Eventi: seminari, convegni

Seminario Prof. Maurizio Falcone: lunedì 3 giugno ore 14:30

Seminario Prof. Maurizio Falcone

Adaptive Filtered Schemes for First Order Evolutive Hamilton-Jacobi Equations

Lunedì 3 giugno ore 14:30 - Laboratorio HPC (DISIM, Piano terra, Coppito 1)

Abstract

M. Falcone, G. Paolucci (Dipartimento di Matematica, Università di Roma "La Sapienza")
and S. Tozza (INDAM and Università di Roma "La Sapienza")


The accurate numerical solution of Hamilton-Jacobi (HJ) equations is a challenging topic of growing importance in many fields of application, e.g. control theory, KAM theory, image processing and material science.
This is a delicate issue due to the lack of regularity of viscosity solutions [1], so the construction of high-order methods can be rather difficult. In fact simple monotone schemes are at most first order accurate so monotonicity should be abandoned and the proof of high-order convergence becomes very challenging. Several methods have been proposed (e.g. ENO and WENO schemes [6]) but a precise convergence result is still missing and their implementation is often tricky [3].

More recently, the simple idea of filtered schemes for HJ equations has been proposed. They are obtained coupling two different schemes: one is monotone (and convergent) and the other is high-order accurate. The filtered scheme switches from one scheme to the other according to a filter function and a switching parameter, this feature is crucial to prove high-order convergence where the solution is regular. Then it seems natural to adapt the choice of this parameter to the regularity of the solution in the cell via a smoothness indicator improving the standard filtered scheme (where this parameter is constant [2]) by an adaptive and automatic choice at every iteration.

Here we introduce a smoothness indicator to select the regions where we have to update the regularity threshold and we adapt the parameter in time and space.
We present a general convergence result and some error estimates for the new adaptive filtered scheme in 1D [4]. Finally, we show a 2D application to the segmentation problem in image processing [5].

 

References

[1] G. Barles, Solutions de viscositè des equations de Hamilton-Jacobi, Springer Verlag, 1994.
[2] O. Bokanowski, M. Falcone, S. Sahu, An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations, SIAM Journal on Scientific Computing 38:1 (2016), A171–A195.
[3] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, 2013.
[4] M. Falcone, G. Paolucci, and S. Tozza, A High-Order Scheme for Image Segmentation via a modified Level-Set method. 2018, submitted. arXiv:1812.03026.
[5] M. Falcone, G. Paolucci, and S. Tozza, Convergence of Adaptive Filtered schemes for first order evolutive Hamilton-Jacobi equations. 2018, submitted, arXiv: 1812.02140
[6] G. Jiang, D.-P. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing 21 (2000), 2126–2143.

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