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Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments Miloslav Capek 1 1 Department of Electromagnetic Field, Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz March 5, 2019

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Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments

Miloslav ˇ Capek1

1Department of Electromagnetic Field,

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

March 5, 2019 Sensing, Imaging, and Machine Learning Workshop Lund University

Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 1 / 25

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Outline

  • 1. Shape Synthesis
  • 2. Local Step (TS)
  • 3. Global Step (GA)
  • 4. Preliminary Results
  • 5. 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 dofs, GA+TS.

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Shape Synthesis

Analysis × Synthesis

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Shape Synthesis

Analysis × Synthesis

Analysis (A) ◮ Shape Ω is given, BCs are known, determine EM quantities. p = LJ (r) = A

  • Ω, Ei

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Shape Synthesis

Analysis × Synthesis

Analysis (A) ◮ Shape Ω is given, BCs are known, determine EM quantities. p = LJ (r) = A

  • Ω, Ei

? Synthesis (S ≡ A−1) ◮ EM behavior is specified, neither Ω nor BCs are known.

  • Ω, Ei

= A−1p = Sp

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Shape Synthesis

Shapes Known to Be Optimal (In Certain Sense)

w L L/2 s

(a) (b)

Possible parametrization (unknowns: s, w, i.e., number of meanders).

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Shape Synthesis

Shapes Known to Be Optimal (In Certain Sense)

w L L/2 s

(a) (b)

Possible parametrization (unknowns: s, w, i.e., number of meanders).

0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 20 50 100 200 500 1000 ka p ≡ Qrad Qlb,TM

rad

Qrad Objective function p is Q-factor Qrad compared to its bound Qlb,TM

rad

.

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Shape Synthesis

Synthesis – Difficulties

How to get

  • Ω, Ei

= A−1p = Spuser? Questions inherently related to synthesis are [Deschapms, IEEE TAP 1972]:

  • 1. Can puser be chosen arbitrary?
  • 2. If puser is such that there exists a solution
  • Ω, Ei

, is that solution unique?

  • 3. If puser is known only approximately, which is

always the case, is the corresponding solution for

  • Ω, Ei

close to the exact one?

  • 4. If
  • Ω, Ei

is not exactly realized what effect will this have on A

  • Ω, Ei

?

A discretized meanderline antenna “M1” from [Best, IEEE APM 2015].

Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 5 / 25

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SLIDE 9

Shape Synthesis

Synthesis – Difficulties

How to get

  • Ω, Ei

= A−1p = Spuser? Questions inherently related to synthesis are [Deschapms, IEEE TAP 1972]:

  • 1. Can puser be chosen arbitrary? No.
  • 2. If puser is such that there exists a solution
  • Ω, Ei

, is that solution unique? No.

  • 3. If puser is known only approximately, which is

always the case, is the corresponding solution for

  • Ω, Ei

close to the exact one? No.

  • 4. If
  • Ω, Ei

is not exactly realized what effect will this have on A

  • Ω, Ei

? Potentially huge.

A discretized meanderline antenna “M1” from [Best, IEEE APM 2015].

Generally, infinitely many possibilities and local minima → need for shape discretization.

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Shape Synthesis

Shape Synthesis: Rigorous Definition

For a given impedance matrix Z ∈ CN×N, matrices A, {Bi}, {Bj}, a given excitation vector V ∈ CN, find a vector x such that minimize IHA (x) I subject to IHBi (x) I = pi IHBj (x) I ≤ pj Z (x) I = V x ∈ {0, 1}N ◮ Structure perturbation, ◮ combinatorial optimization, ◮ x serves as a characteristic function, ◮ A =

  • xxT

⊗ A.

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Shape Synthesis

Shape Synthesis: Rigorous Definition

For a given impedance matrix Z ∈ CN×N, matrices A, {Bi}, {Bj}, a given excitation vector V ∈ CN, find a vector x such that minimize IHA (x) I subject to IHBi (x) I = pi IHBj (x) I ≤ pj Z (x) I = V x ∈ {0, 1}N ◮ Structure perturbation, ◮ combinatorial optimization, ◮ x serves as a characteristic function, ◮ A =

  • xxT

⊗ A. minimize IHA (x) I subject to IHBi (x) I = pi IHBj (x) I ≤ pj Z (x) I = V x ∈ [0, 1]N ◮ Modulation of material’s parameters, ◮ relaxation of the combinatorial approach, ◮ alternatively, Aii = Aii + xiR0, ◮ close to topology optimization.

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Shape Synthesis

Initial Setup

Initial shape Ω.

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Shape Synthesis

Initial Setup

Initial shape Ω. A discretized region ΩT .

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Shape Synthesis

Initial Setup

Initial shape Ω. A discretized region ΩT . Basis functions {ψn} applied.

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Shape Synthesis

Shape Synthesis: Combinatorial Optimization Approach

◮ Each of N basis functions (dofs) is taken as {0, 1} unknown (not present/present). ◮ Feeding (Ei) is specified at the beginning. ◮ Fixed mesh grid ΩT : matrix operators calculated the only time.

Edge removal

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9

· · · Y22 Y23 · · · Y2N Y32 Y33 · · · Y2N . . . . . . . . . ... . . . YN2 YN3 · · · YNN                       A basis function removal.

I = Z−1V = YV

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Shape Synthesis

Shape Synthesis: Combinatorial Optimization Approach

◮ Each of N basis functions (dofs) is taken as {0, 1} unknown (not present/present). ◮ Feeding (Ei) is specified at the beginning. ◮ Fixed mesh grid ΩT : matrix operators calculated the only time. ◮ Suffers from curse of dimensionality, 2N possible solutions for N dofs. ◮ There is no “good” algorithm (working in polynomial time). Let us change the paradigm. . .

Edge removal

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9

· · · Y22 Y23 · · · Y2N Y32 Y33 · · · Y2N . . . . . . . . . ... . . . YN2 YN3 · · · YNN                       A basis function removal.

I = Z−1V = YV

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Local Step (TS)

Local Step: Topology Sensitivity

Current density if n ∈ {1, . . . , N} dof is removed/added: IfB =

  • If + ζf1I1

· · · If + ζfnIn · · · If + ζfNIN

  • .

Dofs to be removed/added.

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SLIDE 18

Local Step (TS)

Local Step: Topology Sensitivity

Current density if n ∈ {1, . . . , N} dof is removed/added: IfB =

  • If + ζf1I1

· · · If + ζfnIn · · · If + ζfNIN

  • .

An objective function p is defined as a quadratic form p (I) = IHAI IHBI and is calculated with a Hadamard product (vectorization) x (IfB) = diag

  • IH

fBAIfB

  • ⊘ diag
  • IH

fBBIfB

  • .

Dofs to be removed/added.

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Local Step (TS)

Local Step: Topology Sensitivity

Current density if n ∈ {1, . . . , N} dof is removed/added: IfB =

  • If + ζf1I1

· · · If + ζfnIn · · · If + ζfNIN

  • .

An objective function p is defined as a quadratic form p (I) = IHAI IHBI and is calculated with a Hadamard product (vectorization) x (IfB) = diag

  • IH

fBAIfB

  • ⊘ diag
  • IH

fBBIfB

  • .

Topology sensitivity is introduced as [Capek, et al., IEEE TAP 2019.] τ fB (p, ΩT ) = p (IfB) − p (If) ≈ ∇p (If) .

Dofs to be removed/added.

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Local Step (TS)

Greedy Algorithm – Example: Rectangular Plate

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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

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Local Step (TS)

Greedy Search Based on Topology Sensitivity

◮ Gradient-based method! ◮ Topology sensitivity τ (p) is calculated fast.

◮ Suitable for massive parallelization (granularity). ◮ Dynamic programming keeps the size of matrix operators as small as possible.

◮ A shape is represented by a gene g of length N − F (F is the number of feeders).

◮ Greedy algorithm through all the nearest neighbors (g : H (gi, g) = 1).

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 Greedy search with nearest neighbors (N = 4), 2N possible solutions, N 2N−1 possible connections.

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Local Step (TS)

Monte Carlo Analysis (Two Plates of 2:1 Aspect Ratio)

◮ A set of 5 · 104 random seeds, an objective function (p) is Q-factor.

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

Number of local improvements to find local minimum.

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

Q-factor normalized to its lower bound.

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Local Step (TS) Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 13 / 25

i = 0, Q/QTM

lb

= 119.35

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Local Step (TS)

Local Step as a Data-Mining Tool

plate 4 × 8 6 × 12 8 × 16 dofs, 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 Qmin/QTM

lb

1.18 1.12 1.11 Computer: CPU Threadripper 1950 (3.4 GHz), 128 GB RAM. ◮ Algorithm fully parallelized, 16 physical cores used (almost linear speed-up). ◮ How to ensure that every unique shape (gene) g is calculated just once?

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Local Step (TS)

Data Clustering – Are There Any Similarities?

10 20 30 40 50 60 70 1.2 1.4 1.6 1.8 2 final samples (tolerance in Q-factor 1 · 10−7) Qfinal/QTM

lb

Rectangular plate with 8 × 4 grid, minimization of Q-factor normalized to its lower bound and selection of pivots for k-means clustering (evaluated for Hamming distance).

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Local Step (TS)

Data Clustering – Are There Any Similarities?

50 100 150 S1 = 1959 solution (s) µ1 = 50.8, σ1 = 1.15 30 60 90 120 150 180 1 gs(n) 50 100 150 S2 = 24893 µ2 = 57.0, σ2 = 0.54 30 60 90 120 150 180 50 100 150 S3 = 23148 µ3 = 83.3, σ3 = 5.53 30 60 90 120 150 180

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Global Step (GA)

Genetic Algorithm with Topology Sensitivity

◮ Local step (topology sensitivity) is incorporated into a heuristic algorithm.

◮ Perfectly suitable since their representation via genes g.

◮ Each agent of genetic algorithm is refined by greedy algorithm

◮ Genetics explores subspace containing local minima only!

◮ Improved genes are returned to genetics for selection phase (roulette) and cross-overs. Entirely new paradigm: ◮ Local mutations equivalent to adding noise into a code with error-correction. ◮ Cross-overs shall respect geometrically continuous regions. ◮ Selection phase can be a subject of clustering (K-means,. . . ).

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Preliminary Results

Example: A Rectangular Plate

50 100 150 200 250 300 1.5 2 2.5

1st it. 25th it. 2nd it. 100th it. 5th it. 300th it.

agents Qfinal/QTM

lb

Agents returned to GA (SOGA, 12 × 6 grid, ka = 1/2, 320 agents, tuning penalization).

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Preliminary Results

Example: A Rectangular Plate (Results)

Found structure (SOGA, p = Q, 20 × 10 grid, ka = 1/2, tuning penalization), Qfinal/QTM

lb

≈ 1.1.

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Preliminary Results

Example: A Spherical Shell (Results)

SOGA, N = 1295, ka = 0.2, 192 agents, 300 iterations, Qmin/Qlb = 1.27, Qmin/QTM

lb

= 0.85. ◮ An antenna reaching the Q-factor bound on a spherical shell is known to be a helix [Best].

Bottom view. Front view. Top view.

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Preliminary Results

Local Step as a Useful Filter?

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20 40 60 80 100 120 140 160 dof (iteration i = 1) (from TS to GA) agents (from GA to TS)

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Concluding Remarks

Surrogated Model (Reduces Number of Unknowns)

Asymptotic complexity within the i-th iteration (for just one agent): O (τ i) ∝ MB (i) E2 (i) , where M is the number of matrix operators, B is the number of investigated edges, and E is the number of enabled basis functions. How to accelerate?

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Concluding Remarks

Surrogated Model (Reduces Number of Unknowns)

Asymptotic complexity within the i-th iteration (for just one agent): O (τ i) ∝ MB (i) E2 (i) , where M is the number of matrix operators, B is the number of investigated edges, and E is the number of enabled basis functions. How to accelerate?

  • 1. Not all edges are subjects to addition/removal.
  • 2. Hadamard products/low-rank representation of matrix operators.
  • 3. External archive for calculated genes (use them as pivots).
  • 4. Reduction of the number of dofs (surrogated models).

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Concluding Remarks

Reduction of the Complexity

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

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Concluding Remarks

Reduction of the Complexity

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

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Concluding Remarks

Reduction of the Complexity

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

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Concluding Remarks

Moving in the Solution Space

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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Concluding Remarks

Moving in the Solution Space

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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Concluding Remarks

Moving in the Solution Space

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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Concluding Remarks

Concluding Remarks

Topics of ongoing research ◮ To learn more about (structure of) the problem.

◮ Do an exhaustive search for N ≈ 35.

◮ To take advantage of the bounds.

◮ Improved branch-and-cut?

◮ To precisely classify the problem using graph theory.

◮ Solution space can be represented as an N-cube graph QN which forms Levy

  • family. These graphs have many well-known properties.

◮ To utilize the combinatorial optimization approaches (branch-and-cut, etc.).

◮ To determine M ≪ N pivots minimizing the maximum Hamming distance to any node. ◮ To apply clustering and other machine learning techniques to support diversity and convergence of the genetics. ◮ To introduce multicriteria version of the topology sensitivity τ (p).

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Questions

Questions?

Miloslav ˇ Capek miloslav.capek@fel.cvut.cz March 5, 2019 version 1.1 The presentation is available at

◮ capek.elmag.org Acknowledgment: This work was supported by the Ministry of Education, Youth and Sports through the project CZ.02.2.69/0.0/0.0/16 027/0008465.

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