Comparative analysis of numerical methods for determining parameters of chemical reactions from experimental data

The aim of this paper is to select an optimal numerical method for determining the parameters of chemical reactions. The importance of the topic is due to the modern needs of industry, such as the improvement of chemical reactors and oil or gas processing. The paper deals with the problem of determining reaction rate constants using gradient methods and stochastic optimization algorithms. To solve an forward problem, implicit methods for solving stiff ODE systems are used. A correlation method of practical identifiability of the required parameters is used. The genetic algorithm, particle swarm method, and fast annealing method are implemented to solve an inverse problem. The gradient method for the solution of the inverse problem is implemented, and a formula for gradient of the functional is given with the corresponding adjoint problem. We apply an identifiability analysis of the unknown coefficients and arrange the coefficients in the order of their identifiability. We show that the best approach is to apply global optimization methods to find the interval of global solution and after that we refine inverse problem solution using gradient approach.

On the modeling of ultrasound wave propagation in the frame of inverse problem solution

In this paper we consider the inverse problem of detecting the inclusions inside the human tissue by using the acoustic sounding wave. The problem is considered in the form of coefficient inverse problem for first-order system of PDE and we use the gradient descent approach to recover the coefficients of that system. The important part of the sceme is the solution of the direct and adjoint problem on each iteration of the descent. We consider two finite-volume methods of solving the direct problem and study their Influence on the performance of recovering the coefficients.

Estimation of transient boundary heat flux by using hybrid algorithm in a two dimensional laminar duct flow

Conjugate gradient method (CGM) with adjoint problem is a popular optimization algorithm in solving the inverse heat transfer problems. It starts with the initial guess solution of unknown parameters to be estimated which would be updated in an iterative procedure. The initial guess solution is one of the significant factors for the accuracy of estimation. In the current study, the Jaya algorithm has been developed to provide the initial guess solution to CGM. The resultant algorithm is called hybrid algorithm. The test problem considered here for the study is the estimation of transient boundary heat flux for two-dimensional hydrodynamically developed and thermally developing forced convective laminar duct flow. The hybrid algorithm is found to be robust and accurate than CGM.

A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

SIMULTANEOUS DETERMINATION OF A SOURCE TERM AND DIFFUSION CONCENTRATION FOR A MULTI-TERM SPACE-TIME FRACTIONAL DIFFUSION EQUATION

An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

Numerical Experiments to Identify Suspended Matter Flow Rate over the Seabed in a Model of Impurity Transport

The paper deals with variation assimilation of model data on the concentration of suspended matter in the upper layer of the Sea of Azov. Such information is used during practical evaluation of identification algorithms to test the assimilation of concentration values derived from satellite information. Combined use of surface concentration estimates and modeling results based on the transport model is of interest for determining the strength of sources of suspended matter inflow. The test problem has been solved of determination of the required parameter in the sea bottom boundary condition when parameterizing the sediment inflow (agitation) from bottom sediments due to dynamic processes in the sea bottom layer. Two approaches to search for the required constant for the parameterization used in the calculations are implemented. A variational identification algorithm based on adjoint problem solving is used in determining the spatially variable flow of suspended matter on the seabed. The assimilation of measurement data into a model of passive admixture transport allows to determine the spatial structure of such flows at a given time interval. When implementing the variational identification algorithm, gradient methods are used to find optimal estimates by minimizing the quadratic functional of the prognosis quality. The solution of the adjoint problem is used to construct the gradient of the prognosis quality functional. Descent is performed in the direction of this gradient. During realization of variational procedure the main problem, the adjoint problem and the problem in variations, which is necessary to determine an iteration parameter, are solved. The flow fields and turbulent diffusion coefficients used in the calculations were obtained using a dynamic model of the Sea of Azov in sigma coordinates under exposure to intense easterly wind.

THE PROBLEM OF THE COMBINED USE OF FILTRATION THEORY AND MACHINE LEARNING ELEMENTS FOR SOLVING THE INVERSE PROBLEM OF RESTORING THE HYDRAULIC CONDUCTIVITY OF AN OIL FIELD

This article presents the methodology involving the combined use of machine learning elements and a physically meaningful filtration model. The authors propose using a network of radial basis functions for solving the problem of restoring hydraulic conductivity in the interwell space for an oil field. The advantage of the proposed approach in comparison with classical interpolation methods as applied to the problems of reconstructing the filtration-capacitive properties of the interwell space is shown. The paper considers an algorithm for the interaction of machine learning methods, a filtration model, a mechanism for separating input data, a form of a general objective function, which includes physical and expert constraints. The research was carried out on the example of a symmetrical element of an oil field. The proposed procedure for finding a solution includes solving a direct and an adjoint problem.

Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

In this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.