Semi-definite Programming Techniques for Quadratic Inverse Eigenvalue Problem

Transcription

1 Semi-definite Programming Techniques for Quadratic Inverse Eigenvalue Problem Moody T. Chu North Carolina State University March 31, National Chiao Tung University

2 Outline Basic Ideas General Quadratic Model Constraints and Fundamental Questions Some Applications Brief History Model Updating Spectrum Modification Challenges Structure Preservation Symmetry Positive Semi-definiteness Other Difficulties Semi-definite Programming Quick Overview Solvability Test Model Updating Conclusion

3 Outline Basic Ideas General Quadratic Model Constraints and Fundamental Questions Some Applications Brief History Model Updating Spectrum Modification Challenges Structure Preservation Symmetry Positive Semi-definiteness Other Difficulties Semi-definite Programming Quick Overview Solvability Test Model Updating Conclusion

4 Outline Basic Ideas General Quadratic Model Constraints and Fundamental Questions Some Applications Brief History Model Updating Spectrum Modification Challenges Structure Preservation Symmetry Positive Semi-definiteness Other Difficulties Semi-definite Programming Quick Overview Solvability Test Model Updating Conclusion

5 Outline Basic Ideas General Quadratic Model Constraints and Fundamental Questions Some Applications Brief History Model Updating Spectrum Modification Challenges Structure Preservation Symmetry Positive Semi-definiteness Other Difficulties Semi-definite Programming Quick Overview Solvability Test Model Updating Conclusion

6 Outline Basic Ideas General Quadratic Model Constraints and Fundamental Questions Some Applications Brief History Model Updating Spectrum Modification Challenges Structure Preservation Symmetry Positive Semi-definiteness Other Difficulties Semi-definite Programming Quick Overview Solvability Test Model Updating Conclusion

7 Mass-Spring Vibration Equation of motion: m m m m ẍ+ c 1 + c 2 0 c c 2 0 c 2 + c 3 c c 3 c ẋ+ k 1 + k 2 + k 5 k 2 k 5 0 k 2 k 2 + k 3 k 3 0 k 5 k 3 k 3 + k 4 + k 5 k k 4 k x = f(t).

8 Second-Order Differential System Dynamical system with n-degree-of-freedom: Mẍ + (C + G)ẋ + (K + N)x = F. Some interpretations: M := Mass matrix M = M 0. C := Damping matrix C = C. K := Stiffness matrix K = K 0. G := Gyroscopic matrix G = G. N := Circulatory matrix N = N. F := External force.

9 Integrated Circuit Ohm s Law: V = RI = L di dt = 1 C I dt. Kirchhoff s Law: The algebraic sum of the current at any junction must be zero. The algebraic sum of voltage drops around any closed loop must be zero. Very large scale integration.

10 Quadratic Eigenvalue Problem Assume the homogeneous solution x(t): x = e λt u. Look for nontrivial solution to the QEP: Q(λ)u := (λ 2 M + λc + K )u = 0. If M is nonsingular, then there are 2n eigenpairs. Many applications. Many numerical techniques.

11 Direct Problem versus Inverse Problem Direct Problem: Given physical parameters, e.g., mass, length, elasticity, etc., analyze and derive the spectral information and, hence, induce the dynamical behavior. Inverse Problem: Validate, determine, or estimate the parameters of the system according to its observed or expected behavior.

12 A Simplified Interpretation Direct problem Express the behavior in terms of the parameters. Inverse problem Express the parameters in term of the behavior.

13 Structured Problem Be it from finite element discretization or from real physical configuration, the coefficient matrices M, C and K are often structured. The structure of symmetry and positive definiteness in the mass-spring system. The structure of asymmetry and connectivity in the RCL model. M = C = K = L 2 L L 2 L L L , 0 R 2 R 2 0 R 1 + R R 4 0 R 2 R 2 + R 3 0 R R C C3 C C C3 C , How to solve the structured problem?

14 Partial Spectral Information In a large and complicated physical system, it is often impossible to obtain the entire spectral information. Quantities related to high frequency terms generally are susceptible to measurement errors due to the finite bandwidth of measuring devices. Only partial spectral information are of interest and observable. How much eigeninformation is enough?

15 Solvability: Fundamental Questions Determine a necessary or a sufficient condition under which an IEP has a solution. Computability: Develop a scheme through which, knowing a priori that the given spectral data are feasible, the solution to an IEP can be constructed numerically. Sensitivity: Quantify how a solution to an IEP is subject to changes of the spectral data. Applicability: Distinguish whether the given data are exact or approximate, complete or incomplete, and whether only an estimation of the parameters of the system is sufficient. Decide between physical realizability and physical uncertainty which constraint of the problem should be enforced.

16 Vibrating String m 2 m 3 F m 1 x 1 x 2 x 3 m 4 x 4 F h Discretization: Equally spaced particles (with spacing h and mass m i ) on a string. Subject to a constant horizontal tension F.

17 Equation of motion for 4 particles: d 2 x 1 m 1 dt 2 = F x 1 h + F x 2 x 1 h d 2 x 2 m 2 dt 2 = F x 2 x 1 + F x 3 x 2 h h d 2 x 3 m 3 dt 2 = F x 3 x 2 + F x 4 x 3 h h d 2 x 4 m 4 dt 2 = F x 4 x 3 h In matrix form: d 2 x dt 2 = DAx x = [x 2 1, x 2, x 3, x 4 ] T A = D = diag(d 1, d 2, d 3, d 4 ) with d i = F m i h. F x 4 h

18 An MIEP Eigenvalues of DA: Necessarily real and nonnegative. Are the squares of the so called natural frequencies of the system. Inverse problem: Want to place the weight m i so that the system has a prescribed set of natural frequencies. A is symmetric and tridiagonal. D is diagonal. This is a multiplicative inverse eigenvalue problem. What is the range of possible natural frequencies? Can the system have arbitrary natural frequencies?

19 Control Theory ẋ(t) = Ax(t) + Bu(t). x(t) R n denotes the state of a certain physical system. u(t) R m denotes a control to the system. Want to select the input u(t) so that the dynamics of the resulting x(t) is driven into a certain desired state.

20 Feedback Control and Pole Assignment Problem State feedback. u(t) = }{{} F x(t) = ẋ(t) = (A + BF)x(t). gain matrix Output feedback. y(t) = }{{} C x(t) known u(t) = }{{} K y(t) output matrix = ẋ(t) = (A + BKC)x(t). Choose F or K so that the matrices A + BF or A + BKC has a prescribed set of eigenvalues.

21 Stochastic Inverse Spectrum Problem Construct a Markov chain with prescribed spectrum. Row-stochastic structure. No strings of symmetry. Eigenvalues can appear in complex conjugate pairs. A hard problem (Karpelevič 51, Minc 88). Need to know how the eigenvalues of n n stochastic matrices are distributed in C. The set Θ n consisting eigenvalues of n n stochastic matrices is completely characterized. The Karpelevič theorem characterizes only one complex value a time. When will two or more points in Θ n belong to the spectrum of the same stochastic matrix?

22 Basic Ideas Model Updating Structure Preservation Semi-definite Programming Conclusion Karpelevič s Theorem A number λ is an eigenvalue for a stochastic matrix if and only if it belongs to a region Θ n such as the one shown below for n =

23 Applications in Other Disciplines Applied Mechanics and structure design Construct a model of a mass-spring system with prescribed natural frequencies/modes. Applied physics Quantum mechanics. Geophysics. Numerical analysis Computing B-stable RK methods have real poles. Quadratures. Mathematical analysis Inverse Sturm-Liouville problems.

24 Brief History Inverse Sturm-Liouville problem: Ambartsumyan 29 Krein 33 Borg 46, Levinson 49 Gel fand&levitan 51 Kac 66 (Can one hear the shape of a drum?) Hochstadt 73, Barcilon 74, McLaughlin 76, Hald 78 Zayed 82, Issacson et al 83, McLaughlin 86, Andersson 88 Lowe et al 95, Rundell 97 Matrix theory: Downing&Householder 56, Mirsky 58 Hochstadt 67 de Oliveira 70, Hald 72, Golub 73, Friedland 77, de Boor&Golub 78 Biegler-König 81, Shapiro 83, Barcilon 86, Sun 86, Boley&Golub 87 Landau 94, Chu 98 (A list of 39 different types of IEPs.)

25 Applied Mechanics: Barcilon 74 Gottlieb 83, Gladwell 86 Ram 91, Gladwell 96, Nylen&Uhlig 97 Computation: Morel 76, Boley&Golud 78 Nocedal et al 83, Friedland et al 88, Laurie 88 Chu 90, Zhou&Dai 91, Trench 97, Xu 98 Chinese Scholars: Lots of contributions. Particularly strong in solvability and sensitivity analysis. Mostly published in Chinese, not accessible by the West. Quadratic Inverse Eigenvalue Problems: Real challenge. Research barely begins.

26 Establishing an Effective Model Precise mathematical models of physical systems are rarely available. Disturbance to the measurement. Imperfect characterization of the natural phenomenon. Discretization error.. Necessary to update a primitive model to attain consistency with empirical results.

27 The Updating Problem Given A structured quadratic pencil (M 0, C 0, K 0 ), A few of its associated eigenpairs {(λ i, u i )} k i=1 with k < 2n, Newly measured eigenpairs {(σ i, y i )} k i=1. Update (M 0, C 0, K 0 ) to (M, C, K ) so that The same structure is maintained, The subset {(λ i, u i )} k i=1 is replaced by {(σ i, y i )} k i=1 as k eigenpairs of (M, C, K ). Must there be some other relationship between the pencils (M, C, K ) and (M 0, C 0, K 0 )?

28 Structure Preserving The structure imposed on the model depends inherently on the underlying physical system. Physical feasibility requires that structure be preserved. Some examples: M is diagonal and positive; C and K are symmetric and banded; K is positive semi-definite. The inner-connectivity of an electronic circuit. The structure varies from problem to problem. The analysis of a solution varies from problem to problem? Would prefer to see a unified theory.

29 Spurious Eigenstructure Vibration parameters not related to the newly measured parameters should remain invariant. These parameters have already been proved to be acceptable in the original model. Do not wish to introduce new vibrations via updating. Simply do not know of any information about these parameters. No spill-over is desirable. Update (M 0, C 0, K 0 ) to (M, C, K ) so that The same structure is maintained, The subset {(λ i, u i )} k i=1 is replaced by {(σ i, y i )} k i=1 as k eigenpairs of (M, C, K ). The remaining 2n k eigenpairs, unmeasurable and unknown, are the same as those of the original (M 0, C 0, K 0 ). Can the no spill-over phenomenon be achieved?

30 Minimal or Robust Changes The solution to the updating problem is not unique. Wish to optimize the adjustment or robustness. minimize (M,C,K ) subject to M M 0 2 F + C C 0 2 F + K K 0 2 F structural and spectral constraints. Need to know the representation of a solution before we can search for the optimal solution. Would be difficult to parameterize the feasible set. No longer needed with SDP techniques.

31 Setup Describe the partial eigeninformation via (Λ, X) R k k R n k. Λ = diag{λ [2] 1,..., λ[2] l, λ 2l+1,..., λ k }, X = [x 1R, x 1I,..., x lr, x li, x 2l+1,..., x k ].» λ [2] αj β j = j R 2 2, β β j α j 0, for j = 1,..., l, j All given eigenvalues are simple. Look for real and structured matrices M, C and K so that the equation MXΛ 2 + CXΛ + KX = 0, (1) is satisfied. What structure and how many eigenpairs are needed?

32 Maintaining Symmetry (k n ) Assume the QR decomposition X = Q [ R 0 ]. Write Define S := RΛR 1. (Λ, X) (S, R). Q MQ = (This is a fixed matrix.) (Same eigeninformation.) [ ] [ ] [ ] M11 M 12, Q C11 C CQ = 12, Q K11 K KQ = 12. M 21 M 22 C 21 C 22 K 21 K 22 Look for quadratic pencils so that M 11 RΛ 2 + C 11 RΛ + K 11 R = 0, (a square pencil,) M 21 RΛ 2 + C 21 RΛ + K 21 R = 0, (a rectangular pencil.)

33 M = arbitrary SPD. Parametric Solution (Kuo-Lin-Xu 2005) M 11 = arbitrary SPD. (k(k + 1)/2) 2 M 21 = arbitrary = M 12. k(n k) M 22 = M 21 M 1 11 M 12 + arbitrary SPD. ((n k)(n k + 1)/2) 2 C 11 = (M 11 S + S M 11 + R ΩR 1 ). C 21 = arbitrary = C 12. k(n k) C 22 = arbitrary symmetric. ((n k)(n k + 1)/2) 2 K 11 = S M 11 S + R ΩΛR 1. K 21 = K 12 = (M 21S 2 + C 21 S). K 22 = arbitrary symmetric. ((n k)(n k + 1)/2) 2 Ω = Additional free parameters. k {[ ] [ ] } ξ1 η Ω := diag 1 ξl η,..., l, ξ η 1 ξ 1 η l ξ 2l+1,..., ξ k. l

34 Counting Freedom Symmetric M, C and K = 3 2n(n + 1) unknowns. Given k eigenpairs = nk equations. The system will be under-determined and, hence, solvable, so long as k < 3 2 (n + 1) eigenpairs are specified. If k n, Free choice of M, C 21, C 22, K 22 and Ω = 3 n(n + 1) nk free 2 parameters in the general solution.»» M11 M 12 C11 C, 12 M 21 M 22 C 21 C 22» K11 K, 12 K 21 K 22 What are solutions when k > n? Cannot guarantee positive semi-definiteness of C or K. Cannot deal with gyroscopic or other structured systems..

35 Maximal Allowable Eigenpairs (Chu-Datta-Lin-Xu 2008) Provides an upper bound of the allowable eigenpairs, { 3l + 1, if n = 2l, k max = 3l + 2, if n = 2l + 1. Parametric representation of all solutions up to k = k max. Shows that spill-over for damped system is generally unpreventable. No guarantee of positive semi-definiteness or other structures.

36 Maintain Positive Semi-definiteness Can do QIEP for M 0, C 0, and K 0, with complete information of eigenvalues and eigenvectors (Lancaster-Prells 2005). with all eigenvalues and eigenvectors corresponding to real eigenvalues (Lancaster 2007). Can do QIEP for M 0, and K 0, but no C, with partially prescribed eigenpairs (Chu-Kuo-Lin 2004). Cannot maintain SPD structure with fine grains of connectivity.

37 Eigenstructure Completion (Dong-Lin-Chu 2008) Spectral decomposition (Gohbert-Lancaster-Rodman 1980, Chu-Xu 2008) M = (XJH 1 X ) 1, C = MXJ 2 H 1 X M, K = MXJ 3 H 1 X M + CM 1 C. (J, X) = a standard pair eigeninformation. A particular nonsingular H. XH 1 X = 0. H J = (H J). H = H. Block diagonal with at most 2 2 blocks If all eigenvalues of J have negative real part, and if XJ 2 H 1 X 0, then generically M 0, K 0, and C 0.

38 Other Difficulties Most theory and algorithms are structure dependent. Unable to address finer grain of inner-connectivity. Unable to satisfy the requisite nonnegativity. Unable to characterize mixed-type structure. Unable to tackle local" model updating. Would like to have a unified theory and/or robust algorithms. Literature moves in diverse directions.

39 What is SDP? Primal problem: Minimize X C X Subject to A i X = b i, i = 1,..., m X 0, C, A 1,..., A m = given symmetric matrices. = Frobenius inner product. Dual problem: Maximize y,s Subject to b y m i=1 y ia i + S = C, S 0,

40 Powerful Mathematical Programming Tool SDP is versatile in modeling problems arising in broad discipline areas. Combinatorial optimization, Min-max eigenvalue problems,... Controller synthesis, CAD design, Network queueing, QIEPs. SDP has a considerable similarity to the linear programming. Vector of variables in LP Symmetric matrices. Nonnegativity constraints Positive semi-definite constraints. Duality theory, algorithms, convergence and polynomial time-complexity for LP can be extended mechanically" to SDP.

41 Research Activities Profusion of research results: The book, Handbook of Semi-definite Programming, has 877 references. Online bibliography has 1333 references, available at http: //liinwww.ira.uka.de/bbliography/math/psd.html Many well developed MATLAB-based software packages. SeDuMe (Sturm), downloadable at SDPT3 (Toh-Tütüncü-Todd), downloadable at Useful interface. YALMIP (Löfberb), downloadable at

42 Solvability Test % Define symmetric variables sm = sdpvar(n,n); sc = sdpvar(n,n); sk = sdpvar(n,n); % Specify equality and definiteness constraints F = set(sm*x*lambda^2+sc*x*lambda+sk*x == zeros(n,k)); F = F + set(sm >= 0) + set(sc >= 0) + set(sk >= 0); % Select SDPT3 as the solver ops = sdpsettings( solver, sdpt3 );

43 % Invoke SDP solver solvesdp(f,[],ops); % Retrieve numerical solution M = double(sm) C = double(sc) K = double(sk) % Check feasibility checkset(f);

44 Make-or-break Calculation For solvability test, no objective function is needed. The feasibility set is convex, containing at least the trivial solution. An appropriate normalization is needed to assess the final construction. SPD algorithm is simple but efficient. Can be exploited to answer many theoretical questions. How many arbitrarily prescribed eigenstructure (all eigenvalues must have negative real part) can be completed for a QIEP with positive semi-definite coefficient matrices?

45 Random Eigenstructure Completion (n = 10) k c k p k n k t

46 Handling Gyroscopic Constraints : % Define skew-symmetric variables sg = sdpvar{n,n, skew, real ); sn = sdpvar(n,n, skew, real ); % Modify equality constraints F = set(sm*x*lambda^2+(sc+sg)*x*lambda+(sk+sn)*x ==... zeros(n,k)); :

47 Versatility and Generality Any of the skew-symmetric G or N adds n(n 1) 2 extra variables. Would increase solvability by allowing a larger number k of prescribed eigenpairs. Other types of coefficients, such as Toeplitz, Hankel, palindromic, sparsity pattern, or fixed entries can easily be incorporated into the SDP formulations for QIEPs. Being linear in coefficient matrices, the SDP approach can be used for matrix polynomials of general degrees.

48 Connectivity and Nonnegativity Constraints % Define physical parameters sm = sdpvar(4,1); sc = sdpvar(3,1); sk = sdpvar(5,1); % Define connectivity constraints sm = diag(sm); sc = [sc(1)+sc(2) 0 -sc(2) 0; ; -sc(2) 0 sc(2)+sc(3) -sc(3); 0 0 -sc(3) sc(3)]; sk = [sk(1)+sk(2)+sk(5) -sk(2) -sk(5) 0; -sk(2) sk(2)+sk(3) -sk(3) 0; -sk(5) -sk(3) sk(3)+sk(4)+sk(5) -sk(4); 0 0 -sk(4) sk(4)]

49 % Define equality and nonnegativity constraints F = set(sm*x*lambda^2+sc*x*lambda+sk*x == zeros(n,k)); F = F + set(sm > 0) + set(sc > 0) + set(sk > 0); % Normalize the first mass (or others) solvesdp(f,(sm(1,1)-1)^2,ops);

50 Convex Programming The SDP formulation is a convex programming so long as the artificial objective function is convex. The SDP computation tells Either a completely successful reconstruction" with positive physical parameters, or An utterly disastrous failure" giving rise to either the trivial solution or a degenerated system.

51 Model Updating % Define variables/constraints, structured if needed sdpvar um uc uk : F =... % Include rotated Lorentz cones F = F + set(rcone(reshape((m_0-sm),n^2,1),um,1/2)); F = F + set(rcone(reshape((c_0-sc),n^2,1),uc,1/2)); F = F + set(rcone(reshape((k_0-sk),n^2,1),uk,1/2)); % Define objective function and call for SOCP solvers sdpsolve(f,um + uc + uk,ops);

52 Harwell-Boeing BCSST*01 Data Generalized eigenvalue problem of size (λm 0 K 0 )x = 0, Has 24 positive eigenvalues and the remaining 24 at positive infinity. Improperly scaling: M 0 F and K 0 F Eigenvalue computation would be particularly challenging. (λm 0 K 0 )x 2 of the 24 real-valued eigenvalues and eigenvectors computed by MATLAB ranges from to

53 Sparsity Pattern of BCSST*01 Data 0 M 0 0 K nz = nz = 400

54 Updating Procedures Update (λ, x) to (µ, x). µ stands for a one-sided" random perturbation of eigenvalue λ. with σ = 10 p, p = 1,, 10. µ = λ(1 + σ randn(1) ) Two steps involved in the updating procedure: (µ, x) feasibility (M, K ) optimization (M 0, K 0 ). Measure feasibility by (µm K )x 2. Search for minimal update.

55 Performance of Updating BCSST*01 Data Residules over 20 samples of μ Objective values over 20 samples of μ 4 2 log 10 ( (μm K)x 2) log 10 (J (M,K)) p p

56 Conclusion The inverse eigenvalue problem is a diverse area full of research interests and activities. The inverse problem, expressing the parameters in term of the behavior, is just as important as the direct problem in applications. (The AMS has recently added a new number 65F18 for inverse eigenvalue problem to its Subject Classifications.) Model updating can be embedded as a quadratic inverse eigenvalue problem. The general solution to the inverse eigenvalue problem for adjoint quadratic pencils with up to 3 2 (n + 1) given eigenpairs is completely known. Structured problems are more complicated (and needed!) SDP techniques offers a unified and effectual avenue of attack on the QIEPs in general and the MUPs in particular.