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Abstract. Being one of the key tools in conformation dynamics, the identification of metastable states of Markov chains has been subject to extensive research in recent years, especially when the Markov chains represent energy states of biomolecules. Some previous work on this topic involved the computation of the eigenvalue cluster close to one, as well as… (More)

- Elena Virnik
- SIAM J. Scientific Computing
- 2007

We apply algebraic multigrid (AMG) as a preconditioner for solving large singular linear systems of the type (I−T T )x = 0 with GMRES. Here, T is assumed to be the transition matrix of a Markov process. Although AMG and GMRES are originally designed for the solution of regular systems, with adequate adaptation their applicability can be extended to problems… (More)

- Timo Reis, Elena Virnik
- SIAM J. Control and Optimization
- 2009

We propose a model reduction method for positive systems that ensures the positivity of the reduced-order model. In the standard as well as in the descriptor case, for continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to… (More)

- ELENA VIRNIK
- 2008

Let A be a nonnegative idempotent matrix. It is shown that the Schur complement of a submatrix, using the Moore-Penrose inverse, is a nonnegative idempotent matrix if the submatrix has a positive diagonal. Similar results for the Schur complement of any submatrix of A are no longer true in general.

- ELENA VIRNIK, Elena Virnik
- 2008

Abstract. In this paper, we discuss stability properties of positive descriptor systems in the continuous-time as well as in the discrete-time case. We present characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems. We show that if the spectral projector onto the finite deflating subspace of… (More)

- OF SINGULAR M-MATRICES, ELENA VIRNIK
- 1982

We apply algebraic multigrid (AMG) as a preconditioner for solving large singular linear systems of the type (I − TT )x = 0 with GMRES. Here, T is assumed to be the transition matrix of a Markov process. Although AMG and GMRES were originally designed for the solution of regular systems, with adequate adaptation their applicability can be extended to… (More)

We present a new extension of the well-known Perron-Frobenius theorem to regular matrix pairs (E, A). The new extension is based on projector chains and is motivated from the solution of positive differential-algebraic systems or descriptor systems. We present several examples where the new condition holds, whereas conditions in previous literature are not… (More)

Being one of the key tools in conformation dynamics, the identification of meta-stable states of Markov chains has been subject to extensive research in recent years, especially when the Markov chains represent energy states of biomolecules. Some previous work on this topic involved the computation of the eigenvalue cluster close to one, as well as the… (More)

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