DENSERKS: Fortran sensitivity solvers using continuous, explicit Runge-Kutta schemes

DENSERKS is a Fortran sensitivity equation solver package designed for integrating models whose evolution can be described by ordinary differential equations (ODEs). A salient feature of DENSERKS is its support for both forward and adjoint sensitivity analyses, with built-in integrators for both first and second order continuous adjoint models. The software implements explicit Runge-Kutta methods with adaptive timestepping and high-order dense output schemes for the forward and the tangent linear model trajectory interpolation. Implementations of six Runge-Kutta methods are provided, with orders of accuracy ranging from two to eight. This makes DENSERKS suitable for a wide range of practical applications. The use of dense output, a novel approach in adjoint sensitivity analysis solvers, allows for a high-order cost-effective interpolation. This is a necessary feature when solving adjoints of nonlinear systems using highly accurate Runge-Kutta methods (order five and above). To minimize memory requirements and make long-time integrations computationally efficient, DENSERKS implements a two-level checkpointing mechanism. The code is tested on a selection of problems illustrating first and second order sensitivity analysis with respect to initial model conditions. The resulting derivative information is also used in a gradient-based optimization algorithm to minimize cost functionals dependent on a given set of model parameters

The linear barycentric rational method for a class of delay Volterra integro-differential equations

A method for solving delay Volterra integro-differential equations is introduced. It is based on two applications of linear barycentric rational interpolation, barycentric rational quadrature and barycentric rational finite differences. Its zero–stability and convergence are studied. Numerical tests demonstrate the excellent agreement of our implementation with the predicted convergence orders

Superconvergant interpolants for the collocation solution of boundary value ordinary differential equations

Publisher's version/PDFA long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h[superscript 2k]), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C[superscript 0] continuous solution approximation that has a global error of O(h[supercript k+1]). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C[superscript 1] continuous "superconvergent" interpolants whose global errors are O(h[superscript 2k]). The key ideas are to use the theoretical framework of continuous Runge-Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge-Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis

A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes

We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results

The Physics of Financial Networks

As the total value of the global financial market outgrew the value of the real economy, financial institutions created a global web of interactions that embodies systemic risks. Understanding these networks requires new theoretical approaches and new tools for quantitative analysis. Statistical physics contributed significantly to this challenge by developing new metrics and models for the study of financial network structure, dynamics, and stability and instability. In this Review, we introduce network representations originating from different financial relationships, including direct interactions such as loans, similarities such as co-ownership and higher-order relations such as contracts involving several parties (for example, credit default swaps) or multilayer connections (possibly extending to the real economy). We then review models of financial contagion capturing the diffusion and impact of shocks across each of these systems. We also discuss different notions of ‘equilibrium’ in economics and statistical physics, and how they lead to maximum entropy ensembles of graphs, providing tools for financial network inference and the identification of early-warning signals of system-wide instabilities

Adaptive step ODE algorithms for the 3D simulation of electric heart activity with graphics processing units

In this paper we studied the implementation and performance of adaptive step methods for large systems of ordinary differential equations systems in graphics processing units, focusing on the simulation of three-dimensional electric cardiac activity. The Rush-Larsen method was applied in all the implemented solvers to improve efficiency. We compared the adaptive methods with the fixed step methods, and we found that the fixed step methods can be faster while the adaptive step methods are better in terms of accuracy and robustness. (c) 2013 Elsevier Ltd. All rights reserved.This work has been partially funded by Universitat Politecnica de Valencia through Programa de Apoyo a la InvestigaciOn y Desarrollo (PAID-06-11) and (PAID-05-12), by Generalitat Valenciana through projects PROMETEO/2009/013 and Ayudas para la realizacion de proyectos de I+D para grupos de investigacion emergentes GV/2012/039, and by Ministerio Espafiol de Economia y Competitividad through project TEC2012-38142-004.García Mollá, VM.; Liberos Mascarell, A.; Vidal Maciá, AM.; Guillem Sánchez, MS.; Millet Roig, J.; González Salvador, A.; Martínez Zaldívar, FJ.... (2014). Adaptive step ODE algorithms for the 3D simulation of electric heart activity with graphics processing units. Computers in Biology and Medicine. 44:15-26. https://doi.org/10.1016/j.compbiomed.2013.10.023S15264