Vadimas Starikovičius

This paper presents and analyzes robust numerical algorithms for solving inverse problems for parabolic equations, specifically focusing on the determination of an unknown time-dependent source function from an integral flux condition. The study is motivated by mathematical models based on Navier-Stokes equations, particularly those exhibiting Poiseuille-type solutions. We employ a variational approach, formulating the inverse problem as the minimization of a Tikhonov regularization cost functional. Discrete approximation schemes are rigorously derived using finite volume methods in space and both backward Euler and Crank-Nicolson schemes in time. A key contribution of this work is the strict justification of the gradient formula for the cost functional by deriving the adjoint problem directly from the fully discrete scheme, rather than discretizing the continuous adjoint problem. This methodology is extended to problems involving fractional powers of elliptic operators and two-dimensional domains. Numerical experiments are conducted to compare the efficiency of Gradient Descent and Conjugate Gradient methods. The results demonstrate that the Conjugate Gradient method significantly outperforms standard gradient descent, maintaining high accuracy and convergence rates even with the inclusion of regularization terms and complex diffusion operators.

We investigate the ill-posedness of the inverse biosensor problem when the biosensor signals are corrupted by noise. To solve the problem, we employ feed-forward and convolutional neural networks. Computational experiments were performed with different levels of additive and multiplicative noises for the batch and flow injection analysis modes of the biosensor. Obtained results show that the largest errors of recovered concentrations are located on the edges of the training domain. We have found that the inverse problem is less ill-posed in the flow injection analysis mode and concentrations can be reliably recovered for higher levels of noise compared to the batch mode. This finding is confirmed by the application of the DIRECT global optimization method to the considered inverse biosensor problem.

Overview of GPU usage while solving different engineering problems, comparison between CPU and GPU computations and overview of the heat conduction problem are provided in this paper. The Jacobi iterative algorithm was implemented by using Python, TensorFlow GPU library and NVIDIA CUDA technology. Numerical experiments were conducted with 6 CPUs and 4 GPUs. The fastest used GPU completed the calculations 19 times faster than the slowest CPU. On average, GPU was from 9 to 11 times faster than CPU. Significant relative speed-up in GPU calculations starts when the matrix contains at least 4002 floating-point numbers.

Summary: In this paper, we develop and investigate the parallel numerical algorithms for three different state‐of‐the‐art numerical methods for solving the non‐local problems described by fractional powers of elliptic operators. These methods transform the non‐local problem into some local differential problems of elliptic or parabolic type. A two‐level parallelization approach is applied to construct the efficient parallel algorithms using the domain decomposition and master‐slave methods, to deal with the increase in computational complexity. We show and compare the serial and parallel solution times that are required to achieve similar accuracy of the solution using different algorithms. Results of extensive convergence tests are presented solving a three‐dimensional test problem with known decrease of the solution's convergence rate depending on the fractional power coefficient. We analyze and discuss the non‐trivial question, which parallel algorithm is recommended to achieve certain accuracy for the given fractional power coefficient.

Mathematical modelling of cardiac haemodynamics presents a great challenge to the computational scientists due to numerous numerical issues and required computational resources. In this paper, we study the parallel performance of 3D simulation software for the blood flow through the aortic valve. The fluid flow problem with the open aortic valve leaflets is formulated and solved in parallel. The choice between the segregated and coupled numerical schemes is discussed and investigated. We present and compare the parallel performance results of both types of parallel solvers. We investigate their strong and weak scalability.

Summary: Mathematical models with fractional‐order differential operators are computationally expensive due to the non‐local nature of these operators. In this work, we construct and investigate parallel solvers for problems described by fractional powers of elliptic operators, like fractional diffusion. Three state‐of‐the‐art approaches are used to transform the non‐local fractional‐order differential problem into local partial differential equation problems formulated in a space of higher dimension. Numerical schemes and parallel algorithms are developed for all three approaches. The resulting parallel algorithms have very different properties. We investigate the weak and strong scalability of the developed parallel algorithms and compare their parallel performance.