65L05 Initial value problems
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- Hamiltonian matrix (2)
- canonical form (2)
- chemical master equation (2)
- differential-algebraic equation (2)
- index reduction (2)
- strangeness index (2)
- Kronecker chain (1)
- Matrix triples (1)
- Takagi factorization (1)
- chattering (1)
The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted $\ell_1$-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.
The chemical master equation is a fundamental equation in chemical kinetics. It is an adequate substitute for the classical reaction-rate equations whenever stochastic effects become relevant. In the present paper we give a simple argument showing that the solutions of a large class of chemical master equations, including all those in which elementary reactions between two and more molecules do not generate a larger number of molecules than existed before,
are bounded in weighted $\ell_1$-spaces. As an illustration for the implications of this kind of regularity we analyze the effect of truncating the state space. This leads to an error analysis of the finite state projection of the chemical master equation, an approximation that underlies many numerical methods.
We discuss the solution of linear second order differential-algebraic equations with variable coefficients.
Since index reduction and order reduction for higher order higher index differential-algebraic systems do not commute, appropriate index reduction methods for higher order DAEs are required.
We present an index reduction method based on derivative arrays that allows to determine
an equivalent second order system of lower index in a numerical computable way.
For such an equivalent second order system
an appropriate order reduction method allows to formulate a suitable first order DAE system of low index
that has the same solution components as the original second order system.
We consider hybrid systems of differential-algebraic equations and present
a general framework for general nonlinear over- and underdetermined hybrid
systems that allows the
analysis of existence and uniqueness and the application of index reduction
methods for hybrid differential-algebraic systems.
A particular difficulty in the numerical simulation of hybrid systems is
(numerical) chattering, i.e., fast oscillations between modes of operations.
A regularization technique using sliding modes allows to regularize the
system behavior in the case of chattering.
Further, we show how chattering behavior during the numerical solution can
be prevented using sliding mode simulation. The advantage of the sliding mode
simulation is illustrated by numerical examples.
Canonical forms for matrix triples $(A,G,\hat G)$, where
$A$ is arbitrary rectangular and $G$, $\hat G$ are either real symmetric
or skew symmetric, or complex Hermitian or skew Hermitian, are derived.
These forms generalize classical product Schur forms as well as
singular value decompositions.
An new proof for the complex case is given, where there is no need to
distinguish whether $G$ and $\hat G$ are Hermitian or skew Hermitian.
This proof is independent from the results in Bolschakov/Reichstein 1995, where
a similar canonical form has been obtained for the complex case,
and it allows generalization to the real case. Here,
the three cases, i.e., that
$G$ and $\hat G$ are both symmetric, both skew symmetric or one each,
are treated separately.
The classical singular value decomposition for a matrix $A\in\Cmn$ is a
canonical form for $A$ that also displays the eigenvalues
of the Hermitian matrices $AA^\ast$ and $A^\ast A$. In this paper, we develop
a corresponding decomposition for $A$ that provides the Jordan canonical forms
for the complex symmetric matrices $AA^T$ and $A^TA$. More generally, we consider
the matrix triple $(A,G_1,G_2)$, where $G_1\in\CC{m}, G_2\in\CC{n}$
are invertible and either complex symmetric and complex skew-symmetric, and we
provide a canonical form under transformations of the form
$(A,G_1,G_2)\mapsto(X^T A Y, X^T G_1X, Y^T G_2Y)$, where $X,Y$ are nonsingular.
We discuss the eigenvalue problem for
general and structured matrix polynomials which may
be singular and may have eigenvalues at infinity.
We derive staircase
condensed forms that allow deflation of the infinite eigenvalue and
singular structure of the matrix polynomial.
The remaining reduced order staircase form leads to
new types of linearizations which determine the finite eigenvalues and
and corresponding eigenvectors. The new linearizations
also simplify the construction of structure preserving linearizations.