Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family of schemes with similar convergence rates. The ODE interpretation also suggests restarting Nesterov's scheme leading to an algorithm, which can be rigorously proven to converge at a linear rate whenever the objective is strongly convex.

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A differential equation for modeling Nesterov's accelerated gradient method: theory and insights

動注化学療法における pharmacokinetics と臨床

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the behavior of such systems can be obtained by energy arguments. The claim would be that there exists a critical temperature Tc, such that for T > Tc the free energy F (c, T ) of the system is a single welled function of the concentration c of one of the species, whereas for T < Tc the free energy is double welled. Referring to Figure 1, a system which was spatially uniform at temperature T1, when cooled to temperature T2, would find it energetically preferrable to separate itself into two systems, one at concentration cA and one at concentration cB. To be more specific consider now the system at temperature T < Tc. Assume that the free energy F (c) per unit volume (Gibbs free energy or Landau-Ginzburg free energy) of the spatially homogeneous system has the convex/concave shape indicated in Figure 2. More precisely F is concave in the spinodal interval cA < c < c s B and convex elsewhere. The points cA and cB where the supporting tangent touches the graph are sometimes referred to as the binodal points. The derivative f(c) = F (c) is depicted in Figure 3. The free energy of a spatially heterogeneous system would then be given by

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The cahn-hilliard equation

Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions

We present an ordinary differential equations approach to solve general smooth minimization problems including a convergence analysis. Generically often the procedure ends up at a point which fulfills sufficient conditions for a local minimum. This procedure will then be rewritten in the concept of differential algebraic equations which opens the route to an efficient implementation. Furthermore, we link this approach with the classical SQP-approach and apply both techniques onto two examples relevant in applications.

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A dynamical systems approach to constrained minimization

Using dynamical systems methods to solve minimization problems

In pharmacokinetics/pharmacodynamics (PKPD) the measured response is often delayed relative to drug administration, individuals in a population have a certain lifespan until they maturate or the change of biomarkers does not immediately affects the primary endpoint. The classical approach in PKPD is to apply transit compartment models (TCM) based on ordinary differential equations to handle such delays. However, an alternative approach to deal with delays are delay differential equations (DDE). DDEs feature additional flexibility and properties, realize more complex dynamics and can complementary be used together with TCMs. We introduce several delay based PKPD models and investigate mathematical properties of general DDE based models, which serve as subunits in order to build larger PKPD models. Finally, we review current PKPD software with respect to the implementation of DDEs for PKPD analysis.

Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations

We use the qualitative properties of the solution flow of the gradient equation 
$\dot{x} = - \nabla f(x)$
 to compute a local minimum of a real-valued function 
$f$
 . Under the regularity assumption of all equilibria we show a convergence result for bounded trajectories of a consistent, strictly stable linear multistep method applied to the gradient equation. Moreover, we compare the asymptotic features of the numerical and the exact solutions as done by Humphries, Stuart (1994) and Schropp (1995) for one-step methods. In the case of 
$A(\alpha )$
 -stable formulae this leads to an efficient solver for stiff minimization problems.

Johannes Schropp

Papers

Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions

A dynamical systems approach to constrained minimization

Using dynamical systems methods to solve minimization problems

Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations

A note on minimization problems and multistep methods