Valentin Shinkarev

The paper discusses the construction of parallel algorithms and their software implementation for solving ill-posed linear inverse problems. Such problems often reduce to solving large overdetermined systems of linear algebraic equations (SLAEs) with dense matrices. These inverse problems are addressed using regularizing algorithms, a key step of which involves selecting the regularization parameter. In computationally challenging multidimensional inverse problems, the accuracy of the regularized solution can be critically affected by round-off errors accumulated during the computation. Incorporating an estimate of these accumulated errors into the procedure for selecting the regularization parameter can enhance the accuracy of the resulting solution. However, it may also significantly limit the scalability of the corresponding parallel software implementation. This work compares the capabilities of various standards of the Message Passing Interface parallel programming technology to solve the problem of partial compensation for additional computing and communication costs associated with accounting for round-off errors. Examples of the proposed algorithm’s implementation are provided using the Python programming language and the mpi4py package, with their structure designed for straightforward adaptation to C/C++/Fortran. В работе обсуждаются особенности построения параллельных алгоритмов и их программной реализации для решения некорректно поставленных линейных обратных задач, которые сводятся к необходимости решения больших переопределенных систем линейных алгебраических уравнений с плотно заполненной матрицей. Такие обратные задачи решаются с использованием регуляризирующих алгоритмов, одним из этапов реализации которых является выбор параметра регуляризации. В вычислительно сложных многомерных обратных задачах на точность регуляризованного решения могут оказывать критическое влияние ошибки машинного округления, накапливающиеся в процессе счета. Включение оценки накопленных ошибок машинного округления в процедуру выбора параметра регуляризации позволяет повысить точность полученного регуляризованного решения, однако может существенно ограничить масштабируемость соответствующей параллельной программной реализации алгоритма. В работе сравниваются возможности различных стандартов технологии параллельного программирования Message Passing Interface по решению проблемы частичной компенсации дополнительных вычислительных и коммуникационных затрат, связанных с учетом ошибок машинного округления. Приводятся примеры реализации предложенного алгоритма на языке программирования Python с использованием пакета mpi4py, структура которых ориентирована на простую адаптацию для языков C/C++/Fortran.

The paper proposes an algorithm for parallelizing calculations that arise when using highly optimized minimization functions available in many computing packages. The main idea of the proposed algorithm is based on the fact that although the ``inner workings'' of the minimization function used may not be known to the user, it inevitably uses in its work auxiliary functions that implement the calculation of the minimized functional and its gradient, which are usually realized by the user and in many cases can be parallelized. The paper discusses in detail both the parallelization algorithm and its software implementation using MPI parallel programming technology, which can act as template for parallelizing a wide set of applied minimization problems. An example of software implementation of the proposed algorithm is demonstrated using the Python programming language, but can be easily rewritten using the C/C++/Fortran programming languages.

The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of these approaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.

We consider an algorithm for monitoring the process of sputter deposition of optical coatings based on the analysis of sample data of broadband measurements for a fixed set of wavelengths. On the one hand, the considered algorithm uses broadband measurement data, which is an advantage compared to using only monochromatic measurement data for one fixed wavelength, and on the other hand, it is fast compared to algorithms that use complete broadband measurement data. It has been demonstrated that the proposed algorithm allows one to obtain fairly accurate estimates of the layer thicknesses in deposited multilayer optical coatings.

This paper discusses a method for taking into account rounding errors when constructing a stopping criterion for the iterative process in gradient minimization methods. The main aim of this work was to develop methods for improving the quality of the solutions for real applied minimization problems, which require significant amounts of calculations and, as a result, can be sensitive to the accumulation of rounding errors. However, this paper demonstrates that the developed approach can also be useful in solving computationally small problems. The main ideas of this work are demonstrated using one of the possible implementations of the conjugate gradient method for solving an overdetermined system of linear algebraic equations with a dense matrix.