Adaptive partitioning techniques for ordinary differential equations

Many stiff systems of ordinary differential equations (ODEs) modeling practical problems can be partitioned into loosely coupled subsystems. In this paper the objective of the partitioning is to permit the numerical integration of one time step to be performed as the solution of a sequence of small subproblems. This reduces the computational complexity compared to solving one large system and permits efficient parallel execution under appropriate conditions. The subsystems are integrated using methods based on low order backward differentiation formulas.This paper presents an adaptive partitioning algorithm based on a classical graph algorithm and techniques for the efficient evaluation of the error introduced by the partitioning.The power of the adaptive partitioning algorithm is demonstrated by a real world example, a variable step-size integration algorithm which solves a system of ODEs originating from chemical reaction kinetics. The computational savings are substantial. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L06, 65Y05     

Acceleration of the EM algorithm using the vector epsilon algorithm

The expectation–maximization (EM) algorithm is a very general and popular iterative computational algorithm to find maximum likelihood estimates from incomplete data and broadly used to statistical analysis with missing data, because of its stability, flexibility and simplicity. However, it is often criticized that the convergence of the EM algorithm is slow. The various algorithms to accelerate the convergence of the EM algorithm have been proposed. The vector ε algorithm of Wynn (Math Comp 16:301–322, 1962) is used to accelerate the convergence of the EM algorithm in Kuroda and Sakakihara (Comput Stat Data Anal 51:1549–1561, 2006). In this paper, we provide the theoretical evaluation of the convergence of the ε-accelerated EM algorithm. The ε-accelerated EM algorithm does not use the information matrix but only uses the sequence of estimates obtained from iterations of the EM algorithm, and thus it keeps the flexibility and simplicity of the EM algorithm.     

Hydrodynamic limit for ∇φ interface model on a wall

 We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational inequality. Received: 10 January 2002 / Revised version: 18 August 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85 Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational inequality     

Statistical algorithms for solving the Cauchy problem for second-order parabolic equations: The “dual” scheme

This paper is a continuation of [A. S. Sipin, “Statistical Algorithms for Solving the Cauchy Problem for Second-Order Parabolic Equations,” Vestn. S.-Peterburg. Univ., Mat. Mekh. Astron., No. 3, 65–74 (2011)]. A new algorithm of the Monte Carlo method for solving the Cauchy problem for a second-order parabolic equation with smooth coefficients is considered. Unbiased estimators for functionals of the solutions of this problem are constructed. Unlike in the paper cited above, the “dual” scheme of constructing unbiased estimators for functionals of the solutions of an integral equation equivalent to the Cauchy problem is considered. This simplifies the modeling procedure, because the boundaries of the spectrum for the matrix of the leading coefficients in the equation are not required to be known.     

On the uniqueness of the solution of dual equation of a singular Sturm–Liouville problem

In this paper we consider differential systems having a singularity and one turning point. First, by a replacement, we transform the system to a linear second-order equation of Sturm–Liouville type with a singularity. Using the infinite product representation of solutions provided in [8], we obtain the dual equation, then we investigate the uniqueness of the solution for the dual equation of the inverse spectral problem of Sturm–Liouville equation. This result is necessary for expressing inverse problem of indefinite Sturm–Liouville equation.     

Some geometric aspects of the nonlinear O(3) sigma-model in the dimension (2+0)

On the basis of the gauge equivalence between an elliptic analog of the O(3) sigma-model and thesinh-Gordon equation proved by the author, connection formulas are obtained. Their interpretation in terms of differential geometry is given. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 20–27. Translated by E. Sh. Gutshabash.     

Second order Runge–Kutta methods for Stratonovich stochastic differential equations

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly. AMS subject classification (2000)  65C30, 65L06, 60H35, 60H10     

Mathematical modeling of demographic processes

The article examines a mathematical model that describes the dynamics of the total population and its age structure. Time-dependent birth and death rates are assumed. The mathematical model is a first-order partial differential equation. The analytical solution makes it possible to determine the age distribution at each time instant depending on the birth and death functions and the initial distribution. The model can be used for demographic planning and forecasting. It has been applied to analyze the demographics of Russia. Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 50–65, 2008.     

Thirty years of G-stability

The 1976 paper of G. Dahlquist, [13], has had a wide-ranging impact on our understanding of numerical methods for the solution of stiff differential equation systems. The present paper surveys some of the work of Dahlquist in this area. It also shows how this has led to contributions by other authors. In particular, the paper contains a review of non-linear stability for Runge–Kutta and general linear methods. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L05, 65L06, 65L20     

An airfoil optimization technique for wind turbines

Optimization algorithms coupled with computational fluid dynamics are used for wind turbines airfoils design. This differs from the traditional aerospace design process since the lift-to-drag ratio is the most important parameter and the angle of attack is large. Computational fluid dynamics simulations are performed with the incompressible Reynolds-averaged Navier–Stokes equations in steady state using a one equation turbulence model. A detailed validation of the simulations is presented and a computational domain larger than suggested in literature is shown to be necessary. Different approaches to parallelization of the computational code are addressed. Single and multiobjective genetic algorithms are employed and artificial neural networks are used as a surrogate model. The use of artificial neural networks is shown to reduce computational time by almost 50%.     

Nonlinear 0–1 programming: II. Dominance relations and algorithms

A nonlinear 0–1 program can be restated as a multilinear 0–1 program, which in turn is known to be equivalent to a linear 0–1 program with generalized covering (g.c.) inequalities. In a companion paper [6] we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0–1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time to an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12–15, and the difference in their performance grows with the number of such terms. Research supported by the National Science Foundation under grant ECS 7902506 and by the Office of Naval Research under contract N00014-75-C-0621 NR 047-048.     

Application of the dual operator method to an equation describing the behavior of a boundary function

The dual operator is an analogue of the conjugate operator in linear theory. In this study the dual operator is applied to a second-order differential equation describing the behavior of the zero-order boundary function in the boundary function method used to derive the asymptotic solution of the singularly perturbed integro-differential plasma-sheath equation. This approach produces is a three-point difference scheme. The results of a numerical solution of the Cauchy problem are reported. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 49–60, 2007.     

Drug Release Kinetics from Biodegradable Polymers via Partial Differential Equations Models

In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9–14, 2008; Lao et al. in Eur. J. Pharm. Biopharm. 70:796–803, 2008). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269–2286, 2011) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.     

Dissipativity of differential equations and the corresponding difference equations

We establish conditions under which the existence of a bounded solution of a difference equation yields the existence of a bounded solution of the corresponding differential equation. We investigate the relationship between the dissipativities of differential and difference equations in terms of Lyapunov functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1249–1256, September, 2006.     

Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution, for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.     

A primal–dual simplex algorithm for bi-objective network flow problems

In this paper we develop a primal–dual simplex algorithm for the bi-objective linear minimum cost network flow problem. This algorithm improves the general primal–dual simplex algorithm for multi-objective linear programs by Ehrgott et al. (J Optim Theory Appl 134:483–497, 2007). We illustrate the algorithm with an example and provide numerical results.     

Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension

We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. In particular we define regularization operators which, combined with the standard interpolators, enable us to prove discrete Poincaré–Friedrichs inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension.     

Robust Dynamics and Control of a Partially Observed Markov Chain

In a seminal paper, Martin Clark (Communications Systems and Random Process Theory, Darlington, 1977, pp. 721–734, 1978) showed how the filtered dynamics giving the optimal estimate of a Markov chain observed in Gaussian noise can be expressed using an ordinary differential equation. These results offer substantial benefits in filtering and in control, often simplifying the analysis and an in some settings providing numerical benefits, see, for example Malcolm et al. (J. Appl. Math. Stoch. Anal., 2007, to appear). Clark’s method uses a gauge transformation and, in effect, solves the Wonham-Zakai equation using variation of constants. In this article, we consider the optimal control of a partially observed Markov chain. This problem is discussed in Elliott et al. (Hidden Markov Models Estimation and Control, Applications of Mathematics Series, vol. 29, 1995). The innovation in our results is that the robust dynamics of Clark are used to compute forward in time dynamics for a simplified adjoint process. A stochastic minimum principle is established.     

An inverse problem for a system with a moving boundary

The problem of estimating the right-hand side of a nonlinear parabolic equation is considered. A finite-step algorithm based on the model positional control method and the finite element method is proposed. The algorithm is robust to informational noise and computational errors. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 23–33.