DD, MG and Deflation Reinhard Nabben Comparison of Projection Methods TU Berlin derived from Deflation, Domain Deflation Comparison Decomposition and Multigrid Methods Deflation vs. additive coarse grid corrections Deflation vs balancing Reinhard Nabben Comparison of TU Berlin Projection methods Numerical Y.A. Erlangga TU Berlin, K. Vuik , TU Delft, J. Tang TU Delft comparison Conclusion supported by Deutsche Forschungsgemeinschaft (DFG) 20.08.2007
DD, MG and Outline Deflation Reinhard Nabben TU Berlin Deflation Deflation Comparison Deflation vs. additive coarse grid corrections Comparison Deflation vs. additive coarse grid corrections Deflation vs balancing Comparison of Deflation vs balancing Projection methods Numerical comparison Comparison of Projection methods Conclusion Numerical comparison Conclusion
DD, MG and Deflation Reinhard Nabben Ax = b TU Berlin Deflation A is sparse and symmetric positive definite; condition Comparison number of A is huge, due to large jumps in the coefficients Deflation vs. additive coarse grid corrections Deflation vs balancing Applications: Comparison of ◮ reservoir simulations Projection methods ◮ porous media flow Numerical comparison ◮ elasticity Conclusion ◮ more
DD, MG and Deflation Deflated CG Reinhard Nabben TU Berlin Nicolaides 1987, Mansfield 1988, 1990, Kolotilina 1998, Deflation Vuik, Segal, and Meijerink 1999, Morgan 1995, Saad, Comparison Deflation vs. Yeung, Erhel, and Guyomarch 2000, Frank and Vuik additive coarse grid corrections 2001, Blaheta 2006 Deflation vs balancing Comparison of Projection methods Deflation and restarted GMRES Numerical comparison Morgan 1995, Erhel, Burrage, and Pohl 1996, Chapman Conclusion and Saad 1997, Eiermann, Ernst, and Schneider 2000, Morgan 2002 Clemens et al. 2003,2004, de Sturler et al. 2006, Aksoylu, H. Klie, and M.F . Wheeler 2007
DD, MG and Deflation with eigenvectors Deflation Reinhard Nabben TU Berlin Deflation u T Au i = λ i u i Z = [ u 1 , . . . , u r ] i u j = δ ij Comparison Deflation vs. additive coarse grid corrections Deflation vs P = I − AZ ( Z T AZ ) − 1 Z T , Z ∈ R n × r , balancing Comparison of Projection methods spectrum ( PA ) = { 0 , . . . , 0 , λ r + 1 , . . . λ n } Numerical comparison Conclusion
DD, MG and Deflation with general vectors Deflation Reinhard Nabben TU Berlin Deflation E = Z T AZ Z = [ z 1 , . . . , z r ] rankZ = r Comparison Deflation vs. additive coarse grid corrections Deflation vs P = I − AZE − 1 Z T , Z ∈ R n × r , balancing Comparison of Projection methods Numerical comparison PAZ = 0 Conclusion spectrum ( PA ) = { 0 , . . . , 0 , µ r + 1 , . . . µ n }
DD, MG and Deflation for linear systems Deflation Reinhard Nabben TU Berlin Z ∈ R n × r Z = [ z 1 , . . . , z r ] rankZ = r Deflation Comparison Deflation vs. P = I − AZE − 1 Z T Ax = b additive coarse grid corrections Deflation vs balancing x = ( I − P T ) x + P T x We have: Compute both! Comparison of Projection methods Numerical 1. ( I − P T ) x = Z ( Z T AZ ) − 1 Z T b comparison preconditioner M − 1 : 2. Solve PA ˜ x = Pb Conclusion M − 1 PA ˜ x = M − 1 Pb 3. Build P T ˜ x = P T x
DD, MG and Another Deflation Variant (Kolotilina; Saad et al.) Deflation Reinhard Nabben TU Berlin Choose a random vector, ¯ Deflation x , Comparison Deflation vs. start CG with x 0 := Z ( Z T AZ ) − 1 Z T b + P T ¯ additive coarse x grid corrections Deflation vs balancing for the system Comparison of Projection methods Numerical P T M − 1 Ax = P T M − 1 b , comparison Conclusion
DD, MG and Deflation Deflation M − 1 preconditioner, ILU Z appro. eigenvectors Reinhard Nabben TU Berlin ZE − 1 Z T Deflation Comparison Deflation vs. additive coarse Domain decomposition grid corrections M − 1 add. Schwarz Z grid transfer operator Deflation vs ZE − 1 Z T coarse grid correction balancing Comparison of Projection methods Numerical Multigrid comparison M − 1 smoother Z grid transfer operator Conclusion ZE − 1 Z T coarse grid correction
DD, MG and Additive coarse grid corrections Deflation Reinhard Nabben TU Berlin Z T : R n → R r : Z : R r → R n : Deflation restriction prolongation Comparison Deflation vs. additive coarse grid corrections Z T AZ Galerkin product Deflation vs balancing Z ( Z T AZ ) − 1 Z T Coarse Grid Correction Comparison of Projection methods Preconditioner Numerical comparison M − 1 + Z ( Z T AZ ) − 1 Z T P ad = Conclusion Bramble, Pasciak and Schatz 1986, Dryja and Widlund 1990
DD, MG and Comparison: Deflation vs. additive coarse Deflation grid correction Reinhard Nabben TU Berlin Deflation Comparison M − 1 − M − 1 AZE − 1 Z T M − 1 P Deflation vs. = additive coarse grid corrections M − 1 + ZE − 1 Z T P ad = Deflation vs balancing Comparison of Theorem Projection methods Nabben, Vuik 04 Numerical For all Z with rankZ = r we have: comparison Conclusion λ n ( M − 1 PA ) ≤ λ n ( P ad A ) λ r + 1 ( M − 1 PA ) ≥ λ 1 ( P ad A ) Thus: cond eff ( M − 1 PA ) ≤ cond ( P ad A )
DD, MG and abstract Balancing Neumann-Neumann Deflation preconditioner Reinhard Nabben TU Berlin Mandel 1993, Dryja and Widlund 1995, Mandel and Deflation Brezina 1996 Comparison Deflation vs. additive coarse M − 1 Neumann-Neumann preconditioner grid corrections Deflation vs balancing Comparison of P B = ( I − ZE − 1 Z T A ) M − 1 ( I − AZE − 1 Z T ) + ZE − 1 Z T , Projection methods Numerical comparison Conclusion E = Z T AZ , Z ∈ R n × r P = I − AZE − 1 Z T , P B = P T M − 1 P + ZE − 1 Z T .
DD, MG and Deflation and abstract balancing Deflation preconditioner Reinhard Nabben TU Berlin Deflation Theorem Comparison Nabben, Vuik 06 Deflation vs. additive coarse ◮ cond eff ( M − 1 PA ) ≤ cond ( P B A ) grid corrections Deflation vs balancing Comparison of Projection methods Numerical comparison Conclusion
DD, MG and Deflation and abstract balancing Deflation preconditioner Reinhard Nabben TU Berlin Deflation Theorem Comparison Nabben, Vuik 06 Deflation vs. additive coarse ◮ cond eff ( M − 1 PA ) ≤ cond ( P B A ) grid corrections Deflation vs spectrum ( M − 1 PA ) = { 0 , . . . , 0 , µ r + 1 , . . . , µ n } balancing ◮ Comparison of spectrum ( P B A ) = { 1 , . . . , 1 , µ r + 1 , . . . , µ n } Projection methods Numerical comparison Conclusion
DD, MG and Deflation and abstract balancing Deflation preconditioner Reinhard Nabben TU Berlin Deflation Theorem Comparison Nabben, Vuik 06 Deflation vs. additive coarse ◮ cond eff ( M − 1 PA ) ≤ cond ( P B A ) grid corrections Deflation vs spectrum ( M − 1 PA ) = { 0 , . . . , 0 , µ r + 1 , . . . , µ n } balancing ◮ Comparison of spectrum ( P B A ) = { 1 , . . . , 1 , µ r + 1 , . . . , µ n } Projection methods ◮ For ˜ x 0 , D = x 0 , B � x − x k , D � A ≤ � x − x k , B � A . Numerical comparison Conclusion
DD, MG and Deflation and abstract balancing Deflation preconditioner Reinhard Nabben TU Berlin Deflation Theorem Comparison Nabben, Vuik 06 Deflation vs. additive coarse ◮ cond eff ( M − 1 PA ) ≤ cond ( P B A ) grid corrections Deflation vs spectrum ( M − 1 PA ) = { 0 , . . . , 0 , µ r + 1 , . . . , µ n } balancing ◮ Comparison of spectrum ( P B A ) = { 1 , . . . , 1 , µ r + 1 , . . . , µ n } Projection methods ◮ For ˜ x 0 , D = x 0 , B � x − x k , D � A ≤ � x − x k , B � A . Numerical comparison x and x 0 , B = ZE − 1 Z T b + P T ¯ ◮ ˜ x 0 , D = ¯ x then Conclusion x k , D = x k , B .
DD, MG and Implementation of the Balancing Deflation preconditioner Reinhard Nabben TU Berlin Solve Ax = b Deflation Take x 0 , B = ZE − 1 Z T b + P T ¯ x Comparison Deflation vs. additive coarse grid corrections Deflation vs Then the balancing preconditioner P T M − 1 P + ZE − 1 Z T balancing Comparison of can be implemented with the use of Projection methods P T M − 1 Numerical comparison Conclusion only. Mandel 93, Toselli, Widlund 04. Motivation: Save of work per iteration
DD, MG and Our more detailed analysis shows Deflation Reinhard Nabben ◮ better effective condition number TU Berlin ◮ better clustering Deflation ◮ better A-norm of the error Comparison Deflation vs. additive coarse grid corrections Deflation vs balancing Comparison of Projection methods Numerical comparison Conclusion
DD, MG and Deflation Remarks from others Reinhard Nabben TU Berlin ◮ Balancing is robust w.r.t. inexact solves, deflation not. Deflation Comparison Deflation vs. additive coarse grid corrections Deflation vs balancing Comparison of Projection methods Numerical comparison Conclusion
DD, MG and Comparison of Projection methods Deflation Reinhard Nabben TU Berlin Deflation, Variant 1 Deflation Comparison Deflation vs. M − 1 P additive coarse grid corrections Deflation vs Deflation, Variant 2 balancing Comparison of Projection methods P T M − 1 Numerical comparison Conclusion
DD, MG and Deflation Balancing method Reinhard Nabben TU Berlin Deflation P T M − 1 P + ZE − 1 Z T Comparison Deflation vs. additive coarse Reduced balancing method, Variant 1 grid corrections Deflation vs balancing Comparison of P T M − 1 P Projection methods Numerical Reduced balancing method, Variant 2 comparison Conclusion P T M − 1
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