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[CT108]

Session Date & Time : 1E (Aug.21, 17:40-19:20)
Type : Contributed Talk
Abstract : Spectral methods solve elliptic partial differential equations (PDEs) numerically. Their main advantage is spectral convergence, i.e., error decays exponentially when the solution is analytic. We present numerical schemes for solving some time-dependent linear PDEs utilizing the ultraspherical spectral method in space and time, thus portraying overall spectral convergence. Moreover, they lead to sparse and well-conditioned linear systems. We compare their performance with existing spectral schemes and explore their parallelization in time.
Classification : 65M70, 65L05, 35K20, 35L20, 41A10
Author(s) :
- Avleen Kaur (University of Saskatchewan)
- S, H. Lui (University of Manitoba)

Session Date & Time : 1E (Aug.21, 17:40-19:20)
Type : Contributed Talk
Abstract : This talk describes a scalable sparse direct solver for linear systems that arise from the discretization of elliptic PDEs in 2D or 3D. The scheme uses a decomposition of the domain into thin subdomains, or ``slabs''. The general framework is easier to optimize for modern heterogeneous architectures than than traditional multi-frontal schemes. Crucial to the scalability, are novel randomized algorithms that recover structure from matrix-free samples and reduce the dimensionality of large dense matrices.
Classification : 65M70, 65M55, 65M22, 65M06
Author(s) :
- Anna Yesypenko (University of Texas at Austin)
- Per-Gunnar Martinsson (University of Texas at Austin)

Session Date & Time : 1E (Aug.21, 17:40-19:20)
Type : Contributed Talk
Abstract : An anisotropic PDE model for the classification of grey scale images has been proposed. A fully automatic classification of any image is achieved by a suitably designed multi-modal well potential function. The model is solved by the spectrally accurate pseudo-spectral method and has been successfully tested on several bench-mark problems for its superior performance measured in terms on standard metrics like SNR, PSNR, SSIM etc.
Classification : 65M70
Author(s) :
- Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)

Session Date & Time : 1E (Aug.21, 17:40-19:20)
Type : Contributed Talk
Abstract : The KdV equation plays an important role in describing several physical phenomena. It is solved numerically using the septic Hermite collocation method. The method is found to be unconditionally stable, fifth order convergent in space and second-order convergent in the time direction. The algorithm is implemented on various test problems. The results are compared with literature data. The simulation results shows that SHCM is highly applicable and giving better results than the previous ones.
Classification : 65M70, 65N12, Numerical solution of PDEs
Author(s) :
- Vijay Kumar Kukreja (Head, Department of Mathematics, SLIET Longowal - 148106 (Punjab))
- Archna Kumari Kaundal (Sant Longowal Institute of Engineering and Technology, Longowal)

Session Date & Time : 1E (Aug.21, 17:40-19:20)
Type : Contributed Talk
Abstract : In this talk, we present a new deep learning based algorithm to solve high-dimensional nonlinear parabolic PDEs and PIDEs, extending the Deep Splitting algorithm developed in Beck, Becker, Cheridito, Jentzen, Neufeld in 2021. We use random neural networks, instead of fully trained neural networks, which immensely speeds-up the algorithm. Moreover, we provide a convergence analysis of the algorithm and demonstrate its application to high-dimensional problems in finance.
Classification : 65M75, 91G20, Deep Learning method for nonlinear PDEs
Author(s) :
- Ariel Neufeld (NTU Singapore)
- Philipp Schmocker (NTU Singapore)
- Sizhou Wu (NTU Singapore)