Solving the structural inverse gravity problem by the modified gradient methods

Мартышко, П. С.; Akimova, Elena N.; Misilov, Vladimir E.

doi:10.1134/s1069351316050098

Cited by 8 publications

(5 citation statements)

References 3 publications

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“…Previously, to solve Equations ( 2) and (3), we used the conjugate gradient method (CGM) with the weighting factors. 13 This method has the following form:…”

Section: Gradient Methods With Weighting Factorsmentioning

confidence: 99%

“…The vector 𝛾 of weighting factors is calculated for the components of the combined solution vector z using the approximate separate fields F 1 , F 2 , … , F L corresponding to the surfaces S l , l = 1..L (see Martyshko et al 13 )…”

Section: Choosing the Weighting Factors And Regularization Parametersmentioning

confidence: 99%

“…Previously, to solve Equations () and (), we used the conjugate gradient method (CGM) with the weighting factors 13 . This method has the following form:

\begin{matrix} right z^{k + 1} & left = z^{k} - ψ \frac{⟨ p^{k}, S (z^{k}) ⟩}{{‖A^{'} (z) p^{k}‖}^{2}} p^{k}, p^{k} = v^{k} + β^{k} p^{k - 1}, p^{0} = v^{0}, \\ right β^{k} & left = \max \{\frac{⟨ v^{k}, v^{k} - v^{k - 1} ⟩}{{‖v^{k - 1}‖}^{2}}, 0\}, v = γ \circ S (z), \\ right S (z) & left = A^{'} {(z)}^{*} (A (z) - F), \end{matrix}

where z k is the solution estimate at the k th iteration, ψ is the damping factor, γ is the vector of the weighting factors, A ′ ( z ) is the Jacobian matrix of the discretized integral operator A ( z ), and ∘ is the operation of componentwise vector multiplication.…”

Section: Statement Of the Inverse Gravimetry Problemmentioning

confidence: 99%

“…The vector γ of weighting factors is calculated for the components of the combined solution vector z using the approximate separate fields F 1 , F 2 , … , F L corresponding to the surfaces S l ,

l = 1 . . L

(see Martyshko et al 13 )

F_{l} = [f_{l, 1}, f_{l, 2}, \dots, f_{l, n}] \to [γ_{l, 1}, γ_{l, 2}, \dots, γ_{l, n}], γ_{l, j} = \frac{{|f_{l, j}|}^{μ}}{\max_{i = 1 . . n} \{{|f_{l, i}|}^{μ}\}}, j = 1 . . n,

where 1 ≤ μ ≤ 2 is the smoothing parameter.…”

Section: Statement Of the Inverse Gravimetry Problemmentioning

confidence: 99%

“…In previous works, [12][13][14][15][16] for solving the problem in the case of multiple layers, we constructed the algorithms based on the steepest descent and the conjugate gradient methods with the weighting factors. These methods allow one to find multiple surfaces from the base equation simultaneously.…”

Section: Introductionmentioning

confidence: 99%

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Regularized gradient algorithms for solving the nonlinear gravimetry problem for the multilayered medium

Akimova

Misilov

Sultanov

2020

Math Methods in App Sciences

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We present new numerical algorithms for solving the structural inverse gravimetry problem for the case of multiple surfaces. The inverse problem of finding the multiple surfaces that divide the constant density layers is an ill‐posed one described by a nonlinear integral equation of the first kind. To solve it, it is necessary to apply the regularization ideas. The new regularized variants of the gradient type methods with the weighting factors are constructed, namely, the steepest descent and conjugate gradient method. We suggest the empirical rule for choosing the regularization parameters. On the basis of the constructed methods, we elaborate the parallel algorithms and implement them in the multicore CPU using the OpenMP technology. A set of experiments with the disturbed data is performed to test the gradient algorithms and study performance of the developed code. For the test problems with quasi‐real data, these new regularized algorithms increase the accuracy and speed up computation in comparison with the unregularized ones. By using the 8‐core CPU, we achieve the speedup of 8 times.

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“…Previously, to solve Equations ( 2) and (3), we used the conjugate gradient method (CGM) with the weighting factors. 13 This method has the following form:…”

Section: Gradient Methods With Weighting Factorsmentioning

confidence: 99%

Section: Choosing the Weighting Factors And Regularization Parametersmentioning

confidence: 99%

“…Previously, to solve Equations () and (), we used the conjugate gradient method (CGM) with the weighting factors 13 . This method has the following form: