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Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models.
We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model.
The authors propose a recycling MINRES scheme for a solution of subsequent self-adjoint linear systems as appearing, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with a preconditioner and arbitrary inner products. Numerical experiments with nonlinear Schrödinger equations indicate a substantial decrease in computation time when recycling is used.
We consider a shape implant design problem that arises in the context of facial surgery. We introduce a reformulation as an optimal control problem, where the control acts as a boundary force. The state is modelled as a minimizer of a polyconvex hyperelastic energy functional. We show existence of optimal solutions and derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results are presented.
The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some
problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are
sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schroedinger and the drift-diffusion equation
but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.
Flux variability analysis (FVA) is an important tool to further analyze the results obtained by flux balance analysis (FBA) on genome-scale metabolic networks. Standard FVA may predict unbounded fluxes through some reactions in the network even if the nutrient uptake rate is bounded. These fluxes violate the second law of thermodynamics. They may be eliminated by extending flux variability analysis with thermodynamic constraints.
We present a new algorithm for efficient flux variability (and flux balance) analysis with thermodynamic constraints, suitable for analyzing genome-scale metabolic networks. We first show that flux balance analysis with thermodynamic constraints is NP-hard. Then we derive a theoretical tractability result, which can be applied to metabolic networks in practice. We use this result to develop a new constraint programming algorithm Fast-tFVA for fast flux variability analysis with thermodynamic constraints (tFVA). Computational comparisons with previous methods demonstrate the efficiency of the new method. For tFVA, a speed-up of factor 30-300 is achieved.
In an analysis of genome-scale metabolic networks in the BioModels database, we found that in 485 out of 716 networks additional irreversible or fixed reactions could be detected.
This article describes Fortran 77 subroutines for computing eigenvalues and invariant subspaces
of Hamiltonian and skew-Hamiltonian matrices. The implemented algorithms are based on orthogonal
symplectic decompositions, implying numerical backward stability as well as symmetry
preservation for the computed eigenvalues. These algorithms are supplemented with balancing and
block algorithms, which can lead to considerable accuracy and performance improvements. As a
by-product, an efficient implementation for computing symplectic QR decompositions is provided.
We demonstrate the usefulness of the subroutines for several, practically relevant examples.
Stewart's recently introduced Krylov-Schur algorithm
is a modification of the implicitly restarted Arnoldi algorithm which
employs reordered Schur decompositions to perform restarts and de-
ations in a numerically reliable manner. This paper describes a variant
of the Krylov-Schur algorithm suitable for addressing eigenvalue
problems associated with products of large and sparse matrices. It
performs restarts and de
ations via reordered periodic Schur decompositions
and, by taking the product structure into account, it is
capable to achieve qualitatively better approximations to the eigenvalues
of small magnitude.
The role of larger bulges in the QR algorithm is controversial. Large bulges are infamous
for having a strong, negative influence on the convergence of the implicit shifted QR algorithm.
This paper provides a new explanation of this shift blurring effect by connecting the computation of
the first column of the shift polynomial to the notoriously ill-conditioned pole placement problem.
To avoid shift blurring, modern variants of the QR algorithm employ chains of tightly coupled tiny
bulges instead of one large bulge. It turns out that larger bulges still play a positive role in these
variants; a slight increase of the bulge sizes often results in considerable performance improvements.
We investigate the condition number for a complex eigenvalue of a real matrix under
real perturbations. Based on an explicit formula, it is shown that this number is never
smaller than 1/
p
2 times the corresponding condition number with respect to complex
perturbations. This result can be generalized to the condition number of an arbitrary
complex-valued function under real perturbations. This extends to related condition
numbers.
We give an algorithm to compute N steps of a convolution quadrature approximation
to a continuous temporal convolution using only O(N logN) multiplications and O(logN) active
memory. The method does not require evaluations of the convolution kernel, but instead O(logN)
evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the
stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integrodifferential
equations of convolution type. In a numerical example we apply it to solve a subdiffusion
equation with transparent boundary conditions.
In this paper we propose a new finite element realization of the Perfectly Matched
Layer method (PML-method). Our approach allows to deal with a wide class of
polygonal domains and with certain types of inhomogeneous exterior domains.
Among the covered inhomogeneities are open waveguide structures playing an essential
role in integrated optics. We give a detailed insight into implementation
aspects. Numerical examples show exponential convergence behavior to the exact
solution with the thickness of the PML sponge layer.
Lyapunov and exponential dichotomy spectral theory is extended
from ordinary differential equations (ODEs) to nonautonomous
differential-algebraic equations (DAEs). By using orthogonal
changes of variables, the original DAE system is transformed into
appropriate condensed forms, for which concepts such as Lyapunov
exponents, Bohl exponents, exponential dichotomy and spectral
intervals of various kinds can be analyzed via the resulting
underlying ODE. Some essential differences between the spectral
theory for ODEs and that for DAEs are pointed out. Numerical
methods for computing the spectral intervals associated with
Lyapunov and Sacker-Sell (exponential dichotomy) spectra are
derived by modifying and extending those methods proposed for ODEs. Perturbation theory and error analysis are discussed, as
well. Finally, some numerical examples are presented to illustrate
the theoretical results and the properties of the numerical
methods.
A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework
(2007)
We give a mathematically rigorous definition of a grid for algorithms solving
partial differential equations. Unlike previous approaches, our grids have a
hierarchical structure. This makes them suitable for geometric multigrid
algorithms and hierarchical local grid refinement. The description is also
general enough to include geometrically nonconforming grids. The definitions
in this article serve as the basis for an implementation of an abstract grid
interface as C++ classes in the DUNE.
In a companion paper [Matheon-Preprint 403] we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this
definition we derive an efficient interface specification as a set of C++ classes.
This interface separates the applications from the grid data structures.
Thus, user implementations become independent of the underlying grid
implementation. Modern C++ template techniques are used to provide an
interface implementation without big performance losses.
The implementation is realized as part of the
software environment DUNE.
Numerical tests demonstrate the flexibility and the efficiency of our approach.
In many applications such as data compression, imaging or
genomic data analysis,
it is important to approximate a given $m\times n$ matrix $A$
by a matrix $B$ of rank at most $k$ which is much smaller than $m$ and $n$.
The best rank $k$ approximation can be determined via
the singular value decomposition
which, however, has prohibitively
high computational complexity and storage requirements
for very large $m$ and $n$.
We present an optimal least squares algorithm for computing a rank $k$
approximation to an $m\times n$ matrix $A$ by reading
only a limited number of rows and columns of $A$.
The algorithm has complexity $\mathcal O(k^2\max(m,n))$ and
allows to iteratively improve given rank $k$
approximations by reading additional rows and
columns of $A$. We also show how this approach can be extended
to tensors and present numerical results.
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 different characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems. We show that if the spectral projector onto the right finite deflating subspace of the matrix pair $(E,A)$ is non-negative, then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case. Furthermore, we provide sufficient conditions that guarantee entry-wise non-negativity along with positive semi-definiteness of solutions of generalised projected Lyapunov equations. As an application of the framework established throughout this paper, we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case.
Adjoint Broyden a la GMRES
(2007)
It is shown here that a compact storage implementation of a quasi-Newton
method based on the adjoint Broyden update reduces in the affine
case exactly to the well established GMRES procedure. Generally,
storage and linear algebra effort per step are small multiples of $n\cdot k$,
where $n$ is the number of variables and $k$ the number of steps taken
in the current cycle. In the affine case the storage is exactly $(n+k)\cdot k$
and in the nonlinear case the same bound can be achieved if adjoints,
i.e. transposed Jacobian-vector products are available. A
transposed-free variant that relies exclusively on Jacobian-vector
products (or possibly their approximation by divided differences)
requires roughly twice the storage and turns out to be somewhat slower
in our numerical experiments reported at the end.
Structured eigenvalue conditioning and backward error of a class of polynomial eigenvalue problems
(2007)
Characterisations of simple eigenvalues of complex matrix polynomials with *-even/odd and *-palindromic/antipalindromic structures that have the same normwise condition number with respect to structure preserving and arbitrary perturbations are obtained. Here * denotes either the transpose T or the conjugate tranpose *. In the process we obtain formulae for the normwise structured condition number of simple eigenvalues of T-palindromic/antipalindromic and *-even/odd polynomials. Moreover, conditions under which the normwise structured backward error of approximate eigenvalues of such polynomials is equal to the unstructured error are also derived. These lead to complete characterisations of approximate eigenvalues that have the same structured and unstructured backward errors for the *-even/odd and T-even/odd polynomials.