Arnoldi algorithms with structured orthogonalization

We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDN), which are formulated as semi-explicit differential algebraic equations (DAEs). The solution can be decomposed to a sum of two projections, one in the range of the system operator and the other in its null space. The range projection can be computed with one shift-and -invert Krylov subspace method. The other projection can be computed with the algebraic equations. Differing from the ordinary Arnoldi method, the orthogonality in the Krylov subspace is replaced with the semi-inner product induced by the positive semi-definite system operator. With proper adjustment, numerical ranges of the Krylov operator lie in the right half plane, and we obtain theoretical convergence analysis for the modified Arnoldi algorithm in computing phi-functions. Lastly, simulations on RLC networks are demonstrated to validate the effectiveness of the Arnoldi algorithm with structured-orthogonalization.

[1]  T. Ericsson.  A Generalised Eigenvalue Problem and The Lanczos Algorithm , 1986 .

[2]  B. Parlett,et al.  How to implement the spectral transformation , 1987 .

[3]  Michel Crouzeix,et al.  Numerical range and functional calculus in Hilbert space , 2007 .

[4]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[5]  Gerhard Wanner,et al.  Dahlquist’s classical papers on stability theory , 2006 .

[6]  Sani R. Nassif,et al.  Power grid analysis benchmarks , 2008, 2008 Asia and South Pacific Design Automation Conference.

[7]  C. Pearcy.  An elementary proof of the power inequality for the numerical radius. , 1966 .

[8]  T. Göckler.  Rational Krylov subspace methods for phi-functions in exponential integrators , 2014 .

[9]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[10]  Satoru Iwata,et al.  Index characterization of differential-algebraic equations in hybrid analysis for circuit simulation , 2010, Int. J. Circuit Theory Appl..

[11]  Chung-Kuan Cheng,et al.  Transient Circuit Simulation for Differential Algebraic Systems using Matrix Exponential , 2018, 2018 IEEE/ACM International Conference on Computer-Aided Design (ICCAD).

[12]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

[13]  N. Higham.  Functions Of Matrices , 2008 .

[14]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[15]  H. de la Cruz,et al.  Efficient computation of phi-functions in exponential integrators , 2020, J. Comput. Appl. Math..

[16]  L. Trefethen.  Approximation Theory and Approximation Practice (Other Titles in Applied Mathematics) , 2012 .

[17]  Chung-Kuan Cheng,et al.  Stability and Convergency Exploration of Matrix Exponential Integration on Power Delivery Network Transient Simulation , 2020, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[18]  Volker Grimm.  Resolvent Krylov subspace approximation to operator functions , 2012 .

[19]  Charles R. Johnson.  NUMERICAL DETERMINATION OF THE FIELD OF VALUES OF A GENERAL COMPLEX MATRIX , 1978 .

[20]  Arjan van der Schaft,et al.  Surveys in Differential-Algebraic Equations I , 2013 .

[21]  Lothar Reichel,et al.  Error Estimates and Evaluation of Matrix Functions via the Faber Transform , 2009, SIAM J. Numer. Anal..

[22]  Karl Meerbergen,et al.  Implicitly restarted Arnoldi with purification for the shift-invert transformation , 1997, Math. Comput..

[23]  Quan Chen,et al.  Time-Domain Analysis of Large-Scale Circuits by Matrix Exponential Method With Adaptive Control , 2012, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[24]  M. Hochbruck,et al.  Rational approximation to trigonometric operators , 2008 .

[25]  R. Freund.  Krylov-subspace methods for reduced-order modeling in circuit simulation , 2000 .

[26]  Jacob K. White,et al.  A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits , 1996, Proceedings of International Conference on Computer Aided Design.

[27]  L. Tuckerman,et al.  A method for exponential propagation of large systems of stiff nonlinear differential equations , 1989 .

[28]  P. Rentrop,et al.  The Drazin inverse in multibody system dynamics , 1993 .

[29]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[30]  Joost Rommes,et al.  Exploiting structure in large-scale electrical circuit and power system problems , 2009 .

[31]  Axel Ruhe,et al.  The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems , 1980 .

[32]  Axel Ruhe.  Rational Krylov sequence methods for eigenvalue computation , 1984 .

[33]  Marlis Hochbruck,et al.  Exponential Rosenbrock-Type Methods , 2008, SIAM J. Numer. Anal..

[34]  R. Winkler.  Stochastic differential algebraic equations of index 1 and applications in circuit simulation , 2003 .

[35]  R. Varga,et al.  Chebyshev rational approximations to e−x in [0, +∞) and applications to heat-conduction problems , 1969 .

[36]  Marlis Hochbruck,et al.  Residual, Restarting, and Richardson Iteration for the Matrix Exponential , 2010, SIAM J. Sci. Comput..

[37]  Xiang Zhang,et al.  Simulation Algorithms With Exponential Integration for Time-Domain Analysis of Large-Scale Power Delivery Networks , 2015, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[38]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[39]  J. Mashreghi,et al.  On mapping theorems for numerical range , 2015, 1510.08132.

[40]  Volker Grimm,et al.  Automatic Smoothness Detection of the Resolvent Krylov Subspace Method for the Approximation of C0-Semigroups , 2017, SIAM J. Numer. Anal..