Data Mining With Inversion-Free Evaluation of Nearest Neighbors in - PowerPoint PPT Presentation

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

Outline 1. Shape Synthesis 2. Local Step (TS) 3. Global Step (GA) 4. Preliminary Results 5. Concluding Remarks (Sub-)optimal solution of Q-factor minimization over triangularized grid, 753 dofs, GA+TS. This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e. , A ( r , t ) = Re { A ( r ) exp (j ωt ) } . Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 2 / 25

Shape Synthesis Analysis × Synthesis Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 3 / 25

Shape Synthesis Analysis × Synthesis Analysis ( A ) ◮ Shape Ω is given, BCs are known, determine EM quantities. � Ω, E i � p = L J ( r ) = A Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 3 / 25

Shape Synthesis Analysis × Synthesis ? Synthesis ( S ≡ A − 1 ) Analysis ( A ) ◮ Shape Ω is given, BCs are known, ◮ EM behavior is specified, neither Ω nor determine EM quantities. BCs are known. � Ω, E i � � Ω, E i � = A − 1 p = S p p = L J ( r ) = A Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 3 / 25

Shape Synthesis Shapes Known to Be Optimal (In Certain Sense) L/ 2 w s L (a) (b) Possible parametrization (unknowns: s , w , i.e. , number of meanders). Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 4 / 25

Shape Synthesis Shapes Known to Be Optimal (In Certain Sense) L/ 2 1000 Q lb , TM rad w 500 Q rad s 200 p ≡ Q rad 100 L 50 20 10 (a) (b) 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 Possible parametrization (unknowns: ka s , w , i.e. , number of meanders). Objective function p is Q-factor Q rad compared to its bound Q lb , TM . rad Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 4 / 25

Shape Synthesis Synthesis – Difficulties � Ω, E i � = A − 1 p = S p user ? How to get Questions inherently related to synthesis are [Deschapms, IEEE TAP 1972]: 1. Can p user be chosen arbitrary? � Ω, E i � 2. If p user is such that there exists a solution , is that solution unique? 3. If p user is known only approximately, which is always the case, is the corresponding solution for � Ω, E i � close to the exact one? A discretized meanderline antenna “M1” � Ω, E i � 4. If is not exactly realized what effect will from [Best, IEEE APM 2015]. � Ω, E i � this have on A ? Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 5 / 25

Shape Synthesis Synthesis – Difficulties � Ω, E i � = A − 1 p = S p user ? How to get Questions inherently related to synthesis are [Deschapms, IEEE TAP 1972]: 1. Can p user be chosen arbitrary? No. � Ω, E i � 2. If p user is such that there exists a solution , is that solution unique? No. 3. If p user is known only approximately, which is always the case, is the corresponding solution for � Ω, E i � close to the exact one? No. A discretized meanderline antenna “M1” � Ω, E i � 4. If is not exactly realized what effect will from [Best, IEEE APM 2015]. � Ω, E i � this have on A ? Potentially huge. Generally, infinitely many possibilities and local minima → need for shape discretization. Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 5 / 25

Shape Synthesis Shape Synthesis: Rigorous Definition For a given impedance matrix Z ∈ C N × N , matrices A , { B i } , { B j } , a given excitation vector V ∈ C N , find a vector x such that I H A ( x ) I minimize I H B i ( x ) I = p i subject to I H B j ( x ) I ≤ p j Z ( x ) I = V x ∈ { 0 , 1 } N ◮ Structure perturbation, ◮ combinatorial optimization, ◮ x serves as a characteristic function, � xx T � ◮ � A = ⊗ A . Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 6 / 25

Shape Synthesis Shape Synthesis: Rigorous Definition For a given impedance matrix Z ∈ C N × N , matrices A , { B i } , { B j } , a given excitation vector V ∈ C N , find a vector x such that I H A ( x ) I I H A ( x ) I minimize minimize I H B i ( x ) I = p i I H B i ( x ) I = p i subject to subject to I H B j ( x ) I ≤ p j I H B j ( x ) I ≤ p j Z ( x ) I = V Z ( x ) I = V x ∈ { 0 , 1 } N x ∈ [0 , 1] N ◮ Structure perturbation, ◮ Modulation of material’s parameters, ◮ combinatorial optimization, ◮ relaxation of the combinatorial approach, ◮ alternatively, � ◮ x serves as a characteristic function, A ii = A ii + x i R 0 , � xx T � ◮ � A = ⊗ A . ◮ close to topology optimization. Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 6 / 25

Shape Synthesis Initial Setup Initial shape Ω . Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 7 / 25

Shape Synthesis Initial Setup Initial shape Ω . A discretized region Ω T . Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 7 / 25

Shape Synthesis Initial Setup Initial shape Ω . A discretized region Ω T . Basis functions { ψ n } applied. Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 7 / 25

Shape Synthesis Shape Synthesis: Combinatorial Optimization Approach ◮ Each of N basis functions (dofs) is taken Edge removal   as { 0 , 1 } unknown (not present/present). 0 0 0 · · · 0 ◮ Feeding ( E i ) is specified at the beginning.   ψ 7   0 Y 22 Y 23 · · · Y 2 N     ψ 2 ψ 6 ◮ Fixed mesh grid Ω T : matrix operators   0 Y 32 Y 33 · · · Y 2 N   ψ 1 calculated the only time.   ψ 3 ψ 9 . . . .  ...  . . . .   . . . .   ψ 5 ψ 8 0 Y N 2 Y N 3 · · · Y NN ψ 4 A basis function removal. I = Z − 1 V = YV Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 8 / 25

Shape Synthesis Shape Synthesis: Combinatorial Optimization Approach ◮ Each of N basis functions (dofs) is taken Edge removal   as { 0 , 1 } unknown (not present/present). 0 0 0 · · · 0 ◮ Feeding ( E i ) is specified at the beginning.   ψ 7   0 Y 22 Y 23 · · · Y 2 N     ψ 2 ψ 6 ◮ Fixed mesh grid Ω T : matrix operators   0 Y 32 Y 33 · · · Y 2 N   ψ 1 calculated the only time.   ψ 3 ψ 9 . . . .  ...  . . . .   . . . .   ψ 5 ψ 8 0 Y N 2 Y N 3 · · · Y NN ψ 4 ◮ Suffers from curse of dimensionality, 2 N possible solutions for N dofs. A basis function removal. ◮ There is no “good” algorithm (working in polynomial time). I = Z − 1 V = YV Let us change the paradigm. . . Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 8 / 25

Local Step (TS) Local Step: Topology Sensitivity Current density if n ∈ { 1 , . . . , N } dof is removed/added: � � I f B = I f + ζ f 1 I 1 · · · I f + ζ fn I n · · · I f + ζ fN I N . Dofs to be removed/added. Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 9 / 25

Local Step (TS) Local Step: Topology Sensitivity Current density if n ∈ { 1 , . . . , N } dof is removed/added: � � I f B = I f + ζ f 1 I 1 · · · I f + ζ fn I n · · · I f + ζ fN I N . An objective function p is defined as a quadratic form p ( I ) = I H AI I H BI and is calculated with a Hadamard product (vectorization) � � � � I H I H x ( I f B ) = diag f B AI f B ⊘ diag f B BI f B . Dofs to be removed/added. Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 9 / 25

Local Step (TS) Local Step: Topology Sensitivity Current density if n ∈ { 1 , . . . , N } dof is removed/added: � � I f B = I f + ζ f 1 I 1 · · · I f + ζ fn I n · · · I f + ζ fN I N . An objective function p is defined as a quadratic form p ( I ) = I H AI I H BI and is calculated with a Hadamard product (vectorization) � � � � I H I H x ( I f B ) = diag f B AI f B ⊘ diag f B BI f B . Topology sensitivity is introduced as [Capek, et al. , IEEE TAP 2019.] τ f B ( p, Ω T ) = p ( I f B ) − p ( I f ) ≈ ∇ p ( I f ) . Dofs to be removed/added. Miloslav ˇ Capek Data Mining With Inversion-Free Evaluation of Nearest Neighbors in Method of Moments 9 / 25