S. H. Lui

Position:: Professor & Head
Office:: 536 Machray Hall; University of Manitoba; Winnipeg, Manitoba; Canada R3T 2N2; (204) 474-9693 (Head's Office)
E-mail:: luish@cc.umanitoba.ca
Education:: B.Sc., M.Sc. (Toronto); Ph.D. (California Institute of Technology)

I have some projects in numerical analysis of PDEs (spectral methods for PDEs) suitable for graduate students. If interested, please contact me by email for more details and/or apply for MSc or PhD.

Research Areas:

Numerical Analysis, Applied PDEs

1) Numerical PDEs.

One recent direction is spectral methods for time dependent PDEs that are spectrally accurate in both space and time for smooth solutions. Classical discretization schemes use a low-order finite difference scheme for the time derivative and spectral scheme for spatial derivatives. This is not ideal since the time discretization error overwhelms the spatial discretization error. We have shown space-time spectral convergence for classical linear PDEs such as the heat, wave, Airy, beam and Stokes equations, and also demonstrated numerically full spectral convergence for common nonlinear PDEs. Condition number estimates are given for all linear spectral operators mentioned above. One drawback of space-time spectral methods is that time-stepping is no longer possible - unknowns over all time must be solved at the same time.

2) Domain decomposition methods (DDMs) for PDEs.

DDMs have become an exciting and dynamic research area in scientific computation. The idea is very simple - split up the domain of the problem into many subdomains and then solve the PDE in each subdomain. The global solution is obtained by pasting together the limits of sequences of subdomain solutions. Benefits include parallel computation and isolation of difficult subdomains containing, for instance, boundary layers or geometric singularities. My focus has been on proving convergence of such sequences of subdomain solutions to the global solution of nonlinear PDEs.

Recently, I have been looking into DDMs where Robin or higher order boundary conditions are used along the artificial interface. The main problem is how to optimize the parameters in the boundary conditions for fastest convergence.

3) Pseudospectra.

When a matrix is normal, its eigenvalues determine the stability properties of the system associated with the matrix. When the matrix is not normal, then the eigenvalues are no longer reliable indicators. The pseudospectrum of a matrix is a generalization of the spectrum of a matrix. It gives a quantitative estimate of departure from non-normality and can give some stability information for non-normal matrices. My interests include fast computation of pseudospectra as well as a generalization of the spectral mapping theorem from the spectrum of an operator to its pseudospectrum.

4) Nonlinear Elliptic PDEs.

Let L denote the (negative) Laplacian and p>0. Consider positive solutions of L u = u^p on a bounded domain with a homogeneous Dirichlet boundary condition. This nonlinear elliptic PDE and its many variations have played a central in the development of the calculus of variations, bifurcation theory, critical Sobolev exponent, Pohazaev identity, etc. In this research, we consider L u = |D u|^p and study existence, regularity and singularities of solutions. A family of deadcore solutions are found when p<1.

5) Optimization

Broyden's method and BFGS are quasi-Newton iterative methods to solve a system of nonlinear equations and to find a local minimum of a function, respectively. With the exception of Kantorovich's convergence theory for Newton's method, most convergence theories assume existence of a solution and bounds on the nonlinear function and its derivative in some neighbourhood of the solution. These conditions cannot be checked in practice. We derive some convergence theories where all assumptions can be verified, and the existence of a solution and rate of convergence are consequences of the theory.

Another research direction is ODE formulations of iterative methods. The hope is to gain new insights into a convergence theory of the methods.

Recent Publications:

Numerical PDEs

9) New lower bounds on the minimum singular value of a matrix (with A. Kaur), Linear Alg. Appl., 666 (2023) p62-95.
8) Space-time spectral method for the Stokes problem (with A. Kaur), Applied Numer. Math., 188 (2023) p206-234.
7) Spectral collocation in space and time for linear PDEs (with S. Nataj), J. Comput. Phys., 424 (2021) 109843, 22 pages
6) Chebyshev spectral collocation in space and time for the heat equation (with S. Nataj), Elect. Trans. Numer. Anal., 52 (2020), pp295-319
5) Legendre spectral collocation in space and time for PDEs. Numer. Math., 136 (2017), pp.75-99
4) Third-order multirate Runge-Kutta schemes for accelerating finite volume schemes of 3D time dependent Maxwell's equations (with M. Kotovshchikova and D. Firsov, Comm. Appl. Math. Comput. Sci., 15 (2020), pp65--87.
3) A Third order finite volume Weno scheme for Maxwell's equations on tetrahedral meshes (with M. Kotovshchikova and D. Firsov, Comm. Appl. Math. Comput. Sci., 13 (2018), pp87--106
2) A Numerical Study of the Dirichlet and Neumann Eigenvalue Problem of the Laplacian on Cusp Domains, J. Comput. Methods in Sci & Eng., 13 (2013) pp. 433-437.
1) Spectral Domain Imbedding for Elliptic PDEs in Complex Geometry, J. Comput. and Appl. Math. 225 (2009), pp541-557.

Numerical PDEs Book

Numerical Analysis of PDEs, Wiley, 2011

Domain Decomposition Methods

7) Adaptive Wavelet Schwarz methods for the Navier-Stokes equation (with S. Dahlke, D. Lellek and R. Stevenson), Numer. Func. Anal. Optim., 37 (2016), pp.1213-1234.
6) An optimized Schwarz method for domains with an arbitrary interface, J. Comput. Applied Math., 235 (2010), pp. 301-314.
5) An optimized Schwarz method for PDEs with discontinuous coefficients (with O. Dubois), Numer. Algorithms., 51 (2009), pp. 115-131.
4) A Lions nonoverlapping domain decomposition method for domains with an arbitrary interface, IMA J. Numer. Anal., DOI:10.1093/imanum/drm011 (2008).
3) On Linear Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs, Numer. Math., 93 (2002), pp. 109-129.
2) On Schwarz Alternating Methods for the Incompressible Navier Stokes Equation, SIAM J.S.C., 22 (2001), pp. 1974-1986.
1) On Schwarz Alternating Methods for Nonlinear Elliptic PDEs, SIAM J.S.C., 21 (2000), pp.1506-1523.

Pseudospectra

5) Pseudospectrum and condition spectrum (with K. Kumar), Operators & Matrices, 9 (2015), pp. 121-145.
4) On some properties of the pseudo spectral radius (with K. Kumar), Elect. J. Lin. Alg., 27 (2014), pp. 342-353.
3) Pseudospectral Mapping Theorem II, ETNA, 38, (2011), pp.168--183.
2) A Pseudospectral Mapping Theorem, Math. Comp., 72 (2003), pp. 1841-1854.
1) Computation of Pseudospectra by Continuation, SIAM J.S.C., 18 (1997), pp. 565-573.

Nonlinear Elliptic PDEs

3) A nonlinear elliptic problem involving the gradient on a half space (with A. Aghajani and C. Cowan), Discrete and Continuous Dynamical Systems, 43 (2023), pp378--391.
2) Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball (with A. Aghajani and C. Cowan), J. Diff. Eqns, 264 (2018), pp2865-2896.
1) Existence and regularity of solutions of advection problem (with A. Aghajani and C. Cowan), Nonlinear Analysis, 166 (2018), pp. 19-47.

Optimization

2) Superlinear convergence of the methods of Broyden and BFGS based on assumptions about the initial point (with S. Nataj), J. Comput. Appl. Math., 385 (2021) 113204, pp.1-21.
1) Superlinear convergence of nonlinear conjugate gradient methods and scaled memoryless BFGS methods based on assumptions about the initial point (with S. Nataj), Appl. Math. Comput., 369 (2020) 124829