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COMMON PART

Project Number21-71-00039

Project titleDevelopment of new numerical methods and software for the wave propagation modeling in inhomogeneous unbounded domains based on modern optimization methods

Project LeadLytaev Mikhail

AffiliationSt. Petersburg Federal Research Center of the Russian Academy of Sciences,

Implementation period | 07.2021 - 06.2023 |

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-218 - Mathematical simulation of physical phenomena

Keywordsfinite-difference methods, Helmholtz equation, optimization of numerical schemes, stochastic optimization, pseudo-differential operators, radio channel modelling, diffraction, underwater acoustics, radio physics, discrete dispersion analysis

PROJECT CONTENT

Annotation

The research is aimed at improving the efficiency of computer simulation methods for wave propagation in inhomogeneous unbounded regions. Such problems arise in computational hydroacoustics, tropospheric radio wave propagation, geophysics, optics, and quantum mechanics. Despite the different nature of the physical phenomena occurring in these areas, the underlying mathematical models are to a certain extent universal. The main attention in this study will be paid to the radio wave propagation near the Earth's surface and the propagation of acoustic waves in an inhomogeneous underwater environment.
The relevance of computer modeling of the radio wave propagation is dictated by the constant appearance of new wireless radio communication systems, the growth of their density, and the expansion of the radio frequency spectrum used. It is necessary to simultaneously take into account the irregular terrain, urban development, vegetation, and inhomogeneities of the refractive index of the troposphere. Computational underwater acoustics is required for remote sensing of the ocean environment, underwater navigation and communications, acoustic noise monitoring, and bioacoustics. When modeling the propagation of the acoustic waves, it is necessary to take into account spatial variations of the sound speed and density, inhomogeneities of the bottom relief, and the rough sea surface.
There is a wide variety of wave propagation effects, such as interference, diffraction, refraction, scattering, multiple re-reflection, etc. A distinctive feature is also the huge size of the computational domain compared to the wavelength. Often, these tasks need to be solved in real (or close to real) time. All this creates quite strict requirements for the performance of the corresponding numerical methods.
The existing numerical methods for solving this class of problems have two significant disadvantages, which are fully manifested when implementing as part of complex software systems. First, they have a number of artificial computational parameters that are usually selected manually by experts. Secondly, they do not always take into account the specific parameters of the environment in an optimal way, which leads to a significant overspend of computing resources and a decrease in the relevance of the results obtained.
In this study, we propose to develop new numerical methods for solving the one-way Helmholtz equation (a high-order parabolic equation) using modern stochastic optimization methods. It is planned to use the theory of pseudo-differential and pseudo-difference operators, discrete dispersion analysis to obtain relations that reflect the relationship between the parameters of the numerical approximation, the parameters of the environment, and the required accuracy of calculations. Then the specified relations are used for a strict formulation of the numerical scheme parameters optimization problem. Given the complexity of such problems, it is proposed to use stochastic optimization methods to solve them. The parameters of the numerical scheme will be calculated for specific parameters of the propagation medium using optimization methods in such a way as to minimize the required calculation time and memory. In addition, it will give an opportunity to automate the process of selecting the parameters of the numerical scheme. Optimization and automation of numerical methods will lead to wider use of complex mathematical models in applied software systems.
It should be noted that the application of stochastic methods to optimize numerical schemes is a fundamentally new scientific direction. Thus, the results obtained can be further generalized to solve other problems described by differential equations.

Expected results

1. It is planned to develop new numerical methods for solving the Helmholtz equation in inhomogeneous unbounded domains with automatic selection of optimal parameters of the numerical scheme depending on the input data. Based on the developed methods, it is planned to create an open-source software library for conducting computational experiments and solving specific problems of tropospheric radio wave propagation and computational hydroacoustics.
2. Theoretical significance consists in the development of the theory of construction, analysis and optimization of numerical schemes for solving differential equations.
3. The ability to automatically adjust the parameters of numerical schemes depending on the input data without the intervention of the user (expert) will contribute to more active and effective use of complex mathematical models in complex software systems. In particular, the results obtained can be used in software packages for planning wireless communication networks, calculating radio visibility zones, and solving direct and inverse problems of computational hydroacoustics.
4.Many research teams from different countries (in particular, ETH Zurich, Bergische Universität Wuppertal, MIT, Brown University, POI FEB RAS, MPEI etc.) are engaged in solving these problems, which indicates the world level of the planned research.

REPORTS

Annotation of the results obtained in 2022

PyWaveProp - an open source Python 3 software library was developed. Key features of this library:
- Radio wave propagation modelling over irregular terrain, tropospheric channel and vegetation;
- Modelling of the radio wave diffraction over the Earth's surface;
- Modelling transparent boundaries using the discrete non-local boundary conditions;
- Arbitrary operating frequency and radiation pattern of transmitting antennas;
- Automatic generation of the computational grid;
- Automatic fitting of the artificial computational parameters (order and method of approximation, propagation coefficient, parameters of non-local boundary conditions, backscattering, maximum propagation angle) based on input data;
- Arbitrary output result grid selection;
- Possibility of manual adjustment of all computational parameters;
- Approximation of discrete and semi-discrete propagation operators of arbitrary order of accuracy: Padé approximation, rational interpolation, Numerov's scheme, differential evolution method;
- Discrete dispersion analysis of the numerical scheme depending on the parameters and its visualization;
- Simulation of sound propagation in an underwater channel, taking into account the inhomogeneities of the sound speed, density and topography of the bottom;
- Python 3 wrappers over the PETOOL and RAM programs for comparative analysis;
- Visualization of simulation results - spatial distribution of acoustic and electromagnetic wave fields;
- Simulation of backscattering from inhomogeneities and multiple reflections;
- Software implementation of the method for solving the modelling multiple knife-edge diffraction problem;
- Software implementation of the wavenumber integration method;
- Linear approximation of the terrain within the framework of the Greene and Claerbout methods.
The source code of the library and documentation with usage examples are available in the repository https://github.com/mikelytaev/wave-propagation.
Documentation is also available at https://wave-propagation.readthedocs.io/.
Paper entitled "Mesh Optimization for the Acoustic Parabolic Equation" has been published in the Journal of Marine Science and Engineering (Q1). The paper is devoted to the optimization of the computational grid of the numerical scheme for solving the one-way Helmholtz equation, taking into account the specifics of the underwater acoustics propagation problem. In addition, the paper proposes a method for the reference sound speed optimization, which makes it possible to increase the performance of the numerical scheme by 3-5 times and increase the limits of its applicability (maximum propagation angle), without modifying the topology of the numerical scheme.
The paper "Reducing the numerical dispersion of the one-way Helmholtz equation via the differential evolution method" was prepared and accepted for publication in the Journal of Computational Science (Q1). The paper analyzes various configurations of the differential evolution method. The results of the analysis showed that:
- The best results are shown by the randtobest1exp strategy and the choice of a random mutation from the range [0.5;1] at each step.
- The optimal population size is 10-15. The choice of a larger number of individuals does not significantly affect the accuracy, but greatly increases the rate of convergence of the method.
- The convergence rate can be increased by several times by using the multi-threaded calculations. Using 16 threads on a regular Intel Core i9-11900K processor allowed to increase the average convergence rate from 33 to 7 seconds.
- The method makes it possible to effectively take into account the entire spectrum of visible propagation angles (up to 90 degrees).
The results were reported at the following international conferences:
- International Conference on Computational Science (ICCS) 2022 (June 21-23, 2022, London, category A conference). Research "Numerical Approximation of the One-Way Helmholtz Equation Using the Differential Evolution Method" is presented online, the corresponding paper is published in the conference proceedings (LNCS).
- International Conference on Computational Science and its Applications (ICCSA) 2022 (July 4-7, 2022, Malaga, Spain). Research "Interval Approximation of the Discrete Helmholtz Propagator for the Radio-Wave Propagation Along the Earth's Surface" is presented online.
- International Conference on Computational Science and its Applications (ICCSA) 2023 (July 3-6, 2023, Athens). Research "Computational grid optimization for the 3D higher-order parabolic equation" was accepted for presentation and publication. The corresponding article will be published in the conference proceedings (LNCS).

Publications

**1.** *Lytaev M.* **Reducing the numerical dispersion of the one-way Helmholtz equation via the differential evolution method** Journal of Computational Science, - (year - 2023)

**2.** *lytaev M.S.* **Numerical Approximation of the One-Way Helmholtz Equation Using the Differential Evolution Method** Lecture Notes in Computer Science, volume 13350 pp 205-218 (year - 2022) https://doi.org/10.1007/978-3-031-08751-6_15

**3.** *Lytaev M.S.* **Mesh Optimization for the Acoustic Parabolic Equation** Journal of Marine Science and Engineering, Vol. 11, Iss. 3, No. 496 (year - 2023) https://doi.org/10.3390/jmse11030496

**4.** *lytaev M.S.* **Computational grid optimization for the 3D higher-order parabolic equation** Lecture Notes in Computer Science Литературная серия, - (year - 2023)

Annotation of the results obtained in 2021

1. A novel approximation of the one-way Helmholtz equation based on genetic and evolutionary algorithms has been developed.
Rational approximation of the propagation operator was taken as a basis. The coefficients of the numerical scheme and the computational grid sizes are calculated based on the minimization of the discrete dispersion relation error. Taking into account the complexity of the obtained optimization problem, the differential evolution method was used to solve it. The proposed method does not require manual selection of the artificial computational parameters of the numerical scheme. The stability of the scheme is provided by an additional condition on the optimization problem. Various formulations of the optimization problem are investigated. A comparison is made with the Pade approximation method and rational interpolation. The effectiveness of the proposed method is demonstrated on a wedge diffraction problem.
It should be noted that the proposed method of the numerical scheme optimization using stochastic methods goes far beyond the solution of the Helmholtz equation. Similarly, it is possible to optimize almost any higher order numerical scheme with a number of coefficients and design parameters. In this case, the required properties (in particular stability) are set a priori. Thus, prerequisites have been created for the emergence of a fundamentally new class of methods for increasing the productivity of numerical methods.
2. A method for rational interpolation of a semi-discrete propagation operator (one-way Helmholtz equation) has been developed [Lytaev M. S. Rational interpolation of the one-way Helmholtz propagator //Journal of Computational Science. 2022].
The relationship between the semidiscrete pseudo differential propagation operator, variations of the refractive index and the maximum propagation angle is established. It is proposed to use a rational approximation of the propagation operator on the interval instead of the Pade approximation in the vicinity of the point. It is shown that the use of approximation on the interval is more natural for this problem and allows one using a more sparse computational grid than using the local Pade approximation. The proposed method differs from the existing ones only by the coefficients of the numerical scheme and does not require any significant changes in the implementations of existing numerical schemes. Analysis of the stability and errors of the proposed approach was carried out. The advantages of the proposed approach in the tropospheric radio wave propagation problems and propagation of acoustic waves in the underwater environment are shown.
3. A method for the numerical solution of the tropospheric radio wave propagation problem based on the Pade approximation of a completely discrete propagation operator has been developed [Lytaev M. S. Rational interpolation of the one-way Helmholtz propagator //Journal of Computational Science. 2022].
The proposed approach is based on the Pade rational approximation the propagation operator, which is applied simultaneously along the longitudinal and transverse coordinates. At the same time, it is possible to take into account the inhomogeneous refractive index of the troposphere. A discrete dispersion analysis of the proposed scheme is carried out. A comparison with other finite-difference methods for solving the parabolic equation and the split-step Fourier method is given. It is shown that the proposed method allows using a more sparse computational grid than the existing finite-difference methods. This, in turn, leads to faster calculations.
4. A method for the numerical solution of the tropospheric radio wave propagation problem based on rational interpolation of a completely discrete propagation operator has been developed.
A rational approximation of a discrete propagation operator in both dimensions is constructed. Instead of the local Pade approximation, the rational interpolation method is used. The results of numerical simulation confirm the advantages of the proposed approach.
5. The method of discrete nonlocal boundary conditions has been modified, which allows taking into account the arbitrary dependence of the reflection coefficient on the incidence angle [Fresnel Reflection Modeling Within the Higher-order Parabolic Equation and Discrete Nonlocal Boundary Conditions // RadarConf 2022].
The possibility of modeling the interface between two media with different dielectric properties is shown. Special attention is paid to the problem of modeling the dielectric properties of the Earth's surface when solving the problem of tropospheric propagation. Instead of the local Leontovich boundary condition, a non-local boundary condition is used. A discrete nonlocal boundary condition is constructed for a finite-difference approximation of the one-way Helmholtz equation.
All methods developed within the framework of this project are implemented in the Python3 software library. The source code is freely available in the repository https://github.com/mikelytaev/wave-propagation . All the computational experiments carried out and presented in the articles were carried out using the specified library.
In general, the proposed method exceed the existing ones by 5-20 times, depending on the propagation conditions. The more complex the conditions (the higher the maximum angle of propagation and the greater variation of the refractive index), the more effective the proposed methods are compared to existing ones. Detailed results are published in the relevant sections of the articles.

Publications

**1.** *Lytaev M.S.* **An Improved Accuracy Split-Step Padé Parabolic Equation for Tropospheric Radio-Wave Propagation** Lecture Notes in Computer Science, Vol. 12949. pp. 418-433. (year - 2021) https://doi.org/10.1007/978-3-030-86653-2_31

**2.** *Lytaev M.S.* **Rational interpolation of the one-way Helmholtz propagator** Journal of Computational Science, Volume 58, February 2022, 101536 (year - 2022) https://doi.org/10.1016/j.jocs.2021.101536

**3.** *Lytaev M.S.* **Fresnel Reflection Modeling Within the Higher-order Parabolic Equation and Discrete Nonlocal Boundary Conditions** 2022 IEEE Radar Conference (RadarConf22), IEEE 2022 (year - 2022) https://doi.org/10.1109/RadarConf2248738.2022.9764174

**4.** *-* **Ученые СПб ФИЦ РАН разработали метод для моделирования распространения радиосигнала, повышающий эффективность зондирования Земли** Сайт РАН, - (year - )