Sparse Interpolation Annie Cuyt and Wen-shin Lee

Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Sparse Interpolation Applications References Annie Cuyt and Wen-shin Lee Canazei Workshop 7–12/9/2014 Universiteit Antwerpen Department of Mathematics and Computer Science 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 1/77 Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Motivation 11001 0 0 1 1 Universiteit Antwerpen 2/77 Sparse Interpolation Motivation Motivation Basics: exponential Basics: polynomial Approximation theory interpolate Applications References f (x) = α1 + α2 x 100 101 011110111110001 1100 1011 10100 ▸ Newton/Lagrange interpolation: 101 samples ▸ only 4 unknowns: α1 , α2 , x 0 , x 100 ! ▸ how to solve it from 4 samples? 11001 0 0 1 1 Universiteit Antwerpen 3/77 Sparse Interpolation Motivation Motivation Basics: exponential Basics: polynomial Approximation theory ▸ exponential analysis Applications ▸ generalized eigenvalue problems References ▸ computer algebra ▸ orthogonal polynomials ▸ signal processing ▸ moment problems ▸ nonlinear approximation theory ▸ many applications . . . 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 4/77 Sparse Interpolation Motivation Motivation Basics: exponential Basics: polynomial Approximation theory xs = s∆, s = 0, 1, 2, . . . Applications References n k ∑ αi xs i = fs , n ≪ max(ki ), ki ∈ N i=1 n1 n2 ∑ αi,1 cos(φi,1 xs ) + ∑ αi,2 sin(φi,2 xs ) = fs i=1 i=1 n ∑ αi exp(φi xs ) = fs , φi ∈ C i=1 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 5/77 Sparse Interpolation Motivation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 1. Univariate exponential sparse interpolation (Exercise) 2. Multivariate polynomial sparse interpolation (Exercise) 3. Connection with rational approximation theory (Exercise) 4. Applications unlimited 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 6/77 Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Basics: exponential 11001 0 0 1 1 Universiteit Antwerpen 7/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Gaspard Riche de Prony [1795] 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 8/77 Sparse Interpolation Motivation Basics: exponential interpolation problem: Basics: exponential n Basics: polynomial Approximation theory ∑ αi exp(φi xs ) = fs , s = 0, . . . , 2n − 1 i=1 Applications References 2π , ω = 2π/M M ∣I(φi )∣ < M/2, Ωi = exp(φi ω), xs = s n fs = ∑ αi Ωsi , s = 0, . . . , 2n − 1 i=1 101 0111101111100 1 0 010 0111 1101 0011001 0 0 1 1 Universiteit Antwerpen ⎧ α1 + ⋅ ⋅ ⋅ + αn = f 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪α1 Ω1 + ⋅ ⋅ ⋅ + αn Ωn = f1 ⎨ ⎪ ⋮ ⎪ ⎪ ⎪ ⎪ ⎪ 2n−1 2n−1 ⎪ = f2n−1 ⎩α1 Ω1 + ⋅ ⋅ ⋅ + αn Ωn e iω ω= 2π M 9/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential finding Ωi : Basics: polynomial Approximation theory Applications n References n n−1 + ⋅ ⋅ ⋅ + b1 z + b0 ∏(z − Ωi ) = z + bn−1 z i=1 n 0 = ∑ αi Ωsi (Ωni + bn−1 Ωn−1 + ⋅ ⋅ ⋅ + b0 ) i i=1 n n−1 n j=0 i=1 = ∑ αi Ωn+s + ∑ bj (∑ αi Ωj+s i i ) i=1 n−1 = fs+n + ∑ bj fs+j j=0 101 0111101111100 1 0 010 0111 1101 0011001 0 0 1 1 Universiteit Antwerpen 10/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Approximation theory Applications References ⎛ f0 ⎜ ⋮ ⎝fn−1 ... ⋰ ... fn−1 ⎞ ⎛ b0 ⎞ ⎛ fn ⎞ ⋮ ⎟⎜ ⋮ ⎟ = −⎜ ⋮ ⎟ ⎝f2n−1 ⎠ f2n−2 ⎠ ⎝bn−1 ⎠ Hankel matrix: (r ) Hn 101 011110111110001 1100 1011 10100 ⎛ fr =⎜ ⋮ ⎝fr +n−1 ... ⋰ ... fr +n−1 ⎞ ⋮ ⎟ fr +2n−2 ⎠ 11001 0 0 1 1 Universiteit Antwerpen 11/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Hadamard polynomial: Approximation theory Applications References RRR f RRR 0 R ⋮ (0) Hn (z) = RRRRR RRRfn−1 RRR 1 R fn−1 ⋮ f2n−2 z n−1 RRR RRR RRR R f2n−1 RRRR R z n RRRR fn ⋮ (0) n ∏(z − Ωi ) = i=1 ... ⋰ ... ... Hn (z) (0) ∣Hn ∣ = z n + bn−1 z n−1 + ⋅ ⋅ ⋅ + b1 z + b0 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 12/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential formally orthogonal polynomial: Basics: polynomial Approximation theory γ ∶ z s → fs , Applications s = 0, 1, . . . n References γ ∶ exp(φi xs ) = Ωsi → ∑ αi Ωsi = fs γ∶ (0) Hn (z) zi (0) ∣Hn ∣ i=1 → 0, i = 0, . . . , n − 1 (0) Hn (z) (0) ⊥γ z i , i = 0, . . . , n − 1 ∣Hn ∣ [Henrici, 1974] 101 0111101111100 1 0 010 0111 1101 0011001 0 0 1 1 Universiteit Antwerpen 13/77 Sparse Interpolation Basics: exponential Motivation (0) Basics: exponential roots of Basics: polynomial Hn (z) (0) from GEP: ∣Hn ∣ Approximation theory Applications References ⎛ 1 ⎜ Ω (0) Hn = ⎜ 1 ⎜ ⋮ ⎝Ωn−1 = (1) Hn 101 011110111110001 1100 1011 10100 ... Ω2 ... ... 1 T Vn D α Vn ⎛Ω 1 = VnT Dα ⎜ ⎝ ⋱ 1 ⎞ ⎛ α1 Ωn ⎟ ⎜ ⎟⎜ ⋮ ⎟⎜ n−1 ⎠ ⎝ Ωn ⎞ ⎛1 Ω 1 ⋱ ⎟ ⎜ ⋮ Ω2 ⎟⎜ ⎟⎜ ⋮ αn ⎠ ⎝1 Ωn ... ... Ωn−1 1 ⎞ ⋮ ⎟ ⎟ ⋮ ⎟ n−1 ⎠ Ωn ⎞ ⎟ Vn Ωn ⎠ 11001 0 0 1 1 Universiteit Antwerpen 14/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Approximation theory Applications (1) References det (Hn ⎛ ⎛Ω 1 − λ (0) − λHn ) = det ⎜VnT Dα ⎜ ⎝ ⎝ = 0 for λ = Ωi , ⋱ ⎞ ⎞ ⎟ Vn ⎟ Ωn − λ⎠ ⎠ i = 1, . . . , n [Hua and Sarkar, 1990] 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 15/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Approximation theory finding φi : Applications References exp(φi ) = exp(R(φi ))e iI(φi ) M ∣I(φi )∣ < ∶ 2 arg(Ωi ) = arg(exp(φi ω)) 2π = I(φi ) ∈ ]−π, π[ M 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 16/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Approximation theory finding αi : Applications References n s+j ∑ αi Ωi = fs+j , s = 0, . . . , n − 1, 0≤j ≤n i=1 j ⎛ Ω1 ⎜ ⋮ ⎝Ωj+n−1 1 ... ... Ωjn ⎞ ⎛α1 ⎞ ⎛ fj ⎞ ⋮ ⎟⎜ ⋮ ⎟ = ⎜ ⋮ ⎟ Ωj+n−1 ⎠ ⎝αn ⎠ ⎝fj+n−1 ⎠ n remaining interpolation conditions are linearly dependent 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 17/77 Sparse Interpolation Basics: exponential Motivation Basics: exponential Basics: polynomial Approximation theory finding n: Applications References (r ) N < n ∶ ∣HN ∣ ≡/ 0, (r ) N = n ∶ ∣HN ∣ ≠ 0 (r ) N > n ∶ ∣HN ∣ ≡ 0, 101 011110111110001 1100 1011 10100 r = 0, 1, . . . if Ωi ≠ Ωj for i ≠ j [Kaltofen and Lee, 2003] r = 0, 1, . . . 11001 0 0 1 1 Universiteit Antwerpen 18/77 Sparse Interpolation Example: exponential Motivation Basics: exponential Basics: polynomial 4 Approximation theory φ(x) = ∑ αi exp(φi x) i=1 Applications References α1 = 1 φ1 = 0 α2 = 2.4 φ2 = −5 + 19.97i α3 = −2.1 φ3 = 3 + 45i α4 = 0.2 φ4 = 5.3i 2π evaluate at xs = s 100 , M = 100, ∣I(φi )∣ < 50 sequence f0 , . . . , f7 , . . . is linearly generated 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 19/77 Sparse Interpolation Example: exponential Motivation 102 Basics: exponential Basics: polynomial 10−4 Approximation theory Applications References 10−10 10−16 2 4 (0) Figure: HN ̃j − Ωj ∣ ∣Ω 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen ∣Ωj ∣ ≤ 2 × 10−12 , 6 singular, N = 6 ∣φ̃j − φj ∣ ≤ 2 × 10−12 ∣φj ∣ 20/77 Sparse Interpolation Exercise: exponential Motivation Basics: exponential φ(x) = α1 exp(φ1 x) + α2 exp(φ2 x) Basics: polynomial Approximation theory Applications 4 unknowns φ1 , φ2 , α1 , α2 identify φ(x) from References φ(0) = α1 + α2 φ′ (0) = α1 φ1 + α2 φ2 φ′′ (0) = α1 φ21 + α2 φ22 φ′′′ (0) = α1 φ31 + α2 φ32 solution: (0) ∣H3 ∣ 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen RRR φ RR 0 = RRRR φ′0 RRR ′′ RRφ0 φ′0 φ′′0 φ′′′ 0 φ′′0 RRRR RRR φ′′′ 0 RR = 0 RRR φ′v 0 RR symbolically n=2 21/77 > ########################### ## Exercise: exponential ## ########################### > restart: > with(LinearAlgebra): > Phi[0] := alpha[1] * exp(phi[1]*x) + alpha[2] * exp(phi[2]*x); f x 1 F0 := a 1 e > for i from 1 to 4 do Phi[i] := diff(Phi[i-1], x); od; f x 1 F1 := a 1 f1 e 2 f1 x F2 := a 1 f1 e F3 := 4 f1 x (1) f x 2 C a 2 f2 e 2 f2 x C a 2 f2 e 3 f1 x a 1 f1 e C a 2 F4 := a 1 f1 e > for i from 0 to 4 do f[i] := simplify(subs(x=0, Phi[i])); od; f x 2 C a2 e 3 f2 x f2 e 4 f2 x C a 2 f2 e (2) f0 := a 1 C a 2 f1 := a 1 f1 C a 2 f2 2 2 f2 := a 1 f1 C a 2 f2 f3 := 3 a 1 f1 C a 2 f2 f4 := 4 a 1 f1 C a 2 3 f2 4 (3) > H[3] := HankelMatrix([f[0],f[1],f[2],f[3],f[4]],3); (4) a1 C a2 H3 := 2 2 3 3 a 1 f1 C a 2 f2 a 1 f1 C a 2 f2 2 2 a 1 f1 C a 2 f2 a 1 f1 C a 2 f2 a 1 f1 C a 2 f2 2 a 1 f1 C a 2 2 f2 > Determinant(H[3]); 3 a 1 f1 C a 2 3 f2 4 a 1 f1 C a 2 0 (4) 4 f2 (5) > H[2] := HankelMatrix([f[0],f[1],f[2]],2); H2 := a1 C a2 a 1 f1 C a 2 f2 2 2 a 1 f1 C a 2 f2 a 1 f1 C a 2 f2 (6) > F2 := Matrix([[f[2]],[f[3]]]); 2 F2 := 2 a 1 f1 C a 2 f2 3 a 1 f1 C a 2 3 f2 (7) > B:=LinearSolve(H[2],-F2); B := > B_poly := z^2 + B(2)*z + B(1); > z_root:=solve(B_poly=0, z); f1 f2 Kf1 K f2 (8) B_poly := z2 C Kf1 K f2 z C f1 f2 (9) z_root := f2 , f1 (10) > V := Transpose(VandermondeMatrix([z_root[1],z_root[2]],2,2)); 1 1 V := f2 f1 (11) > F0 := Matrix([[f[0]],[f[1]]]); F0 := a1 C a2 a 1 f1 C a 2 f2 (12) > A:=LinearSolve(V,F0); A := > alpha[1] := Pi; alpha[2] := 5; phi[1] := 2; phi[2] := -1/7; a2 a1 (13) a 1 := p a 2 := 5 f1 := 2 1 7 f2 := K > for i from 0 to 4 do f[i] := simplify(subs(x=0, Phi[i])); od; (14) f0 := p C 5 5 7 5 f2 := 4 p C 49 5 f3 := 8 p K 343 5 f4 := 16 p C 2401 f1 := 2 p K (15) > H[3] := HankelMatrix([f[0],f[1],f[2],f[3],f[4]],3); 5 pC5 2pK 7 H3 := 2pK 5 7 4pC 5 5 4pC 8pK 49 343 > Determinant(H[3]); 5 49 8pK 5 343 5 49 4pC (16) 5 16 p C 2401 0 (17) > H[2] := HankelMatrix([f[0],f[1],f[2]],2); 2pK 5 7 4pC 5 49 pC5 H2 := 2pK 5 7 (18) > F2 := Matrix([[f[2]],[f[3]]]); 4pC 5 49 8pK 5 343 F2 := (19) > B:=LinearSolve(H[2],-F2); 2 7 K B := 13 7 (20) K > B_poly := z^2 + B(2)*z + B(1); B_poly := z2 K 13 2 zK 7 7 (21) > z_root:=solve(B_poly=0,z); (22) 1 7 z_root := 2, K (22) > V := Transpose(VandermondeMatrix([z_root[1],z_root[2]],2,2)); 1 1 V := (23) 1 7 2 K > F0 := Matrix([[f[0]],[f[1]]]); pC5 F0 := 2pK 5 7 (24) > A:=LinearSolve(V,F0); A := p 5 (25) Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Basics: polynomial 11001 0 0 1 1 Universiteit Antwerpen 22/77 Sparse Interpolation Basics: polynomial Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Michael Ben-Or and Prasoon Tiwari [1988] 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 23/77 Sparse Interpolation Motivation Basics: polynomial interpolation problem: Basics: exponential Basics: polynomial ∑ Approximation theory (k1 ,...,kd )∈K αk1 ,...,kd x1k1 ⋯xdkd , #K = n Applications References evaluate at (x1 , . . . , xd ) = (ω1s , . . . , ωds ) ωj = exp(2πi/pj ), p(ω1s , . . . , ωds ) = fs , (i) Ωi = k ω1 1 pj > ∂j p(x1 , . . . , xd ), 0 ≤ s ≤ 2n − 1 (i) k ⋯ωdd , pj mutually prime e iωj i = 1, . . . , n ωj = 101 011110111110001 1100 1011 10100 2π pj 11001 0 0 1 1 Universiteit Antwerpen [Giesbrecht, Labahn, and Lee, 2006] 24/77 Sparse Interpolation Basics: polynomial Motivation Basics: exponential finding Ωi : Basics: polynomial Approximation theory Applications n References n n−1 + ⋅ ⋅ ⋅ + b1 z + b0 ∏(z − Ωi ) = z + bn−1 z i=1 n 0 = ∑ αi Ωsi (Ωni + bn−1 Ωn−1 + ⋅ ⋅ ⋅ + b0 ) i i=1 n n−1 n j=0 i=1 = ∑ αi Ωn+s + ∑ bj (∑ αi Ωj+s i i ) i=1 n−1 = fs+n + ∑ bj fs+j j=0 101 0111101111100 1 0 010 0111 1101 0011001 0 0 1 1 Universiteit Antwerpen 25/77 Sparse Interpolation Basics: polynomial Motivation Basics: exponential Basics: polynomial Approximation theory Applications References ⎛ f0 ⎜ ⋮ ⎝fn−1 ... ⋰ ... fn−1 ⎞ ⎛ b0 ⎞ ⎛ fn ⎞ ⋮ ⎟⎜ ⋮ ⎟ = −⎜ ⋮ ⎟ ⎝f2n−1 ⎠ f2n−2 ⎠ ⎝bn−1 ⎠ Ωi : zeros of formally orthogonal Hadamard polynomial n ∏(z − Ωi ) = i=1 101 011110111110001 1100 1011 10100 (0) Hn (z) (0) ∣Hn ∣ 11001 0 0 1 1 Universiteit Antwerpen 26/77 Sparse Interpolation Basics: polynomial Motivation (i) Basics: exponential (i) finding (k1 , . . . , kd ): Basics: polynomial Approximation theory Applications reverse Chinese remainder theorem References m = p1 ⋯pd Ωi = ω k(i) , (i) k(i ) = k1 (i) m kj pj (i) 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen kj ω = exp ⎛ 2πi ⎞ , ⎝ ∏dj=1 pj ⎠ m (i) m + ⋅ ⋅ ⋅ + kd p1 pd mod pj = k(i ) mod pj , < pj , gcd(pj , m/pj ) = 1, j = 1, . . . , d 27/77 Sparse Interpolation Basics: polynomial Motivation Basics: exponential Basics: polynomial Approximation theory finding αi = αk (i) ,...,k (i) : 1 d Applications References n s+j ∑ αi Ωi = fs+j , s = 0, . . . , n − 1, 0≤j ≤n i=1 j ⎛ Ω1 ⎜ ⋮ ⎝Ωj+n−1 1 ... ... Ωjn ⎞ ⎛α1 ⎞ ⎛ fj ⎞ ⋮ ⎟⎜ ⋮ ⎟ = ⎜ ⋮ ⎟ j+n−1 ⎠ ⎝α ⎠ ⎝f ⎠ Ω n j+n−1 n remaining interpolation conditions are linearly dependent 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 28/77 Sparse Interpolation Basics: polynomial Motivation Basics: exponential Basics: polynomial finding n: Approximation theory Applications References floating-point arithmetic: (r ) ∣HN ∣ ≡ 0, N > n, r = 0, 1, . . . exact arithmetic: increase n till δs ∶= fs + bn−1 fs−1 + ⋅ ⋅ ⋅ + b0 fs−n equals 0, s ≥ 2n [Massey, 1969] 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 29/77 Sparse Interpolation Example: polynomial Motivation Basics: exponential p(x, y ) = x 5 y + 2.2x 4 y 4 − 0.5xy 11 + 0.1xy 12 Basics: polynomial Approximation theory Applications p1 = 6, p2 = 13, ω1 = exp(2πi/6), ω2 = exp(2πi/13), References p(ω1s , ω2s ), s = 0, . . . , 7 (floating-point) s = 0, . . . , 7 (exact arithmetic) or p(p1s , p2s ), sequence f0 , f1 , . . . , f7 , . . . is linearly generated and δ8 = 0 Hadamard polynomial: z 4 + (−3.67 + 0.0799i)z 3 + (5.35 − 0.216i)z 2 (−3.67 + 0.216i)z + (0.997 − 0.0805i) 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 30/77 Sparse Interpolation Basics: polynomial Motivation Basics: exponential Basics: polynomial Approximation theory m = 78, Ω1 = ω Applications ω = exp(2πi/78), k(1) = ω 71 = ω 5×13+1×6 Ω2 = ω k(2) = ω 76 = ω 4×13+4×6 References Ω3 = ω k(3) = ω 79 = ω 1×13+11×6 Ω4 = ω k(4) = ω 85 = ω 1×13+12×6 K = {(5, 1), (4, 4), (1, 11), (1, 12)} ⇒ terms x 5 y , x 4 y 4 , xy 11 , xy 12 Vandermonde system 4 s+j ∑ αi Ωi = fs+j , s = 0, . . . , n − 1, 0≤j ≤n i=1 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 31/77 Sparse Interpolation Exercise: polynomial Motivation Basics: exponential Basics: polynomial p(x, y ) = (x − 3)5 (y + 5) + 2.2(x − 3)4 (y + 5)4 Approximation theory − 0.5(x − 3)(y + 5)11 + 0.1(x − 3)(y + 5)12 Applications = u 5 v + 2.2u 4 v 4 − 0.5uv 11 + 0.1uv 12 , References u = x − 3, v = y + 5 p1 = 6, p2 = 13, ω1 = exp(2πi/6), ω2 = exp(2πi/13), u= ω1s , p(ω1s + 3, ω2s − 5), 101 011110111110001 1100 1011 10100 v = ω2s s = 0, . . . , 7 11001 0 0 1 1 Universiteit Antwerpen 32/77 > ########################## ## Exercise: polynomial ## ########################## > restart: > with(LinearAlgebra): > read "./ExpLog.mm": read "./RevChRem.mm": > Digits:= 16: > poly_uv := u^5*v + 2.2*u^4*v^4-0.5*u*v^11+0.1*u*v^12; 5 4 4 11 12 poly_uv := u v C 2.2 u v K 0.5 u v C 0.1 u v > poly_xy := subs(u=x-3,v=y+5, poly_uv); poly_xy := x K 3 5 y C 5 C 2.2 x K 3 4 yC5 4 K 0.5 x K 3 yC5 11 (1) C 0.1 x K 3 yC5 12 (2) > expand(poly_xy); K1.464750 105 x K 1.45595805 107 y C 1.101600 105 C 72900.0 x2 K 3.21998325 107 y2 C x5 y C 1085.0 x4 y K 13110.0 x3 y C 59130.0 x2 y 6 4 4 4 3 3 4 4 2 3 3 2 4 3 2 (3) 2 3 C 4.7644175 10 x y C 2.2 x y C 44.0 x y K 26.4 x y C 330.0 x y K 528.0 x y C 118.8 x y K 3960.0 x y C 2376.0 x y C 6.4450749 106 x y4 C 17820.0 x2 y2 C 1.07374355 107 x y3 C 1.07065475 107 x y2 C 5.5 x y11 C 137.5 x y10 C 2062.5 x y9 C 20625.0 x y8 C 1.443750 105 x y7 C 7.218750 105 x y6 C 2.5781250 106 x y5 C 0.1 x y12 C 5 x5 C 1300.0 x4 K 16050.0 x3 K 1.93357593 107 y4 K 3.22229985 107 y3 K 16.5 y11 K 412.5 y10 K 6187.5 y9 K 61875.0 y8 K 4.331250 105 y7 K 2.1656250 106 y6 K 7.7343750 106 y5 K 0.3 y12 > p1 := 6; p2 := 13; > w1 := evalf(exp(2*Pi*I/p1)); w2 := evalf(exp(2*Pi*I/p2)); p1 := 6 p2 := 13 (4) w1 := 0.5000000000000000 C 0.8660254037844385 I w2 := 0.8854560256532100 C 0.4647231720437684 I (5) > for i from 0 to 7 do f[i] := evalf(subs(x=w1î+3, y=w2î-5, poly_xy)): od; f0 := 2.8 C 0. I f1 := 2.602841017834476 K 0.8741573457604577 I f2 := 2.064816858508122 K 1.589980617941464 I f3 := 1.329941999443174 K 2.035486797328322 I f4 := 0.5888159453592064 K 2.177111443869176 I f5 := 0.02061376974983473 K 2.067629883507751 I f6 := K0.2610215691867326 K 1.827541993129119 I f7 := K0.2414681965457515 K 1.605741666597298 I (6) > H0[4] := HankelMatrix([f[0],f[1],f[2],f[3],f[4],f[5],f[6]],4); H04 := 2.8 C 0. I, 2.602841017834476 K 0.8741573457604577 I, 2.064816858508122 K 1.589980617941464 I, 1.329941999443174 (7) K 2.035486797328322 I , 2.602841017834476 K 0.8741573457604577 I, 2.064816858508122 K 1.589980617941464 I, 1.329941999443174 K 2.035486797328322 I, 0.5888159453592064 K 2.177111443869176 I , 2.064816858508122 K 1.589980617941464 I, 1.329941999443174 K 2.035486797328322 I, 0.5888159453592064 K 2.177111443869176 I, 0.02061376974983473 K 2.067629883507751 I , 1.329941999443174 K 2.035486797328322 I, 0.5888159453592064 K 2.177111443869176 I, 0.02061376974983473 K 2.067629883507751 I, K0.2610215691867326 K 1.827541993129119 I > H1[4] := HankelMatrix([f[1],f[2],f[3],f[4],f[5],f[6],f[7]],4); H14 := 2.602841017834476 K 0.8741573457604577 I, 2.064816858508122 K 1.589980617941464 I, 1.329941999443174 (8) K 2.035486797328322 I, 0.5888159453592064 K 2.177111443869176 I , 2.064816858508122 K 1.589980617941464 I, 1.329941999443174 K 2.035486797328322 I, 0.5888159453592064 K 2.177111443869176 I, 0.02061376974983473 K 2.067629883507751 I , 1.329941999443174 K 2.035486797328322 I, 0.5888159453592064 K 2.177111443869176 I, 0.02061376974983473 K 2.067629883507751 I, K0.2610215691867326 K 1.827541993129119 I , 0.5888159453592064 K 2.177111443869176 I, 0.02061376974983473 K 2.067629883507751 I, K0.2610215691867326 K 1.827541993129119 I, K0.2414681965457515 K 1.605741666597298 I > eigenvalue_list := Eigenvalues(H1[4],H0[4], output=list); eigenvalue_list := 0.8451900855580868 C 0.5344658261253400 I, 0.8451900855413997 K 0.5344658261265820 I, 0.9870502626364418 K 0.1604112808397337 I, 0.9967573080994497 C 0.08046656865700708 I > total_exp_list := []: for i from 1 to 4 do total_exp_list := [op(total_exp_list), round(ExpLog(eigenvalue_list[i],6*13))]: od: > total_exp_list; 7, 71, 76, 1 (9) (10) > exp_list:=[]: for i from 1 to 4 do exp_list := [op(exp_list), [RevChRem(total_exp_list[i], 6, 6*13), RevChRem(total_exp_list[i], 13, 6*13)]]: od: > exp_list; 1, 12 , 5, 1 , 4, 4 , 1, 11 (11) > V := Transpose(VandermondeMatrix(eigenvalue_list)); V := 1. C 0. I, 1. C 0. I, 1. C 0. I, 1. C 0. I , 0.8451900855580868 C 0.5344658261253400 I, 0.8451900855413997 K 0.5344658261265820 I, 0.9870502626364418 K 0.1604112808397337 I, 0.9967573080994497 C 0.08046656865700708 I , 0.4286925614298439 C 0.9034504346214993 I, 0.4286925614003088 K 0.9034504346057614 I, 0.9485364419500248 K 0.3166679937654143 I, 0.9870502625782285 C 0.1604112807331159 I , K0.1205366802302719 C 0.9927088741336253 I, K0.1205366802550991 K 0.9927088740899947 I, 0.8854560256661491 K 0.4647231719910745 I, 0.9709418173518603 C 0.2393156640939936 I > F := Matrix([[f[0]],[f[1]],[f[2]],[f[3]]]); 2.8 C 0. I F := 2.602841017834476 K 0.8741573457604577 I 2.064816858508122 K 1.589980617941464 I (12) (13) 1.329941999443174 K 2.035486797328322 I > A := LinearSolve(V, F); (14) -11 0.1000000000041224 C 1.216000331768808 10 -11 A := 1.000000000007150 C 2.565636441003167 10 I I (14) 2.200000000261579 K 7.965585370092284 10-11 I -11 K0.5000000002728518 C 4.183975685423307 10 I > var_list := [u,v]: p := 0: for i from 1 to 4 do term := A[i,1]: for j from 1 to 2 do term := term*var_list[j]êxp_list[i,j]: od: p := p + term: od: > p; 0.1000000000041224 C 1.216000331768808 10-11 I u v12 C 1.000000000007150 C 2.565636441003167 10-11 I u5 v (15) C 2.200000000261579 K 7.965585370092284 10-11 I u4 v4 C K0.5000000002728518 C 4.183975685423307 10-11 I u v11 > poly_uv; u5 v C 2.2 u4 v4 K 0.5 u v11 C 0.1 u v12 (16) Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Approximation theory 11001 0 0 1 1 Universiteit Antwerpen 33/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Henri Padé [1892] and Christian Pommerenke [1973] 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 34/77 Sparse Interpolation Approximation theory Motivation Basics: exponential n Basics: polynomial Approximation theory fs = ∑ αi exp(φi xs ), s = 0, 1, . . . , 2n − 1 i=1 Applications ∞ References f (z) = ∑ fj z j , fj = 0, j <0 j=0 m p(z) = ∑ ai z i , i=0 n q(z) = ∑ bi z i i=0 ⎛ ∞ j⎞ i ∑ fj z q(z) − p(z) = ∑ ci z ⎝j=0 ⎠ i≥m+n+1 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 35/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial Approximation theory Applications References ⎧ f0 b0 = a0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪f1 b0 + f0 b1 = a1 ⎨ ⎪ ⋮ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f b ⎩ m 0 + ⋅ ⋅ ⋅ + fm−n bn = am b0 = 1 ⎧ ⎪ fm+1 b0 + ⋅ ⋅ ⋅ + fm−n+1 bn = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ⋮ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩fm+n b0 + ⋅ ⋅ ⋅ + fm bn = 0 H(n) (m+1−n) ⎛b1 ⎞ ⎛fm+1 ⎞ ⎜ ⋮ ⎟ = −⎜ ⋮ ⎟ ⎝bn ⎠ ⎝fm+n ⎠ [m/n](z) ∶= p(z)/q(z) 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 36/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial n Approximation theory Applications fs = ∑ αi exp(φi xs ) i=1 n = ∑ αi Ωsi References i=1 ∞ f (z) = ∑ fj z j j=0 n αi 1 − zΩi i=1 =∑ n = Laplace transform of ∑ αi exp(φi x) i=1 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 37/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial Approximation theory [n − 1/n](z) = p(z)/q(z) Applications References n q(z) = ∏(1 − zΩi ) i=1 = zn (0) Hn (1/z) (0) ∣Hn ∣ = b0 z n + b1 z n−1 + ⋅ ⋅ ⋅ + bn−1 z + 1 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 38/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial (m+1−r ) Approximation theory → Hr Applications regular, r ≥ n ∞ References j ∑ fj x from rational function j=0 (r ) Hν singular, ν > n, r > m + 1 − ν ↓ (r ) Hn regular, r ≥ m + 1 − n 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 39/77 Sparse Interpolation Approximation theory Motivation f (z) + ε(z) meromorphic with poles in 0 ≤ ∣z∣ < R of total multiplicity n [de Montessus de Ballore, 1905] Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 ⇓ [m/n](z) → f (z) + ε(z) uniformly on compact sets excluding poles, with poles of f (z) + ε(z) attracting poles of [m/n](z) according to their multiplicity 11001 0 0 1 1 Universiteit Antwerpen 40/77 Sparse Interpolation Motivation Approximation theory f (z) + ε(z) analytic except for a countable number of poles [Nuttall, 1970] and essential singularities [Pommerenke, 1973] Basics: exponential Basics: polynomial ⇓ Approximation theory [m − 1/m](z) → f (z) + ε(z) in measure on compact sets, i.e. Applications References 101 011110111110001 1100 1011 10100 Λ2 ({z ∶ ∣f (z) + ε(z) − [m − 1/m](z)∣ ≥ τ }) → 0 11001 0 0 1 1 Universiteit Antwerpen 41/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial Approximation theory mathematical (noise free): Applications (0) 1. build Hν , References (0) 2. Hν ν = 0, 1, 2, . . . = UΣV T singular value decomposition ⎛σ1 3. Σ = ⎜ ⎝ ⋱ ⎞ ⎟, σν ⎠ σ1 ≥ σ2 ≥ ⋅ ⋅ ⋅ ≥ σn > σn+1 = ⋅ ⋅ ⋅ = σν = 0 4. find Ωi , φi , αi , i = 1, . . . , n 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 42/77 Sparse Interpolation Approximation theory Motivation Basics: exponential Basics: polynomial Approximation theory Applications numerical (with noise): 1. take ν large enough so that noise is clearly separated from n References (1) (0) 2. solve Hν vi = λi Hν vi , i = 1, . . . , ν, λi = Ωi , i = 1, . . . , n 3. find φi n 4. solve ∑ αi exp(φi xj ) = fj , 0 ≤ j ≤ 2ν − 1 i=1 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 43/77 Sparse Interpolation Example: noise Motivation Basics: exponential Basics: polynomial Approximation theory Applications φ1 = 0, α1 = 1, φ2 = −0.2 + 39.5i, α2 = 2, φ3 = −0.5 + 40i, α3 = 4, φ4 = −1, α4 = 8, References ∥ε(z)∥∞ = 10−2 , 101 011110111110001 1100 1011 10100 2π , 100 M = 100 xs = s uniform random noise 11001 0 0 1 1 Universiteit Antwerpen 44/77 Sparse Interpolation Example: noise Motivation Basics: exponential 102 Basics: polynomial Approximation theory 101 Applications References 100 10−1 10−2 10−3 1 2 3 4 (0) Figure: Singular values Hν 101 011110111110001 1100 1011 10100 5 6 with n = 4, ν = 6 11001 0 0 1 1 Universiteit Antwerpen 45/77 Sparse Interpolation Example: noise Motivation Basics: exponential 103 Basics: polynomial Approximation theory 102 Applications 101 References 100 10−1 10−2 10−3 10 20 30 (0) Figure: Singular values Hν 101 011110111110001 1100 1011 10100 40 50 with n = 4, ν = 50 11001 0 0 1 1 Universiteit Antwerpen 46/77 Sparse Interpolation Exercise: approximation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References φ1 = 1.5i, α1 = 10−3 , φ2 = 12.7i, α2 = 2, φ3 = −0.1 + 40i, α3 = 4, φ4 = −0.3 + 25.2i, α4 = 8, ∥ε(z)∥∞ = 2 × 10−3 , 101 011110111110001 1100 1011 10100 2π , 100 M = 100 xs = s uniform random noise 11001 0 0 1 1 Universiteit Antwerpen 47/77 format long; phi = [1.5*1i, 12.7*1i, -0.1+40*1i, -0.3+25.2*1i]; alpha = [10^(-3), 2, 4, 8]; eps = 2*10^(-3); M = 100; plot_signal pause plot_fft pause % synthesized input data with added noise N = input('Enter the dimension for SVD: '); % (10, 6, 3), (100, 100, 4) randn('seed',0); omega = 2*pi/M*(0:2*N-1); f = syn_exp(alpha, phi, omega); v = randn(size(f))+randn(size(f))*1i; vv = v/norm(v,Inf); f = f + eps*vv; % form Hankel matrices H0 and H1 from y sequence [H0,H1] = mat_ge(f); plot_svd pause % reconstruct the parameters via generalized eigenvalues n = input('Size of the model: '); % compute the generalized eigenvalues and form the Vandermonde system E = eig(H1(1:n,1:n),H0(1:n,1:n)); V = rot90(vander(E)); % amplitudes A = V\f(1:n).'; % frequencies and damping factors alpha_rec = A; phi_rec = log(E)*M/(2*pi); pause % extract the non-zero terms extract % plot computed parameters plot_recontructed_parameters 15 3.5 3 10 2.5 5 2 0 1.5 −5 1 −10 −15 0.5 0 pi pi/2 3pi/2 2pi 102 0 −50 −25 0 25 50 −25 0 25 50 −25 0 25 50 8 6 101 4 10 2 0 0 −50 10−1 0 −0.2 10−2 −0.4 10−3 2 4 6 8 10 −50 15 3.5 3 10 2.5 5 2 0 1.5 −5 1 −10 −15 0.5 0 pi pi/2 3pi/2 2pi 103 0 −50 −25 0 25 50 −25 0 25 50 −25 0 25 50 8 102 6 4 101 2 100 0 −50 10−1 0 10−2 −0.2 10−3 −0.4 10−4 20 40 60 80 100 −50 Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Applications 11001 0 0 1 1 Universiteit Antwerpen 48/77 Sparse Interpolation Applications Motivation Basics: exponential Exponential analysis in physical phenomena: Basics: polynomial ▸ power system transient detection Approximation theory ▸ motor fault diagnosis Applications ▸ drug clearance / glucose tolerance References ▸ magnetic resonance / infrared spectroscopy ▸ vibration analysis ▸ seismic data analysis ▸ music signal processing ▸ corrosion rate / crack initiation ▸ odour recognition with electronic nose ▸ typed keystroke recognition ▸ liquid (explosive) identification ▸ ... 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 49/77 Sparse Interpolation Transients Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Transient detection and characterization 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 50/77 Sparse Interpolation Transients Motivation Basics: exponential Basics: polynomial Approximation theory Applications short lived high frequency signal: References 101 011110111110001 1100 1011 10100 ▸ speech processing ▸ turbulent flow ▸ power lines ▸ ... 11001 0 0 1 1 Universiteit Antwerpen 51/77 Sparse Interpolation Transients Motivation Basics: exponential Basics: polynomial Approximation theory ▸ model with φi = 120πi, n Applications ∑ αi cos(120πx + γi )1[Ai ,Zi [ References i=1 101 011110111110001 1100 1011 10100 ▸ n = 3, αi = 1, γ1,3 = −π/2, γ2 = 3π/4 ▸ [A1 , Z1 [ = [0, 0.0308[ [A2 , Z2 [ = [0.0308, 0.0625[ [A3 , Z3 [ = [0.0625, 0.1058[ ▸ M = 1200 ▸ uniformly distributed noise in [−0.05, 0.05] 11001 0 0 1 1 Universiteit Antwerpen 52/77 Sparse Interpolation Transients Motivation Basics: exponential 1 Basics: polynomial Approximation theory Applications References 0 −1 0 37 75 127 Figure: Given transient signal 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 53/77 Sparse Interpolation Transients Motivation Basics: exponential Basics: polynomial ▸ at each instance: 2 exponential terms Approximation theory ▸ characteristics of terms change Applications ▸ inspect rank of References (1) H4 ⎛f 1 ⎜f =⎜2 ⎜f 3 ⎝f 4 f2 f3 f4 f5 f3 f4 f5 f6 f4 ⎞ f5 ⎟ ⎟ f6 ⎟ f7 ⎠ [A1 , Z1 [ = [0/M, 37/M[, [A2 , Z2 [ = [37/M, 75/M[, [A3 , Z3 [ = [75/M, 127/M[ 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 54/77 Sparse Interpolation Transients Motivation Basics: exponential 4 Basics: polynomial Approximation theory 3 singular values Applications References 2 1 0 31 37 69 75 (r ) Figure: Numerical rank of H4 101 011110111110001 1100 1011 10100 127 evolving over time x 11001 0 0 1 1 Universiteit Antwerpen 55/77 Sparse Interpolation Audio signals Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Reconstructing undersampled audio signals 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 56/77 Sparse Interpolation Audio signals Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 song containing 29 notes of 0.25 seconds each: ▸ M = 44100 (Hz) ▸ 11025 samples per note, 319725 in total ▸ 16.35 ≤ φi ≤ 4978.03, i = 1, . . . , 100 ▸ complex exponential model 11001 0 0 1 1 Universiteit Antwerpen 57/77 Sparse Interpolation Audio signals Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 0 0.05 0.1 0.15 0.2 0.25 Figure: Sampled signal produced by 1 note 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 58/77 Sparse Interpolation Audio signals compressive sensing (optimisation, probabilistic) Motivation Basics: exponential −5 4 −4 x 10 4 2 Basics: polynomial Approximation theory 0 0 −2 −2 −4 −4 References 1 2 3 4 5 6 7 1 2 3 sec 5 6 7 4 5 6 7 4 5 6 7 4 5 6 7 −4 x 10 2 0.5 x 10 1 0 0 −0.5 −1 −1 4 sec −5 1 Applications x 10 2 1 2 3 4 5 6 7 −2 1 2 3 sec sec Figure: 4 runs with 1229 samples 0.4 0.2 0.2 0.1 0 0 −0.1 −0.2 −0.2 1 2 3 4 5 6 7 −0.4 1 2 3 sec sec 0.5 1 0.5 0 0 −0.5 −1 −0.5 1 2 3 4 sec 101 011110111110001 1100 1011 10100 5 6 7 −1 1 2 3 sec 11001 0 0 1 1 Universiteit Antwerpen Figure: 4 runs with 456 samples 59/77 Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Audio signals 0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 0 0.05 0.1 0.15 0.2 0.25 Figure: Full set of samples per note 0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 0 101 0111101111100 1 0 010 0111 1101 0.05 0.1 0.15 0.2 0.25 0011001 0 0 1 1 Universiteit Antwerpen Figure: Sparse interpolation with 7 samples per note 60/77 Sparse Interpolation MCSA Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Preventive diagnosis of a broken rotor bar 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 61/77 Sparse Interpolation MCSA Motivation Basics: exponential Basics: polynomial Approximation theory 3-phase induction motors: ▸ consume 40 − 50% of all electricity in industrialized countries ▸ rotor made up metal bars ▸ stator current signal analysed ▸ broken bar(s) characterized by sideband frequencies ▸ difficult to diagnoze under low or no load Applications References 101 011110111110001 1100 1011 10100 Figure: Stator and rotor 11001 0 0 1 1 Universiteit Antwerpen 62/77 Sparse Interpolation MCSA Motivation Basics: exponential Service Factor is about once Basics: processes, this polynomial 1.15 (a) (a) Approximation theory Applications References (b) Fig. 8. Medium-voltage induction motor with one broken rotor bar. (a) The medium-voltage test motor. (b) The rotor cage with one rotor bar removed. (a) monitoring interval can be changed to as fast as onceFFT every Figure: Stator current 120 seconds when appropriate. This helps safeguard the motor against potential failures on its rotor cage. V. EXPERIMENTAL RESULTS To test the reliability of the fault severity evaluation algorithm on large medium-voltage motors, a 500-hp 4.16kV induction motor is tested under both healthy and faulty Universiteit conditions. Experimental setup and test results are reported Antwerpen in this section. 101 011110111110001 1100 1011 10100 11001 0 0 1 1 (b) Fig. 9. Stator current spectrum. (a) Healthy (b) Motor with one spectra: healthy and with 1 motor. broken bar broken bar. To emulate a broken rotor bar failure, a small part of the rotor bar is completely removed. To avoid the subsequent mechanical unbalance and resultant torque pulsation, the rotor bar is then glued to a close rotor bar. Fig. 8 shows the field photos of the medium-voltage motor with the created one broken rotor bar failure. B. Experimental Results 63/77 Sparse Interpolation MCSA Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: 10% load, 16dB noise, ν = 400 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 64/77 Sparse Interpolation Bio-electrical Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Sparse EEG approximation 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 65/77 Sparse Interpolation Bio-electrical Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Bio-electrical signals: ▸ electrical activity of cells and tissues ▸ clinical studies of health status ▸ ECG, EEG, EMG, EOG, . . . ▸ sparse model (n = 8) is approximate 11001 0 0 1 1 Universiteit Antwerpen 66/77 Sparse Interpolation Bio-electrical Motivation Basics: exponential 40 original reconstructed Basics: polynomial Approximation theory Applications 30 20 References 10 0 −10 −20 0 1 2 3 4 time (s) 5 6 7 8 Figure: Reconstruction of 8 second [1 − 20] Hz bandpass filtered EEG 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 67/77 Sparse Interpolation Bio-electrical Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Sparse EOG approximation 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 68/77 Sparse Interpolation Bio-electrical Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Polysomnogram: ▸ 12 channels ▸ 22 wire attachments to patient ▸ heart rate, leg measurement, airflow (chest, abdomen), chin muscle, EEG, EOG, . . . 11001 0 0 1 1 Universiteit Antwerpen 69/77 Sparse Interpolation Bio-electrical Motivation Basics: exponential 150 original reconstructed Basics: polynomial Approximation theory 100 Applications References 50 0 −50 −100 0 1 2 3 4 time (s) 5 6 7 8 Figure: Reconstruction of 8 second EOG (CC = 99.2%) 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 70/77 Sparse Interpolation Spectroscopy Motivation Basics: exponential Basics: polynomial Approximation theory Applications References Figure: Spectral analysis of FID 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 71/77 Sparse Interpolation Spectroscopy Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 Magnetic resonance spectroscopy: ▸ physical and chemical properties of molecules ▸ a.o. concentration of metabolites in the brain ▸ frequencies clustered → high frequence resolution ▸ free induction decay → time constraint ▸ Fourier methods need additional tools 11001 0 0 1 1 Universiteit Antwerpen 72/77 Sparse Interpolation Spectroscopy Motivation Basics: exponential Basics: polynomial Approximation theory Applications References φ(x) = 5 × 10−2 + 2e (−0.97+i79.94π)x + 4e (−1+i80π)x + 8e −1.1x + ε(x) ∥ε(x)∥∞ = 10−3 , circular Gaussian noise Figure: The real (left) and imaginary (right) part of φ(x) 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 73/77 Sparse Interpolation Spectroscopy Motivation Basics: exponential Basics: polynomial 6 Approximation theory Applications 4 References 2 0 -100 pi -50 pi 0 50 pi 100 pi Figure: Real (black) and imaginary (red) parts of FFT 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen 74/77 Sparse Interpolation Spectroscopy Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 8 6 4 2 0 -100 pi -50 pi 0 50 pi 100 pi -50 pi 0 50 pi 100 pi 0 −0.5 −1 -100 pi Figure: Amplitudes (top) and damping factors (bottom) of φ(x) 101 0111101111100 1 0 010 0111 1101 0011001 0 0 1 1 Universiteit Antwerpen 75/77 Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 References 11001 0 0 1 1 Universiteit Antwerpen 76/77 Sparse Interpolation Motivation Basics: exponential Basics: polynomial Approximation theory Applications References 101 011110111110001 1100 1011 10100 11001 0 0 1 1 Universiteit Antwerpen References M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. In STOC ’88: Proceedings of the twentieth annual ACM symposium on Theory of computing, pages 301–309, New York, NY, USA, 1988. ACM. http://dx.doi.org/10.1145/62212.62241. M. de Montessus de Ballore. Sur les fractions continues algébriques. Rend. Circ. Mat. Palermo, 19:185–257, 1905. http://dx.doi.org/10.1007/BF03014011. R. de Prony. Essai expérimental et analytique sur les lois de la dilatabilité des fluides élastiques et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures. J. Ec. Poly., 1:24–76, 1795. M. Giesbrecht, G. Labahn, and W. Lee. Symbolic-numeric sparse interpolation of multivariate polynomials. In ISSAC’06, 2006. http://dx.doi.org/10.1145/1145768.1145792. 9-12 July. P. Henrici. Applied and computational complex analysis I. John Wiley & Sons, New York, 1974. Y. Hua and T. K. Sarkar. Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Signal Process., 38: 814–824, 1990. http://dx.doi.org/10.1109/29.56027. E. Kaltofen and W.-s. Lee. Early termination in sparse interpolation algorithms. J. Symbolic Comput., 36(3-4):365–400, 2003. http://dx.doi.org/10.1016/S0747-7171(03)00088-9. International Symposium on Symbolic and Algebraic Computation (ISSAC’2002) (Lille). J. L. Massey. Shift-register synthesis and BCH decoding. IEEE Trans. Inform. Theory, 15 (1):122–127, 1969. http://dx.doi.org/10.1109/TIT.1969.1054260. J. Nuttall. The convergence of Padé approximants of meromorphic functions. J. Math. Anal. Appl., 31:147–153, 1970. http://dx.doi.org/10.1016/0022-247X(70)90126-5. H. Padé. Sur la représentation approchée d’une fonction par des fractions rationnelles. PhD thesis, Faculté des sciences de Paris, 1892. C. Pommerenke. Padé approximants and convergence in capacity. J. Math. Anal. 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