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In this talk, we present an adaptive multiscale method for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.
The research is partially supported by the Hong Kobg RGC General Reseach Fund (Project: 400411).

One advantage of the energy-preserving methods is that sometimes the energy gives an a priori estimate for the (numerical) solution. For example, in the cubic nonlinear Schroedinger equation, the quartic energy function (the Hamiltonian) yields an estimate $ \|u\|_{\infty} < \infty $ for all $ t>0 $, with the aid of the discrete Gagliardo--Nirenberg and Sobolev inequalities.
Although such discrete inequalities have been known for the simplest forward (i.e. one-sided) finite difference operator, it remained open for more general operators including the standard central-difference operator, as far as the authors know. Accordingly, the analyses of energy-preserving methods with such operators remained open as well.
Recently, we found a unified way of establishing discrete inequalities for a certain range of central-difference type operators. In this talk, we show some results, and illustrate them through applications to some structure-preserving numerical schemes.
This is a joint work with H. Kojima (U. Tokyo) and D. Furihata (Osaka University).

In this talk we present the development of meshless computational method based on the use of kernel-based functions for solving various physical problems. Properties of some special kernels such as radial basis functions; harmonic kernels; fundamental and particular solutions; and Green’s functions will be discussed. For tackling the well-known ill-conditioned system of equations resulted from solving inverse and ill-posed problems, the method has recently been extended to combine the multiscale support vector approach (MSVA) scheme for approximating the solution of a moderately ill-posed problem on bounded domain. The Vapnik's ε-intensive function is adopted to replace the standard Ɩ2 loss function in using the regularization technique to reduce the error induced by noisy data. Convergence proof for the case of noise-free data is then derived under an appropriate choice of the Vapnik's cut-off parameter and the regularization parameter. For noise data case, we demonstrate that a corresponding choice for the Vapnik's cut-off parameter gives the same order of error estimate as both the à posteriori strategy based on discrepancy principle and the noise-free à priori strategy. Numerical examples are constructed to verify the efficiency of the proposed MSVA approach and the effectiveness of the parameter choices.

For the solution of large sparse box constrained least squares problems (BLS), a new iterative method is proposed using CGLS for the inner iterations, and the modulus iterative method for the outer iterations for the solution of the linear complementarity problem resulting from the Karush-Kuhn-Tucker condition of the BLS problem. Theoretical convergence analysis is presented. Numerical experiments show the efficiency of the proposed method with less iteration steps and CPU time compared to projection methods.

We present a method to obtain the registration (bijective point-wise correspondences) between high-genus surfaces. The high-genus topology of the surfaces poses great challenges for surface registration. Conventional approaches partition surfaces into simply-connected patches and registration is done by a patch-by- patch manner. Consistent cuts are required, which are difficult to obtain and prone to error. This talk presents a method to handle the registration problem for high-genus surfaces without introducing consistent cuts. The key idea is to formulate the problem on the universal covering spaces of the surfaces and registration is obtained using quasi-conformal approaches. Both constrained registration and curvature-matching registration will be considered. In the second part of the talk, some applications of the registration algorithm to medical imaging will be presented. (This research is supported by HKRGC GRF (CUHK Project ID: 404612))

If unconstrained optimization methods are applied to problems with constraints, they may fail to solve the problems by generating sequences which violate the constraints. On the other hand, constraints resulting from problems particularly in numerical linear algebra and in statistics often naturally define submanifolds of the Euclidean space. Therefore, such constrained problems can be regarded as unconstrained problems on Riemannian manifolds and Riemannian unconstrained optimization methods can be applied to them. In this talk, we will briefly review Riemannian optimization theory and will discuss several applications to numerical linear algebra and to statistics such as the singular value decomposition and the canonical correlation analysis together with some numerical experiments.

The LLL algorithm receives a lot of attention as an effective numerical tool for preconditioning an integer least squares problem. However, the workings of the algorithm are not well understood. In this talk, we present a linear algebraic way to analyze the LLL reduction, which leads to a new implementation that performs better than the original LLL scheme.

While nonconvex optimization problems are generally regarded as difficult, matrix eigenvalue problems form an important class of problems that can be solved efficiently. In this work we advocate solving some nonconvex optimization problems via an eigenvalue problem.
This talk focuses on the trust region subproblem (TRS), which arises in an algorithm for nonlinear programming. TRS is usually solved via an iterative process of solving linear systems or eigenvalue problems. An alternative approach is to use semidefinite programming, but this also involves solving linear systems iteratively. In this work we introduce an algorithm that solves just one generalized eigenvalue problem, or more specifically just one eigenpair. Our algorithm allows for a non-standard norm without a change of variables and is suited both to the dense and the large-sparse cases. Experiments suggest that our algorithm is superior to existing ones both in accuracy and especially efficiency, particularly for large-sparse problems. Time permitting, we discuss how the approach can be extended to problems with general quadratic constraints.

Based on joint work with Satoru Adachi, Satoru Iwata and Akiko Takeda.

The proximal point algorithm (PPA) has been well studied in the literature.
In particular, its linear convergence rate has been studied by Rockafellar in 1976 under certain condition. We consider a generalized PPA in the generic setting of finding a zero point of a maximal monotone operator, and show that the condition proposed by Rockafellar can also sufficiently ensure the linear convergence rate for this generalized PPA. Both the exact and inexact versions of this generalized PPA are discussed. The motivation to consider this generalized PPA is that it includes as special cases the relaxed versions of some splitting methods that are originated from PPA. Thus, linear convergence results of this generalized PPA can be used to better understand the convergence of some widely usedalgorithms in the literature. We focus on the particular convex minimization context and specify Rockafellar's condition to see how to ensure the linear convergence rate for some efficient numerical schemes, including the classical augmented Lagrangian method proposed by Hensen and Powell in 1969 and its relaxed version, the original alternating direction method of multipliers (ADMM) by Glowinski and Marrocco in 1975 and its relaxed version (i.e., the generalized ADMM by Eckstein and Bertsekas in 1992). Some refined conditions weaker than existing ones are proposed in these particular contexts.

Recently, semidefinite programming (SDP) and second-order cone programming (SOCP) are studied extensively as surprising extensions of classical linear programming (LP). While SDP and SOCP are similar to LP in many aspects, their behavior can be much more complicated in particular when regularity conditions are not satisfied. One of the typical difficulties arising in SDP and SOCP is occurrence of weak infeasibility where the problem can be arbitrarily close to feasible but infeasible. In this talk, we study the structure of weakly infeasible problems in SDP and SOCP and develop are procedure to test whether a given problem is weakly infeasible or not. Some complexity issue will be also discussed. This is a joint work with Lourenco F. Bruno (Tokyo Institute of Technology) and Masakazu Muramatsu (University of Electro-Communications).

Ground-based astronomy refers to acquiring images of objects in outer space via ground-based telescopes. Because of atmospheric turbulence, images so acquired are blurry. One way to estimate the unknown blur or point spread function (PSF) is by using natural or artificial guide stars. Once the PSF is known, the images can be deblurred using well-known deblurring methods. Another way to estimate the PSF is to make use the aberration of wavefronts received at the telescope, i.e., the phase, to derive the PSF. However, the phase is not readily available; instead only its low-resolution gradients can be collected by wavefront sensors. In this talk, we will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and then the PSF in high accuracy. Our model can be solved efficiently by alternating direction method of multiplier whose convergence has been well established. Numerical results will be given to illustrate that our new model is efficient and give more accurate estimation for the PSF.

In this talk, we propose an algebraic reconstruction method for a biomagnetic inverse source problem called magnetoencephalography (MEG) in which current source in the human brain is estimated from the magnetic field measured ouside the head. When assuming that the source consists of a few multipoles, their positions as well as moments can be reconstructed algebraically from the Hankel tensors composed of the weighted integrals of the MEG data.