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[CT108]

CT108 (1/1) : Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65M) / Hydrodynamic stability (76E)

Session Time & Room : 1E (Aug.21, 17:40-19:20) @E704
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
Format : Talk at Waseda University
Author(s) :
- Avleen Kaur (University of Saskatchewan)
- S, H. Lui (University of Manitoba)

Session Time & Room : 1E (Aug.21, 17:40-19:20) @E704
Type : Contributed Talk
Abstract : In this talk, we first provide a brief introduction to quantum computing from a mathematical perspective. No prior knowledge of quantum computing is necessary. We then introduce a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε and so demonstrate that our quantum Monte Carlo algorithm does not suffer from the curse of dimensionality. This talk is based on a joint work with Yongming Li.
Classification : 65M75, 91G20, Deep Learning method for nonlinear PDEs
Format : Talk at Waseda University
Author(s) :
- Ariel Neufeld (NTU Singapore)
- Philipp Schmocker (NTU Singapore)
- Sizhou Wu (NTU Singapore)

Session Time & Room : 1E (Aug.21, 17:40-19:20) @E704
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
Format : Talk at Waseda University
Author(s) :
- Anna Yesypenko (University of Texas at Austin)
- Per-Gunnar Martinsson (University of Texas at Austin)

Session Time & Room : 1E (Aug.21, 17:40-19:20) @E704
Type : Contributed Talk
Abstract : We investigate a chemically reactive front A+B→C involving the radial miscible displacement in porous media. It is a non-linear phenomenon that is mathematically modeled by Darcy’s law coupled with convection-reaction-diffusion equations. A chemical reaction may result in a change in the viscosity profile, which may lead to the interfacial instability known as viscous fingering, which occurs when a low-viscosity fluid displaces a high-viscosity fluid in a porous medium. The instability enhances the fluid mixing.
Classification : 76Exx, 76Sxx, 76Vxx
Format : Talk at Waseda University
Author(s) :
- Priya Verma (Indian Institute of Technology Ropar)
- Manoranjan Mishra (Indian Institute of Technology Ropar)