  • Y = Z−1 (gi)

τ

  • I, P
  • min

n τn < 0

update Y p∗, Ω∗ yes no yes no Ω

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 15 / 19

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SLIDE 30

Heuristically Restarted Topology Sensitivity

global step

Ω, Einc, ka, P Z, W, . . . i ≤ I generate gi

  • Y = Z−1 (gi)

τ

  • I, P
  • min

n τn < 0

update Y p∗, Ω∗ yes no yes no Ω

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 15 / 19

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SLIDE 31

Heuristically Restarted Topology Sensitivity

global step local step

Ω, Einc, ka, P Z, W, . . . i ≤ I generate gi

  • Y = Z−1 (gi)

τ

  • I, P
  • min

n τn < 0

update Y p∗, Ω∗ yes no yes no Ω

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 15 / 19

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SLIDE 32

Heuristically Restarted Topology Sensitivity

Topology Sensitivity (TS) & Heuristic Algorithm (HA)

TS Local, gradient-based, very fast. HA Robust, able to restart TS. TS & HA Moves only through local minima

  • f an optimization problem!

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 16 / 19

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SLIDE 33

Heuristically Restarted Topology Sensitivity

Topology Sensitivity (TS) & Heuristic Algorithm (HA)

TS Local, gradient-based, very fast. HA Robust, able to restart TS. TS & HA Moves only through local minima

  • f an optimization problem!

Four-arm folded helix3.

  • 0S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE
  • Trans. Antennas Propag., vol. 53, no. 3, pp. 1047–1053, 2005. doi: 10.1109/TAP.2004.842600

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 16 / 19

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SLIDE 34

Heuristically Restarted Topology Sensitivity

Topology Sensitivity (TS) & Heuristic Algorithm (HA)

TS Local, gradient-based, very fast. HA Robust, able to restart TS. TS & HA Moves only through local minima

  • f an optimization problem!

Discretized spherical shell (1536 triangles, 2304 DOF).

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 16 / 19

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SLIDE 35

Heuristically Restarted Topology Sensitivity

TS + HA (SOGA) optimization: Electrical size ka = 0.2 Triangles 1536 DOF 2304 Agents 224 Iterations 500 Evaluated antennas 4.6 · 109 Size of solution space 22303 ≈ 1.63 · 1091 Computational time 205 hours Q/QTM

lb

0.826 Q/QTM+TE

lb

1.205

Computer: CPU Threadripper 1950X (3.4 GHz), 16 cores, 128 GB RAM.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 17 / 19

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SLIDE 36

Concluding Remarks

What has been done ◮ Inversion-free topology sensitivity derived3.

◮ Shape perturbation possible with O (N), shape update only with outer product.

◮ Shape optimization possible via effective topology optimization4.

◮ Monte Carlo analysis based on greedy step through nearest neighbors. ◮ Termination criteria from fundamental bounds.

  • 3M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans.

Antennas Propag., vol. 67, no. 6, pp. 3889 –3901, 2019. doi: 10.1109/TAP.2019.2902749

  • 4M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 18 / 19

slide-37
SLIDE 37

Concluding Remarks

What has been done ◮ Inversion-free topology sensitivity derived3.

◮ Shape perturbation possible with O (N), shape update only with outer product.

◮ Shape optimization possible via effective topology optimization4.

◮ Monte Carlo analysis based on greedy step through nearest neighbors. ◮ Termination criteria from fundamental bounds.

Some ideas of ongoing research ◮ Mature cooperation of heuristics and topology sensitivity. ◮ Neural networks/machine learning techniques. ◮ Increase graph coverage;

◮ massive parallelization, model surrogation, graph simplification.

◮ Multi-objective optimization.

  • 3M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans.

Antennas Propag., vol. 67, no. 6, pp. 3889 –3901, 2019. doi: 10.1109/TAP.2019.2902749

  • 4M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, In press (AWPL). doi: 10.1109/LAWP.2019.2912459

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 18 / 19

slide-38
SLIDE 38

Questions

Questions?

Miloslav ˇ Capek miloslav.capek@fel.cvut.cz July 10, 2019 version 1.0 The presentation is available at

◮ capek.elmag.org Acknowledgment: This work was supported by the Czech Science Foundation (project No. 19-06049S).

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-39
SLIDE 39

Questions

Machine Learning With Topology Sensitivity Procedure

Conventional approach = impedance matrix inversion + fitness function evaluation. O

  • N 3

O (N), O

  • N 2
  • Ω, Ei

J = L−1 Ei J p = P (J) p Topology sensitivity: {gi}

  • topo. sens.

{pi} O (N), O

  • N 2
  • 1. Algorithmic complexity reduction.
  • 2. Local optimization algorithm.
  • 3. Excellent for data mining. . .

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-40
SLIDE 40

Questions

Machine Learning With Topology Sensitivity Procedure

Conventional approach = impedance matrix inversion + fitness function evaluation. O

  • N 3

O (N), O

  • N 2
  • Ω, Ei

J = L−1 Ei J p = P (J) p Learned model: gopt

  • lin. reg. model

puser

?

dof, N 180 414 744 shapes per second 3 · 105 7 · 104 2 · 104

Computer: CPU Threadripper 1950X (3.4 GHz), 128 GB RAM.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-41
SLIDE 41

Questions

Reduction of Graph (Computational) Complexity

Full grid of 21 × 11 pixels (N = 1354).

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-42
SLIDE 42

Questions

Reduction of Graph (Computational) Complexity

Truncated grid of 21 × 11 pixels (N = 954).

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-43
SLIDE 43

Questions

Reduction of Graph (Computational) Complexity

Truncated grid of 21 × 11 pixels with modified mesh (N = 115).

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-44
SLIDE 44

Questions

Surrogated Discretized Models (1484 → 450 DOFS), TS+GA

Rectangular plate 2 : 1, fed in the middle, ka = 1/2. Minimum Q-factor for surrogated model. Maximum radiation efficiency.

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-45
SLIDE 45

Questions

Moving in the Solution Space

g = [1 1 · · · 1 1]T g = [1 1 · · · 0 0]T g = [0 0 · · · 0 0]T g = [0 0 · · · 1 1]T

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-46
SLIDE 46

Questions

Moving in the Solution Space

g = [1 1 · · · 1 1]T g = [1 1 · · · 0 0]T g = [0 0 · · · 0 0]T g = [0 0 · · · 1 1]T

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-47
SLIDE 47

Questions

Moving in the Solution Space

g = [1 1 · · · 1 1]T g = [1 1 · · · 0 0]T g = [0 0 · · · 0 0]T g = [0 0 · · · 1 1]T

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-48
SLIDE 48

Questions

Moving in the Solution Space

g = [1 1 · · · 1 1]T g = [1 1 · · · 0 0]T g = [0 0 · · · 0 0]T g = [0 0 · · · 1 1]T

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-49
SLIDE 49

Questions

Moving in the Solution Space

g = [1 1 · · · 1 1]T g = [1 1 · · · 0 0]T g = [0 0 · · · 0 0]T g = [0 0 · · · 1 1]T

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19

slide-50
SLIDE 50

Questions

Moving in the Solution Space

g = [1 1 · · · 1 1]T g = [1 1 · · · 0 0]T g = [0 0 · · · 0 0]T g = [0 0 · · · 1 1]T

Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 19 / 19