A Hybrid Iterative Method for Elliptic Variational Inequalities of the Second Kind
Yujian Cao , Jianguo Huang , Haoqin Wang
Communications on Applied Mathematics and Computation ›› 2024, Vol. 7 ›› Issue (3) : 910 -928.
A Hybrid Iterative Method for Elliptic Variational Inequalities of the Second Kind
A hybrid iterative method is proposed for numerically solving the elliptic variational inequality (EVI) of the second kind, through combining the regularized semi-smooth Newton method and the Int-Deep method. The convergence rate analysis and numerical examples on contact problems show this algorithm converges rapidly and is efficient for solving EVIs.
Deep learning / Semi-smooth Newton method / Elliptic variational inequality (EVI) / Convergence rate analysis
| [1] |
|
| [2] |
|
| [3] |
|
| [4] |
|
| [5] |
|
| [6] |
Brezzi, F., Hager, W.W., Raviart, P.-A.: Error estimates for the finite element solution of variational inequalities. II. Mixed methods. Numer. Math. 31(1), 1–16 (1978) |
| [7] |
|
| [8] |
Chen, J., Chi, X., E, W., Yang, Z.: Bridging traditional and machine learning-based algorithms for solving PDEs: the random feature method. J. Mach. Learn 1(3), 268–298 (2022) |
| [9] |
|
| [10] |
|
| [11] |
|
| [12] |
|
| [13] |
Cuomo, S., Schiano Di Cola, V., Giampaolo, F., Rozza, G., Raissi, M., Piccialli, F.: Scientific machine learning through physics-informed neural networks: where we are and what’s next. J. Sci. Comput. 92(3), 88 (2022) |
| [14] |
|
| [15] |
E, W.N. Machine learning and computational mathematics. Commun. Comput. Phys. 28(5), 1639–1670 (2020) |
| [16] |
E, W.N., Han, J., Jentzen, A.: Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Nonlinearity 35(1), 278–310 (2022) |
| [17] |
E, W. N., Ma, C., Wu, L.: A priori estimates of the population risk for two-layer neural networks. Commun. Math. Sci. 17(5), 1407–1425 (2019) |
| [18] |
E, W. N., Yu, B.: The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6(1), 1–12 (2018) |
| [19] |
|
| [20] |
|
| [21] |
|
| [22] |
|
| [23] |
|
| [24] |
|
| [25] |
|
| [26] |
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: The 29th IEEE Conference on Computer Vision and Pattern Recognition, IEEE, Las Vegas, 27–30 June, 2016 (2016) |
| [27] |
|
| [28] |
|
| [29] |
|
| [30] |
Hüeber, S., Wohlmuth, B.I.: A primal-dual active set strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Engrg. 194(27/28/29), 3147–3166 (2005) |
| [31] |
|
| [32] |
Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. In: The 3rd International Conference on Learning Representations, ICLR, San Diego, 7–9 May, 2015 (2015) |
| [33] |
|
| [34] |
|
| [35] |
|
| [36] |
|
| [37] |
|
| [38] |
|
| [39] |
|
| [40] |
|
| [41] |
|
| [42] |
|
| [43] |
|
| [44] |
|
Shanghai University
/
| 〈 |
|
〉 |