41 research outputs found
Energy-preserving splitting integrators for sampling from Gaussian distributions with Hamiltonian Monte Carlo method
The diffusive behaviour of simple random-walk proposals of many Markov Chain
Monte Carlo (MCMC) algorithms results in slow exploration of the state space
making inefficient the convergence to a target distribution. Hamiltonian/Hybrid
Monte Carlo (HMC), by introducing fictious momentum variables, adopts
Hamiltonian dynamics, rather than a probability distribution, to propose future
states in the Markov chain. Splitting schemes are numerical integrators for
Hamiltonian problems that may advantageously replace the St\"ormer-Verlet
method within HMC methodology. In this paper a family of stable methods for
univariate and multivariate Gaussian distributions, taken as guide-problems for
more realistic situations, is proposed. Differently from similar methods
proposed in the recent literature, the considered schemes are featured by null
expectation of the random variable representing the energy error. The
effectiveness of the novel procedures is shown for bivariate and multivariate
test cases taken from the literature
Splitting and composition methods for explicit time dependence in separable dynamical systems
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On the dynamics of a generalized predator-prey system with Z-type control.
We apply the Z-control approach to a generalized predator prey system and consider the specific case of indirect control of the prey population. We derive the associated Z-controlled model and investigate its properties from the point of view of the dynamical systems theory. The key role of the design parameter A. for the successful application of the method is stressed and related to specific dynamical properties of the Z-controlled model. Critical values of the design parameter are also found, delimiting the lambda-range for the effectiveness of the Z-method. Analytical results are then numerically validated by the means of two ecological models: the classical Lotka-Volterra model and a model related to a case study of the wolf wild boar dynamics in the Alta Murgia National Park. Investigations on these models also highlight how the Z-control method acts in respect to different dynamical regimes of the uncontrolled model. (C) 2016 The Authors. Published by Elsevier Inc
Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges
The articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI Mathematics journal are here collected [...
IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics
We examine spatially explicit models described by reaction-diffusion partial differential equations for the study of predatorâprey population dynamics. The numerical methods we propose are based on the coupling of a finite difference/element spatial discretization and a suitable partitioned RungeâKutta scheme for the approximation in time. The RK scheme here implemented uses an implicit scheme for the stiff diffusive term and a partitioned RK symplectic scheme for the reaction term (IMSP scheme). We revisit some results provided in literature for the classical LotkaâVolterra system and the RosenzweigâMacArthur model. We then extend the approach to metapopulation dynamics in order to numerically investigate the effect of migration through a corridor connecting two habitat patches. Moreover, we analyze the synchronization properties of subpopulation dynamics, when the migration occurs through corridors of variable size
Non-Standard Discrete RothC Models for Soil Carbon Dynamics
Soil Organic Carbon (SOC) is one of the key indicators of land degradation. SOC positively affects soil functions with regard to habitats, biological diversity and soil fertility; therefore, a reduction in the SOC stock of soil results in degradation, and it may also have potential negative effects on soil-derived ecosystem services. Dynamical models, such as the Rothamsted Carbon (RothC) model, may predict the long-term behaviour of soil carbon content and may suggest optimal land use patterns suitable for the achievement of land degradation neutrality as measured in terms of the SOC indicator. In this paper, we compared continuous and discrete versions of the RothC model, especially to achieve long-term solutions. The original discrete formulation of the RothC model was then compared with a novel non-standard integrator that represents an alternative to the exponential RosenbrockâEuler approach in the literature
âA Fortran90 routine for the solution of orthogonal differential problemsâ
In this paper we describe a Fortran90 routine for the numerical integration of orthogonal differential systems based on the Cayley transform methods. Three different implementations of the methods are given: with restart, with restart at each step and in composed form. Numerical tests will show the performances of the solver for the solution of orthogonal test problems, of orthogonal rectangular problems and for the calculation of Lyapunov exponents in the linear and nonlinear cases, and finally for solving inverse eigenvalue problem for Toeplitz matrices. The results obtained using Cayley methods are compared with those given by Fortran90 version of Munthe-Kaas methods, which have been coded in a similar way
Gradient flow methods for matrix completion with prescribed eigenvalues
AbstractMatrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. The challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable. An intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix. Constructive proofs exist to a certain point (and those proofs, such as the classical SchurâHorn theorem, are amazingly elegant enough in their own right) beyond which very few theories or numerical algorithms are available. In this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries. The gradient flow is proposed as a numerical means to tackle the construction. This approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities
Steady-state invariance in high-order Runge-Kutta discretization of optimal growth models
This work deals with infinite horizon optimal growth models and uses the results in one of
Mercenier and Michel (1994) papers as a starting point. Mercenier and Michel (1994)
provided a one-stage RungeâKutta discretization of the above-mentioned models which
preserves the steady state of the theoretical solution. They call this feature the ââsteadystate
invariance propertyââ. We generalize the result of their study by considering
discrete models arising from the adoption of s-stage RungeâKutta schemes. We show
that the steady-state invariance property requires two different RungeâKutta schemes
for approximating the state variables and the exponential term in the objective
function. This kind of discretization is well-known in literature as a partitioned
symplectic RungeâKutta scheme. Its main consequence is that it is possible to rely on the
well-stated theory of order for considering more accurate methods which generalize the
first order Mercenier and Michel algorithm. Numerical examples show the efficiency and
accuracy of the proposed methods up to the fourth order, when applied to test models
Corridor size variation in spatially explicit/implicit models
We propose spatially implicit models described by ordinary differential equations which inherit the information of spatial explicit metapopulation models described by reaction-diffusion partial differential equations. Numerical simulations confirm that the proposed implicit models can capture the qualitative features of the exlicit ones and may reveal as an effective tool to extract predictive information thorough a further theoretical analysis