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An introduction to the

“Group of High Performance Scientiﬁc Computing”

(HiPerSC)

E. Defez, J.J. Ib´a˜nez, J. Peinado, J. Sastre, M.M. Tung

Group of High Performance Scientiﬁc Computing (HiPerSC)

Universitat Polit`ecnica de Val`encia, Spain

edefez@imm.upv.es, jjibanez@dsic.upv.es, jpeinado@dsic.upv.es,

jorsasma@dcom.upv.es, mtung@mat.upv.es

ICIAM 2019

What is the “Group of High Performance Scientiﬁc

Computing (HiPerSC)”?

About the group

•Research group devoted to scientiﬁc and high performance

computing and its applications.

•The group started working in 2008.

•It is formed by researchers from the:

•Institute of Telecomunications and Multimedia Applications

(ITEAM)

•Institute of Instrumentation for Molecular Image (I3M)

•Department of Computer Systems and Computation (DSIC)

•Institute of Multidisciplinary Mathematics (IMM)

All from the Polytechnic University of Valencia (UPV)

•CONTACT email: hipersc@upv.es

•Twitter: @HiPerSC−UPV

ICIAM 2019

Our work

Basically, from the point of view of applied mathematics, the group has worked

on the development of approximation methods for matrix functions (i.e. matrix

exponential, trigonometric matrix functions sine and cosine, hyperbolic matrix

functions ...) and the numerical solution of diﬀerential systems.

In this way we have found some myths and misconceptions

(widespread belief) that we have helped to clarify...

ICIAM 2019

For a matrix ∈C×, we are interested in approximate the matrix exponential = X

≥0!

We have the Taylor approximation

( ) = X

=0 !(1)

and the [ /] Pad´e approximation

( ) = ( ) ( ( ))−1,( ) = X

=0

( + −)! !

( + )!( −)! ! ,( ) = X

=0

( + −)! !(−)

( + )!( −)! !

(2)

People thought that Pad´e were better than Taylor!!!

References

[1] C. B. Moler, C. F. Van Loan. Nineteen dubious ways to compute the exponential of a

matrix. SIAM Rev., 20(4):801–836, 1978.

[2] C. B. Moler, C. F. Van Loan. Nineteen dubious ways to compute the exponential of a

matrix, twenty-ﬁve years later. SIAM Rev., 45(1):3-–49, 2003.

[3] N. J. Higham. The scaling and squaring method for the matrix exponential revisited,

SIAM J. Matrix Anal. Appl., 26, 1179-–1193, 2005.

[4] A. H. Al-Mohy, N. J. Higham. A new scaling and squaring algorithm for the matrix

exponential, SIAM J. Matrix Anal. Appl., 31(3):970–989, 2009.

ICIAM 2019

Contrary to what is believed, the group showed that computing the

matrix exponential by using matrix polynomial approximations (Taylor and

Hermite polynomials) may be better than using Pad´e approximations regarding

both, accuracy and cost.

References

[5] J. Sastre, J. Ib´a˜nez, E. Defez, P. Ruiz. New scaling-squaring Taylor algorithms for

computing the matrix exponential. SIAM Journal on Scientiﬁc Computing, 37 (1),

(submitted in 2009), A439–A455, 2015.

[6] E. Defez, J. Sastre, J. Ib´a˜nez, P. Ruiz. Eﬃcient orthogonal matrix polynomial based

method for computing matrix exponential. Applied Mathematics and Computation, 217,

6451–6463, 2011.

[7] J. Sastre, J. Ib´a˜nez, E. Defez, P. Ruiz. Accurate matrix exponential computation to

solve coupled diﬀerential models in Engineering. Mathematical and Computer modelling,

54, 1835–1840, 2011.

[8] J. Sastre, J. Ib´a˜nez, P. Ruiz. E. Defez. Accurate and eﬃcient matrix exp onential

computation. International Journal of Computer Mathematics, 91 (1), 97-–112. 2014.

[9] P. Ruiz, J. Sastre, J. Ib´a˜nez, E. Defez. High performance computing of the matrix

exponential. Journal of Computational and Applied Mathematics, 291(1), 370-–379, 2016.

[10] J. Sastre, J. Ib´a˜nez, E. Defez. Boosting the computation of the matrix exp onential.

Applied Mathematics and Computation, 340 (1), 206–220, 2019.

ICIAM 2019

For a matrix ∈C×, we are interested in approximate the matrix cosine deﬁned by

cos ( ) = X

≥0

(−1) 2

(2 )! and the matrix hyperbolic cosine is deﬁned by cosh ( ) = X

≥0

2

(2 )!

We have the respective Taylor approximations

( ) = X

=0

(−1) 2

(2 )! ,( ) = X

=0

2

(2 )! (3)

and the [ /] Pad´e approximations.

People thought that Pad´e were better than Taylor also!!!

References

[11] N. J. Higham, M. I. Smith. Computing the matrix cosine. Numerical Algorithms 34.1,

13–26, 2003.

[12] G. I. Hargreaves, N. J. Higham. Eﬃcient algorithms for the matrix cosine and sine.

Numerical Algorithms 40.4, 383–400, 2005.

[13] A. H. Al-Mohy, N. J. Higham, S. D. Relton. New algorithms for computing the matrix

sine and cosine separately or simultaneously. SIAM Journal on Scientiﬁc Computing 37.1,

A456–A487, 2015.

ICIAM 2019

Again, contrary to what is believed, the group showed that computing

matrix (hyperbolic) cosine by using matrix polynomial approximations (Taylor

and Hermite polynomials) may be better than using Pad´e approximations

regarding both, accuracy and cost.

References for Taylor series

[14] J. Sastre, J. Ib´a˜nez, P. Ruiz, E. Defez, Eﬃcient computation of the matrix cosine.

Applied Mathematics and Computation, 219, 7575-–7585, 2013.

[15] E. Defez, P. Alonso-Jord´a, J. Ib´a˜nez, J. Sastre, J. Peinado. Eﬃcient and accurate

algorithms for computing matrix trigonometric functions. Journal of Computational and

Applied Mathematics, 309, 325-–332, 2017.

[16] J. Sastre, J. Ib´a˜nez, P. Alonso-Jord´a, E. Defez. Two algorithms for computing the

matrix cosine function. Applied Mathematics and Computation, 312 (1), 66–77, 2017.

[17] P. Alonso-Jord´a, J. Peinado, J. Ib´a˜nez, J. Sastre, E. Defez. Computing Matrix

Trigonometric Functions with GPUs through Matlab. The Journal of Supercomputing, 75,

1227—1240, 2019.

[18] J. Sastre, J. Ib´a˜nez, P. Alonso-Jord´a, J. Peinado, E. Defez. Fast Taylor polynomial

evaluation for the computation of the matrix cosine. Journal of Computational and Applied

Mathematics, 354, 641-–650, 2019.

ICIAM 2019

References for Hermite series

[19] E. Defez, J. Sastre, J. Ib´a˜nez, P. Ruiz. Computing matrix functions solving coupled

diﬀerential models. Mathematical and Computer Modelling, 50 (5-6), 831–839, 2009.

[20] E. Defez, J. Sastre, J. Ib´a˜nez, P. Ruiz. Computing matrix functions arising in

engineering models with orthogonal matrix polynomials. Mathematical and Computer

Modelling, 57 (7-8), 1738–1743, 2013.

[21] E. Defez, J. Sastre, J. Ib´a˜nez, J. Peinado, M. M. Tung. A method to approximate the

hyperbolic sine of a matrix. International Journal of Complex Systems in Science, 4 (1),

41–45, 2014.

[22] E. Defez, J. Sastre, J. Ib´a˜nez, J. Peinado. Solving engineering models using hyperbolic

matrix functions. Applied Mathematical Mo delling, 40, 2837—2844, 2016.

[23] E. Defez, J. Ib´a˜nez, J . Peinado, J. Sastre, P. Alonso-Jord´a. An eﬃcient and accurate

algorithm for computing the matrix cosine based on new Hermite approximations. Journal

of Computational and Applied Mathematics, 348 (1), 1–13, 2019.

ICIAM 2019

Myth 4: Taylor es the best polynomial

approximation for the matrix exponential

This Myth is also UNFOUNDED!!!!!!!!

Reference

[25] E. Defez, J. Ib´a˜nez, J. Peinado, P. Alonso-Jord´a, J. M. Alonso. Computing matrix

functions by matrix Bernoulli series. Communication in the 2019 International Conference

on Computational & Mathematical Methods in Science & Engineering. June 30-July 6,

Rota, C´adiz - Spain.

ICIAM 2019

Others contributions of the group

A new class of rational-polynomial approximations.

A new way to calculate non-diagonal rational matrix approximations as a

matrix polynomial plus a matrix rational function, being more eﬃcient than

diagonal rational matrix approximations.

Reference

[26] J. Sastre. Eﬃcient mixed rational and p olynomial approximation of matrix functions.

Applied Mathematics and Computation, 218(24), 11938–11946, 2012.

ICIAM 2019

Others contributions of the group

A new method based on matrix splines to

approximate the matrix exponential.

Reference

[27] E. Defez, J. Ib´a˜nez, J . Sastre, J. Peinado, P. Alonso-Jord´a. A new eﬃcient and

accurate spline algorithm for the matrix exponential computation. Journal of

Computational and Applied Mathematics, 337, 354–365, 2018.

ICIAM 2019

Others contributions of the group: Applications to diﬀerent areas of

Physics: Metamaterials, Acoustics, Quantum Optics, Lattice Simulations

References

[28] M. M. Tung. A fundamental Lagrangian approach to transformation acoustics and

spherical spacetime cloaking. Europhys. Lett. 98, (2012) 34002–34006, 2012.

[29] M. M. Tung, E. B. Weinm¨uller. Gravitational frequency shifts in transformation

acoustics. Europhys. Lett. 101, 54006–54011, 2013.

[30] M. M. Tung, J. Peinado. A covariant spacetime approach to transformation acoustics.

Progress in Industrial Mathematics at ECMI 2012, 335–340, 2014.

[31] M. M. Tung. Modelling acoustics on the Poincar´e half-plane. J. Comput. Appl. Math.

337, 336–372, 2018.

[32] M. M. Tung, E. B. Weinm¨uller. Acoustic metamaterial models on the (2+1)D

Schwarzschild plane, J. Comput. Appl. Math. 346, 162–170, 2019.

[33] E. Defez, M. M. Tung. A new type of Hermite matrix polynomial series. Quaestiones

Mathematicae, 41(2), 205-–212, 2018.

[34] M. M. Tung, J. Ib´a˜nez, E. Defez, J. Sastre. Dynamics of Harmonic Oscillators on the

Lattice. Submitted to Computer Physics Communications (2019).

ICIAM 2019

An introduction to the

“Group of High Performance Scientiﬁc Computing”

(HiPerSC)

E. Defez, J.J. Ib´a˜nez, J. Peinado, J. Sastre, M.M. Tung

Group of High Performance Scientiﬁc Computing (HiPerSC)

Universitat Polit`ecnica de Val`encia, Spain

edefez@imm.upv.es, jjibanez@dsic.upv.es, jpeinado@dsic.upv.es,

jorsasma@dcom.upv.es, mtung@mat.upv.es

ICIAM 2019