Fndng he Rghtmost Egenvalues of Large Sparse Non-Symmetrc Parameterzed Egenvalue Problem Mnghao Wu AMSC Program mwu@math.umd.edu Advsor: Professor Howard Elman Department of Computer Scences elman@cs.umd.edu
Introducton Consder the egenvalue problem A S x λb S x () where AS and BS are large sparse non-symmetrc real N N matrces and S s a set of parameters gven by the underlyng partal dfferental equaton (PDE). I am prmarly nterested n computng the rghtmost egenvalues (namely, egenvalues of the largest real parts) of (). he motvaton les n the stablty analyss of steady state solutons of systems of PDEs of the form: B du dt N N N ( u) f R R u R f, : ()
Introducton B du dt N N N ( u) f R R u R f, Defne the Jacoban matrx for the steady state : () * * u by A f / u ( u ), then * u s stable f the egenvalues of () all have negatve real parts. ypcally, f arses from the spatal dscretzaton of a PDE. When fnte dfferences are used to dscretze a PDE, then often B I and () s called a standard egenproblem. If the equatons are dscretzed by fnte elements, B I and () s called a generalzed egenvalue problem. 3
Introducton Besde the numercal algorthm of computng rghtmost egenvalues, another nterestng problem s how the stablty of steady state soluton wll change as the parameters vary. Example: Olmstead Model (Nonlnear Dffuson Equaton) Raylegh Number Fg. steady state soluton s stable Fg. steady state soluton s NO stable 4
Major Computatonal Dffcultes A and B are large, sparse and non-symmetrc Iteratve methods must be used. Iteratve methods cannot solve generalzed egenvalue problems Ax λ Bx z Matrx transformaton θ s necessary. Rghtmost egenvalues are often hard to fnd by egensolvers Matrx transformaton must map rghtmost egenvalues to wellseparated extremal egenvalues. Rghtmost egenvalues can be complex Complex arthmetc should be consdered. z 5
Methodology Egensolver Matrx ransformaton 6
Egensolver: Arnold ype Algorthm he standard Arnold Algorthm: k K ( A v ) span{ v, Av, A v, K, A }. () ( ) It bulds the Krylov subspace procedure Normalze (.) (.) {( λ, x ),, K, k; V, H } Arnold( A, v, k ) v for to k do :. Form w Form h j, K,. (.3) Compute w w h v. ( QR Gram Schmdt ) (.4) Form h +, w. (.5) Form v w / h. ( Normalzat on) + k () 3 Let H [ h ] where h for j > + and V [ v, K, v ]. k j, j Av v ( 4) Compute the egenpars ( λ, z ) () 5 Compute the approxmate j H j. w, k +, j k j j,, K, k of egenvectors j k x, v V H k k by the QR method z,, K, k. k k 7
Egensolver: Arnold ype Algorthm he Convergence Behavor of the Arnold Algorthm he Arnold algorthm converges to well-separated extremal egenvalues. Fg 3. Equally-spaced Egenvalues Fg 4. Well Separated Egenvalues 8
Egensolver: Arnold ype Algorthm he Convergence Behavor of the Arnold Algorthm heorem. Consder λ and θ ( ) λ, K, N smple and suppose that there s a ζ such that θ ζ > θ ζ L θ N ζ. hen λ ˆ λ θ ζ c θ ζ k. 9
Matrx ransformaton: Cayley ransformaton Motvaton of Matrx ransformaton - teratve method cannot solve generalzed egenvalue problems - egensolvers prefer well-separated extremal egenvalues Shft-nvert ransformaton SI ( A, B; σ ) ( A σb) λ θ λ σ B σ R he generalzed egenvalue problem egenvalue problem SI Ax λbx becomes a standard x θx. σ s called the shft.
Matrx ransformaton: Cayley ransformaton Cayley ransformaton ( ) ( ) ( ) ( ) shft ant shft B A B A I B A SI C + : :, ;, τ σ σ λ τ λ θ λ τ σ τ σ τ σ ( ) ( ) ( ) + τ σ λ λ τ σ Re :, L
Implementaton All the algorthms wll be coded n Matlab Avalable software packages (also wrtten n Matlab) - IFISS ( Incompressble Flud Iteratve Soluton Software ) - Implctly Restarted Arnold Code
3 est and Valdaton All test problems have been solved n the lterature. Stage : est the standard Arnold algorthm code, Shft-nvert and Cayley transformaton code est problem : Olmstead model (Nonlnear Dffuson Equaton) ( ) ( ) ( ) ( ) ( ) + + 3 π π u v u u v u t v b u Ru X u C X v C t u R: Raylegh Number
4 est and Valdaton (Stage ) est problem : ubular Reactor Model ( ) ( ) ( ) ( ) ( ) ( ) ( ) () () + ', ' ', ' exp exp y Pe y Pe y BDy X X Pe t Dy X y X y Pe t y h m h m γ γ β γ γ D: Damköhler Number
5 est and Valdaton Stage : Modfy and test the Implctly Restarted Arnold (IRA) code est problem: x M x C C K λ m n N R C R M K m n n n +, Snce matrx B s sngular, standard Arnold algorthm wll produce spurous egenvalues. IRA should be able to solve ths problem.
est and Valdaton Stage 3: Solve the two-dmensonal double dffusve convecton equaton 6
Project Schedule Before November: solve the Olmstead Model; explore the effect of Raylegh number November: solve the ubular Reactor Model; explore the effect of Damköhler Number December: modfy and test IRA code; wrte mdterm report; gve md-term presentaton January to March: Solve the wo-dmensonal Double Dffusve Convecton problem Aprl: wrte fnal report May: wrte fnal report; gve fnal presentaton 7
References Meerbergen, K & Roose, D 996 Matrx transformaton for computng rghtmost egenvalues of large sparse non-symmetrc egenvalue problems. SIAM J. Numer. Anal. 6, 97-346 Meergergen, K & Spence, A 997 Implctly restarted Arnold wth purfcaton for the shft-nvert transformaton. Math. Comput. 66, 667-698. Olmstead, W. E., Davs, W. E., Rosenblat, S. H., & Kath, W. L. 986 Bfurcaton wth memory. SIAM J. Appl. Math. 4, 7-88. Henemann, R. F., & Poore, A. B. 98 Multplcty, stablty, and osllatory dynamcs of a tubular reactor. Chem. Eng. Sc. 36, 4-49 Stewart, G. W. Matrx algorthms, volume II: egensystems. SIAM 8