A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry

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1. What have the authors contributed in "A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry" ?

2. What are the future works mentioned in the paper "A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry" ?

3. What is the nonlinear inverse gravimetric problem?

4. What are the main categories of methods for nonlinear inverse problems?

Accepted Answer

Based on the Regularized Functional Matching Pursuit ( RFMP ) algorithm for linear inverse problems, the authors present an analogous iterative greedy algorithm for nonlinear inverse problems, called RFMP_NL.. This inverse problem is described by a nonlinear integral operator, for which the authors additionally provide the Fréchet derivative.. Finally, the authors present two synthetic numerical examples to show that it is beneficial to apply the presented method to inverse gravimetric problems.. Furthermore, in contrast to other methods, the algorithm does not require the solution of large linear systems.

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In the future, the authors want to apply the algorithm to realistic data sets, for example, data from satellite missions.. On the one hand, with rising computer power, the authors may be able to accomplish this in reasonable time.

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The operator that maps the function σ to UΣ int,ρ |S for fixed ρ is denoted by Sρ and the operator equationSρ [σ ] = gis called the nonlinear inverse gravimetric problem (with a star-shaped domain).

Accepted Answer

(direct) Tikhonov regularization methods (see Tikhonov and Glasko 1965), multilevel methods (see Kaltenbacher et al. 2008, Chapter 5), and sequential subspace optimization methods (see Wald and Schuster 2017) have been developed for nonlinear inverse problems.

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Due to the ill-posedness and the nonlinearity of the inverse problem, it is not surprising that the approximation error is larger than the residual and the noise level.

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In contrast to the linear inverse gravimetric problem, which is concerned with the inversion of the gravitational potential for the mass density distribution inside a given body of mass, the nonlinear problem is the inversion of the potential for the shape of the body of mass given a mass density model.

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Other iterative methods for nonlinear inverse problems are, for example, the LevenbergMarquardt method (see Levenberg 1944; Marquardt 1963) and the iteratively regularized Gauß-Newton method (see Bakushinsky 1992), and many more.

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Of course, one could also think about stopping the iteration earlier, for example, using a discrepancy principle, which is a common procedure for iterative regularization methods.

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the Fréchet derivative of the operator Sρ : C(S2)→ C(S) is given asS ′ρ [σ ](τ)(y) = ∫ S2 ρ(σ(ξ )ξ ) |σ(ξ )ξ − y| (σ(ξ ))2 τ(ξ )dω(ξ )for all σ ,τ ∈ C(S2) and y ∈ S, assuming there exists C > 0 such that |σ(ξ )ξ − y| >

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The authors chose a dictionary D that consists of spherical harmonics up to degree 9 and AbelPoisson kernels with parameter h = 0.7, which are centered on a Driscoll-Healy grid with parameter m = 25, yielding 100 spherical harmonics and 652 Abel-Poisson kernels to obtain #D = 752 in total.

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The symbol ∼ in the second line should indicate that the authors assumed that the interchanging of differentiation and integration is possible.

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Due to the high computational effort that comes with the implementation of the algorithm, currently, this is only a proof of concept that the algorithm successfully solves the inverse problem, since the computation time for the presented small examples is already around 9 hours on a node with 12 cores.

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the stability of the inverse problem can be described as being weak (cf. Isakov 2006, Section 1.1) due to the logarithmic nature of the estimate, which leads to “numerical difficulties” (Isakov 2006).

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This is the reason why, in practice, one still needs to apply a regularization technique although the nonlinear inverse gravimetric problem is stable in theory.

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An alternative for the numerical calculation of the integral in (23) is to transform the integral to the ball B1 such that∫S2 ∫ σ(ξ ) 0 k(rξ ,y)dr dω(ξ ) = ∫ S2 ∫ 1 0 k(tσ(ξ )ξ ,y)σ(ξ )dt dω(ξ )= ∫ B1 k ( |x|σ ( x |x| ) x |x| ,y ) σ ( x |x| ) 1 |x|2 dxand apply an integration method for the ball, as for example derived by Amna and Michel (2017), but for the simplicity of the implementation, the authors do not pursue this approach.

Accepted Answer

The authors will now use the idea of iteratively minimizing the functional in (19) to obtain a greedy algorithm for nonlinear inverse problems.

Accepted Answer

A substantially different classical category of methods for nonlinear inverse problems are gradient-type methods, in particular, the Landweber method, which can be applied to linear (see Landweber 1951) and nonlinear inverse problems (see Hanke et al. 1995).

A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry

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Most frequently asked questions

1. What have the authors contributed in "A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry" ?

2. What are the future works mentioned in the paper "A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry" ?

3. What is the nonlinear inverse gravimetric problem?

4. What are the main categories of methods for nonlinear inverse problems?

5. Why is the approximation error larger than the residual?

6. What is the definition of the inverse gravimetric problem?

7. What other methods are used for nonlinear inverse problems?

8. What is the common procedure for iterative regularization methods?

9. What is the Fréchet derivative of the operator S?

10. How many spherical harmonics are used in the graph?

11. What is the symbol in the second line?

12. Why is the RFMP_NL a good algorithm for nonlinear inverse problems?

13. Why is the stability of the inverse gravimetric problem weak?

14. What is the reason why one still needs to apply a regularization technique?

15. What is the alternative for the numerical calculation of the integral in 23?

16. What is the simplest way to minimize the Tikhonov functional?

17. What is the difference between the two categories of methods for nonlinear inverse problems?

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References

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