Lanczos-like algorithm for the time-ordered exponential: the \(\ast\)-inverse problem. (English) Zbl 07285958

Giscard, Pierre-Louis; Pozza, Stefano

Appl. Math., Praha 65, No. 6, 807-827 (2020).

Summary: The time-ordered exponential of a time-dependent matrix \(\mathsf{A}(t)\) is defined as the function of \(\mathsf{A}(t)\) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in \(\mathsf{A}(t)\). The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by \(\ast\). Yet, the existence of such inverses, crucial to avoid algorithmic breakdowns, still needed to be proved. Here we constructively prove that \(\ast\)-inverses exist for all non-identically null, smooth, separable functions of two variables. As a corollary, we partially solve the Green’s function inverse problem which, given a distribution \(G\), asks for the differential operator whose fundamental solution is \(G\). Our results are abundantly illustrated by examples.

MSC:

65F60	Numerical computation of matrix exponential and similar matrix functions
47B36	Jacobi (tridiagonal) operators (matrices) and generalizations
65F10	Iterative numerical methods for linear systems
65D15	Algorithms for approximation of functions

Keywords:

time-ordering; matrix differential equation; time-ordered exponential; Lanczos algorithm; fundamental solution

References:

[1]	Benner, P.; Cohen, A.; Ohlberger, M.; (eds.), K. Willcox, Model Reduction and Approximation: Theory and Algorithms, Computational Science & Engineering 15. SIAM, Philadelphia (2017) · Zbl 1378.65010 · doi:10.1137/1.9781611974829
[2]	Blanes, S., High order structure preserving explicit methods for solving linear-quadratic optimal control problems, Numer. Algorithms 69 (2015), 271-290 · Zbl 1317.65144 · doi:10.1007/s11075-014-9894-0
[3]	Bouhamidi, A., Pseudo-differential operator associated to the radial basis functions under tension, ESAIM, Proc. 20 (2007), 72-82 · Zbl 1359.35233 · doi:10.1051/proc:072007
[4]	Corless, M. J.; Frazho, A. E., Linear Systems and Control: An Operator Perspective, Pure and Applied Mathematics, Marcel Dekker 254. Marcel Dekker, New York (2003) · Zbl 1050.93001 · doi:10.1201/9780203911372
[5]	Fasshauer, G. E.; Ye, Q., Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38 (2013), 891-921 · Zbl 1267.41018 · doi:10.1007/s10444-011-9264-6
[6]	Giscard, P.-L., On the solutions of linear Volterra equations of the second kind with sum kernels, (to appear) in J. Integral Equations Appl. Available at https://arxiv.org/abs/1909.04033v2 (2020), 25 pages
[7]	Giscard, P.-L.; Lui, K.; Thwaite, S. J.; Jaksch, D., An exact formulation of the time-ordered exponential using path-sums, J. Math. Phys. 56 (2015), Article ID 053503, 18 pages · Zbl 1316.15010 · doi:10.1063/1.4920925
[8]	Giscard, P.-L.; Pozza, S., Lanczos-like method for the time-ordered exponential, Available at https://arxiv.org/abs/1909.03437 (2019), 26 pages
[9]	Giscard, P.-L.; Pozza, S., Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method, Available at https://arxiv.org/abs/2002.06973 (2020), 23 pages
[10]	Golub, G. H.; Meurant, G., Matrices, moments and quadrature, Numerical Analysis 1993 Pitman Research Notes in Mathematics Series 303. Longman Scientific & Technical, Harlow (1994), 105-156 · Zbl 0795.65019
[11]	Golub, G. H.; Meurant, G., Matrices, Moments and Quadrature With Applications, Princeton Series in Applied Mathematics 30. Princeton University Press, Princeton (2010) · Zbl 1217.65056 · doi:10.1515/9781400833887
[12]	Gripenberg, G.; Londen, S.-O.; Staffans, O., Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications 34. Cambridge University Press, Cambridge (1990) · Zbl 0695.45002 · doi:10.1017/CBO9780511662805
[13]	Kwakernaak, H.; Sivan, R., Linear Optimal Control Systems, Wiley-Interscience, New York (1972) · Zbl 0276.93001
[14]	Liesen, J.; Strakoš, Z., Krylov Subspace Methods: Principles and Analysis, Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013) · Zbl 1307.65001 · doi:10.1093/acprof:oso/9780199655410.001.0001
[15]	Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM Studies in Applied Mathematics 7. SIAM, Philadelphia (1985) · Zbl 0566.65094 · doi:10.1137/1.9781611970852
[16]	Mehring, M.; Weberruss, V. A., Object-Oriented Magnetic Resonance: Classes and Objects, Calculations and Computations, Academic Press, London (2012) · doi:10.1016/C2009-0-21093-2
[17]	Polyanin, A. D.; Manzhirov, A. V., Handbook of Integral Equations, Chapman & Hall/CRC Press, Boca Raton (2008) · Zbl 1154.45001 · doi:10.1201/9781420010558
[18]	Razdolsky, L., Integral Volterra equations, Probability Based High Temperature Engineering Springer, Cham (2017), 55-100 · doi:10.1007/978-3-319-41909-1_2
[19]	Reid, W. T., Riccati matrix differential equations and non-oscillation criteria for associated linear differential systems, Pac. J. Math. 13 (1963), 665-685 · Zbl 0119.07401 · doi:10.2140/pjm.1963.13.665
[20]	Rivas, Á.; Huelga, S. F., Open Quantum Systems: An Introduction, SpringerBriefs in Physics. Springer, Berlin (2012) · Zbl 1246.81006 · doi:10.1007/978-3-642-23354-8
[21]	Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia (2003) · Zbl 1031.65046 · doi:10.1137/1.9780898718003
[22]	Schwartz, L., Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg IX-X. Hermann, Paris (1966), French · Zbl 0149.09501
[23]	Sezer, M., Taylor polynomial solutions of Volterra integral equations, Int. J. Math. Educ. Sci. Technol. 25 (1994), 625-633 · Zbl 0823.45005 · doi:10.1080/0020739940250501
[24]	Tricomi, F. G., Integral Equations, Dover Publications, New York (1985) · Zbl 0078.09404