A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations
Abstract
:1. Introduction
2. Direct Problem
2.1. Governing Equations
2.2. Brief Introduction to Godunov Finite-Difference Scheme
3. Inverse Problem of Recovering the Density
4. Modification of Gradient Descent Method
5. Numerical Results
- Gradient constructed of 1 source, fixed in position (Figure 2). The formula of gradient descent is
- Gradient constructed of 1 source, but its location was changed on each iteration through positions , that are located uniformly on the circle of the radius m. The formula is in the next view
- Gradient constructed of 16 sources from positions with each iteration in a cyclic manner. The final formula is
6. Discussion
Author Contributions
Funding
Conflicts of Interest
Abbreviations
IP | Inverse problem |
DP | Direct problem |
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Approach 1 | Approach 2 | Approach 3 | |
---|---|---|---|
Relative error | 0.121 | 0.050 | 0.039 |
Elapsed time | 40 s | 40 s | 480 s |
Approach 1 | Approach 2 | Approach 3 | |
---|---|---|---|
Relative error | 0.071 | 0.026 | 0.032 |
Elapsed time | 15 min | 15 min | 17 min |
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Klyuchinskiy, D.; Novikov, N.; Shishlenin, M. A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations. Computation 2020, 8, 73. https://doi.org/10.3390/computation8030073
Klyuchinskiy D, Novikov N, Shishlenin M. A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations. Computation. 2020; 8(3):73. https://doi.org/10.3390/computation8030073
Chicago/Turabian StyleKlyuchinskiy, Dmitriy, Nikita Novikov, and Maxim Shishlenin. 2020. "A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations" Computation 8, no. 3: 73. https://doi.org/10.3390/computation8030073
APA StyleKlyuchinskiy, D., Novikov, N., & Shishlenin, M. (2020). A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations. Computation, 8(3), 73. https://doi.org/10.3390/computation8030073