Speaker
Description
It is well known that every matrix $A \in \mathbb{C}^{n \times n}$ can be split into its Hermitian and skew-Hermitian parts, i.e.,
$$A = H + S, \quad
H = \frac{1}{2}(A + A^*), \quad
S = \frac{1}{2}(A - A^*),$$
so that $H = H^{*}$ and
$S=-S^{*}$. This simple, yet fundamental observation has many applications.
For example, Householder used it in The Theory of Matrices (1964) to show that all eigenvalues of $A = H + S$ lie in or on the smallest rectangle with sides parallel to the real and imaginary axes that contains all eigenvalues of $H$ and of $S$. This result, attributed to Bendixson (1902), is very useful
in the analysis of spectral properties of saddle point matrices; see, e.g., [1, Section 3.4] or [2]. It implies that if $H$ is positive definite, then all eigenvalues of $A$ have a positive real part. Therefore such (in general non-Hermitian) matrices A are sometimes called positive real or positive stable, but we call $A = H + S$ positive (semi)definite if $H$ has the corresponding property.
In the first part of this talk we will discuss an important class of practically relevant applications where the splitting $A = H + S$ occurs naturally and has a physical meaning. These applications arise from energy-based modeling using differential algebraic equation (DAE) systems in dissipative Hamiltonian (dH) form, or shortly dHDAE systems. The usefulness and applicability of this modeling approach has been demonstrated in a variety of application areas such as thermodynamics, electromagnetics, fluid mechanics, chemical processes, and general optimization.
In the second part we will study the linear algebraic systems arising in this context, and their solution by Krylov subspace methods. The (non-Hermitian) matrices A that occur are positive definite or positive semidefinite. In the positive definite case we can solve the linear algebraic systems using Krylov subspace methods based on efficient three-term recurrences. Such methods were already derived by Widlund [6] (based on earlier work of Concus and Golub [3]) and Rapoport [5]
in the late 1970s. These methods are not widely used or known, and we will therefore summarize the most important facts about their implementation and their mathematical properties. The semidefinite case can be challenging and requires additional techniques to identify and deal with
the “singular part” of the matrix, while the “positive definite part” can still be treated with the three-term recurrence methods. We will illustrate the performance of the iterative methods and compare them with (preconditioned) GMRES on several computed examples.
The talk is based on joint work with Candan Güdücü, Volker Mehrmann, and Daniel B. Szyld [4].
References
[1] M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems Acta Numer., 14
(2005), pp. 1–137.
[2] M. Benzi and V. Simoncini, On the eigenvalues of a class of saddle point matrices, Numer. Math., 103
(2006), pp. 173–196.
[3] P. Concus and G. H. Golub, A generalized conjugate gradient method for nonsymmetric systems of
linear equations, in Computing methods in applied sciences and engineering (Second Internat. Sympos.,
Versailles, 1975), Part 1, Roland Glowinski and Jaques-Louis Lions, eds., vol. 134 of Lecture Notes in
Econom. and Math. Systems, Springer, Berlin, 1976, pp. 56–65.
[4] C. Güdücü, J. Liesen, V. Mehrmann, and D. B. Szyld, On non-Hermitian positive (semi)definite linear
algebraic systems arising from dissipative Hamiltonian DAEs, arXiv:2111.05616, 2021.
[5] D. Rapoport, A Nonlinear Lanczos Algorithm and the Stationary Navier-Stokes Equation, PhD thesis,
Department of Mathematics, Courant Institute, New York University, 1978.
[6] O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J.
Numer. Anal., 15 (1978), pp. 801–812.
