Numerical algorithms for solving the optimal control problem of simple bioreactors

The modified nonlocal feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion–convection–reaction equations. The analysis of the influence of the convection transport is one the aims of this paper. The control loop is defined using the relation, which shows how the amount of the drug produced in the bioreactor and delivered into a human body depends on the substrate concentration specified on the external boundary of the bioreactor. The system of PDEs is solved by using the finite volume and finite difference methods, the control loop parameters are defined from the analysis of stationary linearized equations. The second aim of this paper is to solve the inverse problem and to determine optimal boundary conditions. These results enable us to estimate the potential accuracy of the proposed devices. &nbsp

A Three-Level Parallelisation Scheme and Application to the Nelder-Mead Algorithm

We consider a three-level parallelisation scheme. The second and third levels define a classical two-level parallelisation scheme and some load balancing algorithm is used to distribute tasks among processes. It is well-known that for many applications the efficiency of parallel algorithms of the second and third level starts to drop down after some critical parallelisation degree is reached. This weakness of the two-level template is addressed by introduction of one additional parallelisation level. As an alternative to the basic solver some new or modified algorithms are considered on this level. The idea of the proposed methodology is to increase the parallelisation degree by using less efficient algorithms in comparison with the basic solver. As an example we investigate two modified Nelder-Mead methods. For the selected application, a few partial differential equations are solved numerically on the second level, and on the third level the parallel Wang's algorithm is used to solve systems of linear equations with tridiagonal matrices. A greedy workload balancing heuristic is proposed, which is oriented to the case of a large number of available processors. The complexity estimates of the computational tasks are model-based, i.e. they use empirical computational data

Numerical simulation of nonlocal delayed feedback controller for simple bioreactors

The new nonlocal delayed feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion-reaction equations. The control loop defines the changes of the substrate concentration delivered into the bioreactor at the external boundary of the bioreactor depending on the difference of measurements of the produced drug delivered into the body and the flux of the drug prescribed by a doctor in accordance with the therapeutic protocol. The system of PDEs is solved by using the finite difference method, the control loop parameters are defined from the analysis of stationary linearized equations. The stability of the algorithm for the inverse boundary condition is investigated. Results of computational experiments are presented and analysed

Editor's letter

„Editor's letter" Mathematical Modelling and Analysis, 14(1), p. 13

The robust finite-volume schemes for modeling nonclassical surface reactions

A coupled system of nonlinear parabolic PDEs arising in modeling of surface reactions with piecewise continuous kinetic data is studied. The nonclassic conjugation conditions are used at the surface of the discontinuity of the kinetic data. The finite-volume technique and the backward Euler method are used to approximate the given mathematical model. The monotonicity, conservativity, positivity of the approximations are investigated by applying these finite-volume schemes for simplified subproblems, which inherit main new nonstandard features of the full mathematical model. Some results of numerical experiments are discussed

The robust finite volume schemes for modeling non-classical surface reactions

A coupled system of nonlinear parabolic PDEs arising in modeling of surface reactions with piecewise continuous kinetic data is studied. The nonclassic conjugation conditions are used at the surface of the discontinuity of the kinetic data. The finite-volume technique and the backward Euler method are used to approximate the given mathematical model. The monotonicity, conservativity, positivity of the approximations are investigated by applying these finite-volume schemes for simplified subproblems, which inherit main new nonstandard features of the full mathematical model. Some results of numerical experiments are discussed

Compact high order finite difference schemes for linear Schrödinger problems on non-uniform meshes

In the present paper a general technique is developed for construction of compact high-order finite difference schemes to approximate Schrödinger problems on nonuniform meshes. Conservation of the finite difference schemes is investigated. Discrete transparent boundary conditions are constructed for the given high-order finite difference scheme. The same technique is applied to construct compact high-order approximations of the Robin and Szeftel type boundary conditions. Results of computational experiments are presente

Parallel algorithms for three-dimensional parabolic and pseudoparabolic problems with different boundary conditions

In this paper, three-dimensional parabolic and pseudo-parabolic equations with classical, periodic and nonlocal boundary conditions are approximated by the full approximation backward Euler method, locally one dimensional and Douglas ADI splitting schemes. The stability with respect to initial conditions is investigated. We note that the stability of the proposed numerical algorithms can be proved only if the matrix of discrete operator can be diagonalized and eigenvectors make a complete basis system.Parallel versions of all algorithms are constructed and scalability analysis is done. It is shown that discrete one-dimensional problems with periodic and nonlocal boundary conditions can be efficiently solved with similar modifications of the parallel Wang algorithm