127,088 research outputs found
Adaptive rational Krylov methods for exponential Runge--Kutta integrators
We consider the solution of large stiff systems of ordinary differential
equations with explicit exponential Runge--Kutta integrators. These problems
arise from semi-discretized semi-linear parabolic partial differential
equations on continuous domains or on inherently discrete graph domains. A
series of results reduces the requirement of computing linear combinations of
-functions in exponential integrators to the approximation of the
action of a smaller number of matrix exponentials on certain vectors.
State-of-the-art computational methods use polynomial Krylov subspaces of
adaptive size for this task. They have the drawback that the required Krylov
subspace iteration numbers to obtain a desired tolerance increase drastically
with the spectral radius of the discrete linear differential operator, e.g.,
the problem size. We present an approach that leverages rational Krylov
subspace methods promising superior approximation qualities. We prove a novel
a-posteriori error estimate of rational Krylov approximations to the action of
the matrix exponential on vectors for single time points, which allows for an
adaptive approach similar to existing polynomial Krylov techniques. We discuss
pole selection and the efficient solution of the arising sequences of shifted
linear systems by direct and preconditioned iterative solvers. Numerical
experiments show that our method outperforms the state of the art for
sufficiently large spectral radii of the discrete linear differential
operators. The key to this are approximately constant rational Krylov iteration
numbers, which enable a near-linear scaling of the runtime with respect to the
problem size
ON THE RELATION BETWEEN GLOBAL PROPERTIES OF LINEAR DIFFERENCE AND DIFFERENTIAL-EQUATIONS WITH POLYNOMIAL COEFFICIENTS .1.
AbstractThis paper is concerned with applications of the Mellin transformation in the study of homogeneous linear differential and difference equations with polynomial coefficients. We begin by considering a differential equation (D) with regular singularities at O and ∞ and arbitrary singularities in the rest of the complex plane, and the difference equation (Δ′) obtained from (D) by a variant of the formal Mellin transformation. We define fundamental systems of solutions of (Δ′), analytic in either a right or a left half plane. by the use of Mellin transforms of microsolutions of (D). The relations between these fundamental systems are expressed in terms of central connection matrices of (D). Second, we study the differential equation (D1) obtained from (D) by means of a formal Laplace transformation and the difference equation (Δ1) obtained from (D1) by a formal Mellin transformation. We use Mellin transforms of "ordinary" solutions of (D1) with moderate growth at ∞ to construct fundamental systems of solutions of (Δ1). The relation between these fundamental systems involves certain Stokes multipliers and a formal monodromy matrix of (D1)
Solvability of the Hamiltonians related to exceptional root spaces: rational case
Solvability of the rational quantum integrable systems related to exceptional
root spaces is re-examined and for is established in the
framework of a unified approach. It is shown the Hamiltonians take algebraic
form being written in a certain Weyl-invariant variables. It is demonstrated
that for each Hamiltonian the finite-dimensional invariant subspaces are made
from polynomials and they form an infinite flag. A notion of minimal flag is
introduced and minimal flag for each Hamiltonian is found. Corresponding
eigenvalues are calculated explicitly while the eigenfunctions can be computed
by pure linear algebra means for {\it arbitrary} values of the coupling
constants. The Hamiltonian of each model can be expressed in the algebraic form
as a second degree polynomial in the generators of some infinite-dimensional
but finitely-generated Lie algebra of differential operators, taken in a
finite-dimensional representation.Comment: 51 pages, LaTeX, few equations added, one reference added, typos
correcte
The Spruce Budworm and Forest: A Qualitative Comparison of ODE and Boolean Models
Boolean and polynomial models of biological systems have emerged recently as viable companions to differential equations models. It is not immediately clear however whether such models are capable of capturing the multi-stable behaviour of certain biological systems: this behaviour is often sensitive to changes in the values of the model parameters, while Boolean and polynomial models are qualitative in nature. In the past few years, Boolean models of gene regulatory systems have been shown to capture multi-stability at the molecular level, confirming that such models can be used to obtain information about the system’s qualitative dynamics when precise information regarding its parameters may not be available. In this paper, we examine Boolean approximations of a classical ODE model of budworm outbreaks in a forest and show that these models exhibit a qualitative behaviour consistent with that derived from the ODE models. In particular, we demonstrate that these models can capture the bistable nature of insect population outbreaks, thus showing that Boolean models can be successfully utilized beyond the molecular level
Numerical Solution of Partial Differential Equations Using Polynomial Particular Solutions
Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning.
The derived polynomial particular solutions are also applied in the localized method of particular solutions to solve large-scale problems. Many numerical experiments have been performed to show the effectiveness of the particular solutions on this algorithm.
As another part of the dissertation, a modified method of particular solutions (MPS) has been used for solving nonlinear Poisson-type problems defined on different geometries. Polyharmonic splines are used as the basis functions so that no shape parameter is needed in the solution process. The MPS is also applied to compute the sizes of critical domains of different shapes for a quenching problem. These sizes are compared with the sizes of critical domains obtained from some other numerical methods. Numerical examples are presented to show the efficiency and accuracy of the method
Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems
In this paper, we provide a system-theoretic treatment of certain
continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor
algebra. In particular, if a system of homogeneous polynomial differential
equations can be represented by an orthogonally decomposable (odeco) tensor, we
can construct its explicit solution by exploiting tensor Z-eigenvalues and
Z-eigenvectors. We refer to such HPDS as odeco HPDS. By utilizing the form of
the explicit solution, we are able to discuss the stability properties of an
odeco HPDS. We illustrate that the Z-eigenvalues of the corresponding dynamic
tensor can be used to establish necessary and sufficient stability conditions,
similar to these from linear systems theory. In addition, we are able to obtain
the complete solution to an odeco HPDS with constant control. Finally, we
establish results which enable one to determine if a general HPDS can be
transformed to or approximated by an odeco HPDS, where the previous results can
be applied. We demonstrate our framework with simulated and real-world
examples.Comment: 8 pages, 4 figure
- …