# Results in Numerical Analysis, obtained at the Institute

## Finite element methods

Some results regarding constructive aspects in solving initial and boundary value problems for partial differential equations were obtained

- E. Burman, M. Nechita, L. Oksanen,
*A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime*, Numer. Math., under review. - C.D. Alecsa, I. Boros, F. Frank, P. Knabner, M. Nechita, A. Prechtel, A. Rupp, N. Suciu,
*Numerical benchmark study for flow in highly heterogeneous aquifers*, Adv. Water Res., 138 (2020), 103558. - E. Burman, M. Nechita, L. Oksanen,
*Unique continuation for the Helmholtz equation using stabilized finite element methods*, J. Math. Pures Appl. 129 (2019), pp. 1-22. - C.I. Gheorghiu,
*A Constructive Introduction to Finite Elements Method*, Editura Quo-Vadis, Cluj-Napoca, 1999

## Spectral methods

- C.I. Gheorghiu,
*Spectral Methods for Non-Standard Eigenvalue Problems*, Springer Briefs in Mathematics, 2014 - C.I. Gheorghiu,
*Spectral collocation solutions to a class of pseudo-parabolic equations*, G. Nikolov et al. (Eds.): NMA 2018, LNCS 11189, pp. 1–8, 2019. - C.I. Gheorghiu,
*Spectral Methods for Differential Problems*, Casa Cărţii de Stiintă, Cluj-Napoca, 2007. - C.I. Gheorghiu and I.S. Pop,
*A modified Chebyshev-Tau method for a hydrodynamic stability problem*, Proceedings of ICAOR 1996, v. II, pp. 119-126. - C.I. Gheorghiu, M.E. Hochstenbach, B. Plestenjak, J. Rommes,
*Spectral collocation solutions to multiparameter Mathieu’s system, system*, Appl. Math. Comput. 218 (2012) 11990-12000. - J.P. Boyd, C.I. Gheorghiu,
*All roots spectral methods: Constraints, floating point arithmetic and root exclusion*, Appl. Math. Lett., 67 (2017) 28–32

## Convergence orders of sequences

Connecting different definitions of (classical) C-orders, Q- and R-orders were obtained, together with a survey of the convergence orders of the basic iterative methods (Newton, secant, successive approximations):

- E. Cătinaş,
*How many steps still left to x*?*, SIAM Rev. 63 (2021) no. 3, pp. 585–624. - E. Cătinaş,
*A survey on the high convergence orders and computational convergence orders of sequences*, Appl. Math. Comput.,**343**(2019) 1-20.

## Numerical optimization

- C.D. Alecsa, T. Pinţa, I. Boros,
*New optimization algorithms for neural network training using operator splitting techniques*, Neural Networks, 126 (2020), 178-190. doi: 10.1016/j.neunet.2020.03.018 - A. Viorel, C.D. Alecsa, T.O. Pinţa,
*Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems*, Discrete & Continuous Dynamical Systems, 41 (2021) 7, 3319-3341, doi: 10.3934/dcds.2020407 - C.D. Alecsa, S.C. László, T. Pinţa,
*An extension of the second order dynamical system that models Nesterov’s convex gradient method*, Appl. Math. Optim.,**84**(2021), pp. 1687–1716. doi: 10.1007/s00245-020-09692-1 - C.-D. Alecsa, S.C. László, A. Viorel,
*A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem*, Numer. Algor. 84 (2020), pp. 485-512.

## Newton and Newton-Krylov methods for nonlinear systems in R^{n}

The high convergence orders of the Newton methods have been characterized, while considering all sources of errors; the Newton methods with large number of unknowns were studied when the linear systems are solved by Krylov methods, results regarding convergence, monotony and asymptotical behavior being obtained.

- E. Cătinaş,
*Inexact perturbed Newton methods, and applications to a class of Krylov solvers*, J. Optim. Theory Appl., vol. 108 (2001) no. 3, pp. 543-570. - E. Cătinaş,
*The inexact, inexact perturbed and quasi-Newton methods are equivalent models*, Math. Comp., 74 (2005) no. 249, pp. 291-301. - E. Cătinaş,
*Methods of Newton and Newton-Krylov type*, Risoprint, Cluj-Napoca, 2007.

## Solving of nonlinear equations by Newton, secant, Chebyshev, Steffensen or Aitken methods

Local and semilocal convergence results were obtained:

- I. Păvăloiu,
*Rezolvarea ecuaţiilor prin interpolare*, Ed. Dacia, 1981, 190 pp . - I. Păvăloiu,
*Sur une generalisation de la methode de Steffensen*, Rev. Anal. Numer. Theor. Approx., v. 21 (1992) no. 1, pp. 59-67. - E. Cătinaş,
*On some iterative methods for solving nonlinear equations*, Rev. Anal. Numer. Theor. Approx., 23 (1994) no. 1, pp. 47-53.

For a series of papers in this field, I. Păvăloiu was awarded the “Gheorghe Lazăr” prize of the Romanian Academy, in 1970.

## Monotone sequences for approximating the solutions of nonlinear equations

Some classes of Steffensen, Aitken and Aitken-Steffensen methods were introduced and studied, leading to sequences approximating bilateraly the solutions of nonlinear equations:

- I. Păvăloiu,
*Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences*, Calcolo, v. 32 (1995) nos 1-2, pp. 69-82. - I. Păvăloiu and E. Cătinaş,
*On a Steffensen-Hermite method of order three*, Applied Mathematics and Computation, v. 215 (2009) no. 7, pp. 2663-2672.

## Iterative methods of interpolatory type, with high efficiency index

Among certain classes of iterative methods of interpolatory type, the methods with high efficiency index were determined:

- I. Păvăloiu,
*On computational complexity in solving equations by interpolation methods*, Rev. Anal. Numer. Theor. Approx., 24 (1995) no. 1, 201-214. - I. Păvăloiu,
*Optimal efficiency indexes for iterative methods of interpolatory-type*, Computer Science Journal of Moldova, 5 (1997) no. 1(13), 20-43.

## Krylov methods for numerical computing of large linear systems in R^{n}

Connections between the residuals and the backward errors of the approximative solutions of certain Krylov methods were found, as well as some results regarding relations satisfied by the errors of these approximative solutions.

- E. Cătinaş,
*Inexact perturbed Newton methods, and applications to a class of Krylov solvers*, J. Optim. Theory Appl., vol. 108 (2001) no. 3, pp. 543-570. - E. Cătinaş,
*On the high convergence orders of the Newton-GMBACK methods*, Rev. Anal. Numer. Theor. Approx., 28 (1999) no. 2, pp. 125-132. - E. Cătinaş,
*Methods of Newton and Newton-Krylov type*, Risoprint, Cluj-Napoca, 2007, ISBN 978-973-751-533-9.

## Iterative methods for numerical solving of eigenvalues/eigenvectors

Simpler convergence conditions were obtained for different methods (Newton, Chebyshev, chord and Steffensen method) for the case when the system of nonlinear equations has as solutions the eigenvalues and eigenvectors of a linear operator.

- I. Păvăloiu, E. Cătinaş,
*Remarks on some Newton and Chebyshev-type methods for approximating the eigenvalues and eigenvectors of matrices*, Computer Science Journal of Moldova, 7 (1999) no. 1(19), 3-17. - I. Păvăloiu, E. Cătinaş,
*On approximating the eigenvalues and eigenvectors of linear continuous operators*, Rev. Anal. Numer. Theor. Approx., 26 (1997) nos. 1-2, 19-28.