accelerate iterative methods

Accelerate Iterative Methods Good Algorithms Mixed Precision - PowerPoint PPT Presentation

Accelerate Iterative Methods Good Algorithms Mixed Precision Iterative Methods Good Preconditioners using High Precision Arithmetic Parallel Algorithms Good Implementations Hidehiko Hasegawa Accurate Computations


  1. Accelerate Iterative Methods • Good Algorithms Mixed Precision Iterative Methods • Good Preconditioners using High Precision Arithmetic • Parallel Algorithms • Good Implementations Hidehiko Hasegawa • Accurate Computations hasegawa@slis.tsukuba.ac.jp Faculty of Library, Information and Media Science, University of Tsukuba JSIAM Applied Mathematics Seminar, Dec. 27, 2013 1 JSIAM Applied Mathematics Seminar, Dec. 27, 2013 2 Convergence history ��������������������������������� Numerical Comparison of � � Accelerating Polynomials in �� ���������������������� Product-type Iterative Methods �������� �� �������� �������� �������� �� �� ��� ��� � ��� ��� ��� ��� ��� ��� ���������� Seventh SIAM Conference on Seventh SIAM Conference on H. Hasegawa, K. Abe, and S.-L. Zhang H. Hasegawa, K. Abe, and S.-L. Zhang Applied Linear Algebra 2000 Applied Linear Algebra 2000

  2. Convergence history of Bi-CG part Convergence of Bi-CG part: Quadruple ( reconstruct Bi-CG using alpha and beta in each methods) ( reconstruct Bi-CG using alpha and beta in each methods) �������������������������������������������������� ��������������������������������������������������������������� � � � � �� �� ���������������������� ���������������������� �� �� �� �� �������� �������� �� �������� �� �������� �������� �������� �������� �������� ��� ��� ��� ��� � ��� ��� ��� ��� ��� ��� � ��� ��� ��� ��� ��� ��� ���������� ���������� Seventh SIAM Conference on H. Hasegawa, K. Abe, and S.-L. Zhang Seventh SIAM Conference on H. Hasegawa, K. Abe, and S.-L. Zhang Applied Linear Algebra 2000 Applied Linear Algebra 2000 Convergence history based on one Bi-CG How Bi-CG part works? ( alpha and beta in Bi-CG are used in all methods) • Bi-CGSTAB converges by an effect of MR part ����������������������������������������� ( Bi-CG part is still unstable) � • GPBi-CG makes Bi-CG part stable � �� • CGS did not converge in Quadruple arithmetic ���������������������� �� • In Quadruple arithmetic, simple Bi-CG is the best ( Bi-CG is much affected by Rounding errors) �� �������� �� �������� • In Quadruple arithmetic, Bi-CG part in Bi- �������� �������� CGSTAB is bad convergence even if Bi-CG ��� converges. ��� � ��� ��� ��� ��� ��� ��� ���������� Seventh SIAM Conference on Seventh SIAM Conference on H. Hasegawa, K. Abe, and S.-L. Zhang H. Hasegawa, K. Abe, and S.-L. Zhang Applied Linear Algebra 2000 Applied Linear Algebra 2000

  3. Convergence history based on one Bi-CG Convergence history based on one Bi-CG (Quadruple arithmetic is used for ALL) (Quadruple arithmetic is used for Bi-CG) ������������������������������������ ���������������������������������������������������� � � � � �� �� ���������������������� ���������������������� �� �� �������� �� �������� �� �������� �������� �� �� �������� �������� �������� ��� �������� ��� ��� ��� � ��� ��� ��� ��� ��� ��� � �� ��� ��� ��� ��� ���������� ���������� Seventh SIAM Conference on H. Hasegawa, K. Abe, and S.-L. Zhang Seventh SIAM Conference on H. Hasegawa, K. Abe, and S.-L. Zhang Applied Linear Algebra 2000 Applied Linear Algebra 2000 How accelerating polynomial works • Qudaruple arithmetic works very well. Utilizing Quadruple-Precision • If enough accuracy was provided, Bi-CG was the Floating Point Arithmetic best. • Bi-CGSTAB and GPBi-CG work well. Operation for the Krylov • In Quadruple arithmetic, sometimes it works as Subspace Methods braking not as accelerating. • GPBi-CG is robust in both two conditions. • CGS does not work in both conditions because of “squared”. Seventh SIAM Conference on SIAM Conference on 12 H. Hasegawa, K. Abe, and S.-L. Zhang Applied Linear Algebra 2000 Applied Linear Algebra 2003

  4. BiCG Gamma = 1.3 CGS Gamma = 1.3 SIAM Conference on 13 SIAM Conference on 14 Applied Linear Algebra 2003 Applied Linear Algebra 2003 BiCG Gamma = 2.5 BiCGSTAB Gamma = 1.3 SIAM Conference on 15 SIAM Conference on 16 Applied Linear Algebra 2003 Applied Linear Algebra 2003

  5. BiCGSTAB Gamma = 2.5 GPBiCG Gamma = 2.5 SIAM Conference on 17 SIAM Conference on 18 Applied Linear Algebra 2003 Applied Linear Algebra 2003 Observations High Precision Arithmetic • Fast and smooth convergence are gained from • Reducing round-off errors More accurate computations. • Required Mantissa is based on the problems: • Accelerating algorithms mathematically BiCG 53 bit for Gamma = 1.3 • Not easy to use 100 bit for 1.7 200 bit for 2.1 200 bit for 2.5 • Required Mantissa depends on Algorithms: BiCG 200 bit and 190 iterations CGS 300 bit and 160 x BiCGSTAB 1500 bit and 210 x GPBiCG 300 bit and 310 (Gamma = 2.5 ) SIAM Conference on 19 JSIAM Applied Mathematics Seminar, Dec. 27, 2013 Applied Linear Algebra 2003 20

  6. High Precision Arithmetic Important points! without any Special Hardware • Symbolic Computation (Computer Algebra) • Full or Partial • Variable length Multiple Precision One Precision or Mixed Precisions – GMP • Computing Environment – MBLAS Compiler/Emulation/Interpreter – exflib • Program Interface, API • Fixed length Multiple Precision – FORTRAN REAL *16 – IEEE – Double-double JSIAM Applied Mathematics Seminar, Dec. 27, 2013 JSIAM Applied Mathematics Seminar, Dec. 27, 2013 21 22 Our Solution: Advantages Utilize Accurate Computations for Iterative Methods • Tough for round-off errors • Small Additional Memory • Use Double-double • Small Additional Communications • Use D-D vectors and Double Matrices • Much Computations (Mixed Precision Arithmetic Operations) • Applicable for ALL Iterative Methods • Accelerate by SIMD (even if serial computation such as ILU) • Restart with different Precision • Automatic Tuning • Good tools JSIAM Applied Mathematics Seminar, Dec. 27, 2013 JSIAM Applied Mathematics Seminar, Dec. 27, 2013 23 24

Recommend


More recommend


Explore More Topics

Stay informed with curated content and fresh updates.