An Algebraic Convergence Theory for Primal and Dual Substructuring Methods by Constraints Jan Mandel, University of Colorado joint work with Clark R. Dohrmann, Sandia National Laboratories Radek Tezaur, University of Colorado TU Ostrava, June 8, 2004 Supported by: National Science Foundation Grant DMS-00742789, Sandia National Laboratiories, and Office of Naval Research Grant N-00014-01-1-0356. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000 .
The methods • FETI-DP (Farhat, Lesoinne, Pierson 2000): substructuring method of the FETI family (Farhat, Roux, 1992) , based on Lagrange Multipliers. Convergence proof Mandel and Tezaur (2001). • BDDC (Dohrmann 2002): method from the Balancing Domain Decomposition family (Mandel 1993), based on Additive Schwarz framework of Neumann Neumann type (Dryja, Widlund 1995); convergence analysis Mandel and Dohrmann 2002. Case of corner constraints only and b 0 = b is identical to a Balancing Domain Decomposition variant by Le Tallec, Mandel, Vidrascu 1998 • both methods for SPD problems from structural mechanics, implemented in the SALINAS code at Sandia, massively parallel
Highlights • Continue algebraic analysis as long as possible before switching to calculus (FEM, functional analysis) arguments, abstract from a specific FEM framework to widen the scope of the methods • Algebraic bounds on the condition number to select components of the methods adaptively (current work with Bedˇ rich Soused´ ık, vistor from CTU Prague)
Components of the methods Substructuring and reduction to interfaces Enforcing intersubdomain continuity FETI − DP ⇐ = = ⇒ BDDC Intersubdomain averaging and weight matrices Augmented constraints and coarse dofs Both the BDDC and FETI-DP methods are build from similar components. For a comparison, all components of both methods need to be identical. The methods then use basically different algebraic algorithms.
Substructuring and reduction to interfaces K i is the stiffness matrix for substructure i , symmetric positive problem in decomposed form v 1 K 1 1 . ... 2 v T Kv − v T f → min , . v = . K = v N K N + continuity of dofs between substructures partition the dofs in each subdomain i into internal and interface (boundary) and eliminate interior dofs � K ii � � � � � K ib v i f i i i i i K i = v i = f i = , , . T v b f b K ib K bb i i i i = ⇒ decomposed problem reduced to interfaces 1 T K ii − 1 2 w T Sw − w T g → min , S i = K bb i − K ib K ib S = diag( S i ) , i i i +continuity of dofs between substructures
Enforcing intersubdomain continuity dual methods (FETI): continuity of dofs between substructures: Bw = 0 � � 1 − 1 0 . . . B = [ B 1 , . . . , B N ] = : W → Λ . . . . . . . . . . . . primal methods (BDD,...): U is the space of global dofs R 1 . . R i : U → W i restriction to substructure i , R = . R N continuity of dofs between substructures: w = Ru for some u ∈ U easy to see: R i R T i = I, range R = null B
Intersubdomain averaging and weight matrices primary diagonal weight matrices D Pi : W i → W i , D P = diag( D Pi ) decomposition of unity: R T D P R = I purpose: average between subdomains to get global dofs: u = R T D P w dual weight matrices D Di : Λ → Λ, defined by d α ij = d α j - d α j : diagonal entry of D Pj associated with the same global dof α - d α ij : diagonal entry of D Di for Ω i and Ω j and the same global dof α define B D = [ D D 1 B 1 , . . . D DN B N ], then B T D is a generalized inverse of B : BB T D B = B associated projections are complementary: B T D B + RR T D P = I Rixen, Farhat, Tezaur, Mandel 1998, Rixen, Farhat 1999, Klawonn, Widlund, Dryja 2001, 2002, Fragakis, Papadrakakis 2003 same identities hold also for other versions of the operators
Augmented constraints and coarse dofs Choose matrix Q T P that selects coarse dofs: u c = Q T P w (e.g. values at corners, averages on sides) Define R ci : all constraint values �→ values that can be nonzero on substructure i , define C i = R ci Q T P R T i : u c = 0 ⇐ ⇒ C i w i = 0 ∀ i Assume the generalized coarse dofs define interpolation: ∀ w ∈ U ∃ u c ∀ i : C i R i w = R ci u c ⇐ ⇒ a coarse dof can only involve nodes adjacent to the same set of substructures Define subspace of vectors with coarse dofs continuous across substructures: � W = { w ∈ W : ∃ u c ∀ i : C i w i = R ci u c } FETI constraints Bw = 0 augmented by Q T D Bw = 0 so that w ∈ � W ⇔ Q T D Bw = 0
FETI Approach S 1 ... w ∈ W = W 1 × . . . × W N , S = S N Continuity constraints between substructures: Bw = 0 � � 1 − 1 0 . . . B = [ B 1 , . . . , B N ] = . . . . . . . . . . . . Problem: E ( w ) = 1 2 w T Sw − w T g → min subject to Bw = 0 Enforce constraints by Lagrange multipliers λ , eliminate w , solve problem for λ by PCG: solving independent problems with S i , possibly singular u 1 u 2 u 3 u 4
FETI-DP Approach � enforce selected constraints directly : Q T D Bw = 0 enforce the rest of the constraints Bw = 0 (or all) by multipliers λ = ⇒ saddle point problem: min max L ( w, λ ) = max min L ( w, λ ) w ∈ � λ λ w ∈ � W W L ( w, λ ) = 1 2 w T Kw − w T f + w T B T λ � � � w : Q T Bw = 0 W = functions that satisfy the augmenting constraints ⇒ dual problem: ∂ F ( λ ) = = 0 , F ( λ ) = min L ( w, λ ) ∂λ w ∈ � W preconditioner M = B D SB T D Q T D Bw = 0 enforces continuity - of values across crosspoints (Farhat, Lesoinne, Le Tallec, Pierson, Rixen 2001) - also of averages across edges and faces (for 3D) (Farhat, Lesoinne, Pierson 2000, Pierson thesis 2000, Klawonn, Widlund, Dryja 2002)
Original FETI-DP Implementation Farhat, Lesoinne, Pierson 2000: w = w c + w r , w c values on corners, w r is “remainding dofs” � same values on corners as new common variables w c : w i,c = R i,c u c enforce the same averages across the edges by new multiplier µ ⇒ local problems: coarse problem: = = ⇒ eliminate u c and µ = ⇒ dual problem eliminate w r � � E ( R c u c + w r , λ ) + ( R c u c + w r ) T B T Q D µ F ( λ ) = min u c max min → max µ w r coarse problem for u c , µ from min max = ⇒ indefinite, sparse
Alternative FETI-DP Setting Klawonn, Widlund, Dryja 2002: � W Π ⊕ � � = W W ∆ � W Π is “ primal space” = basis functions: one per crosspoint, edge, face W ∆ = ⊗ � � � W ∆ i , W ∆ i = { w i ∈ W i : C i w i = 0 } is the “dual space” E ( w ) + w T B T λ F ( λ ) = min � �� � u ∈ � W L ( u,λ ) E ( w Π + w ∆ ) + ( w Π + w ∆ ) T B T λ = min min � �� � w Π ∈ � w ∆ ∈ � W Π W ∆ L ( w Π + w ∆ ,λ )
Functions from primal and dual spaces corner primal basis function edge primal basis function 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 20 10 8 15 20 20 6 15 15 10 4 10 10 5 2 5 5 0 0 0 0 Basis functions of the primal (coarse) space � W Π (Klawonn, Widlund, Dryja 2002), 5-point Laplacian 45 30 40 20 35 30 10 25 0 20 −10 15 20 10 5 15 0 20 10 15 20 20 15 10 5 15 10 10 5 5 5 0 0 0 0 Functions from � W ∆ with zero corner values, and zero corner values and edge averages
Nested minimization 0.8 0.6 0.4 0.2 0 1 0.5 1 0.5 0 0 −0.5 −0.5 −1 −1 W = � � W Π ⊕ � W ∆ = ⇒ min L ( w, λ ) = min min L ( w ∆ + w Π , λ ) w ∈ � w Π ∈ � w ∆ ∈ � W W Π W ∆
Coarse basis functions need not be continuous In Klawonn, Windlund, Dryja 2002, the primal basis functions were assumed continuous, B � W Π = 0 - needed for the condition number bounds. ⇒ we can take any � But F is defined by nested minimization = W Π such that � W = � W Π ⊕ � W ∆ . More convenient � W Π for computations is by energy minimization, works for an arbitrary C : � � w ∈ W : C i w i = R ci u c , w T � W Π = i K i w i → min 1.2 1.4 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 10 10 8 8 10 10 6 8 6 8 6 6 4 4 4 4 2 2 2 2 0 0 0 0
A better FETI-DP Implementation Dual problem: ∂ F ( λ ) = 0 where ∂λ 1 2 w T Sw − w T g + w T B T λ F ( λ ) = min w ∈ � � �� � W L ( w,λ ) = min min L ( w ∆ + w Π , λ ) w Π ∈ � w ∆ ∈ � W Π W ∆ Eliminating w ∆ = ⇒ solving N independent constrained problems � � � � � � C T f − B T w i S i i λ i = 0 ς i 0 C i If there are enough corners = ⇒ constraints ( w i ) j = 0 = ⇒ SPD problem These are the same constrained local problems as in BDDC. Minimizing over w Π ∈ � W Π = ⇒ coarse problem is SPD, and also sparse
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