We establish some explicit expressions for norm-wise, mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix $A⊗B$ and more general matrix function compositions involving Kronecker products. The condition number for the weighted least squares problem (WLS) involving a Kronecker product is also discussed.

Total variation (TV) and wavelet $L_1$ norms have often been used as regularization terms in image restoration and reconstruction problems. However, TV regularization can introduce staircase effects and wavelet regularization some ringing artifacts, but hybrid TV and wavelet regularization can reduce or remove these drawbacks in the reconstructed images. We need to compute the proximal operator of hybrid regularizations, which is generally a sub-problem in the optimization procedure. Both TV and wavelet $L_1$ regularisers are nonlinear and non-smooth, causing numerical difficulty. We propose a dual iterative approach to solve the minimization problem for hybrid regularizations and also prove the convergence of our proposed method, which we then apply to the problem of MR image reconstruction from highly random under-sampled k-space data. Numerical results show the efficiency and effectiveness of this proposed method.

An artificial boundary condition method, derived in terms of infinite Fourier series, is applied to solve a class of quasi-Newtonian Stokes flows. Based on the natural boundary reduction involving an artificial condition on the artificial boundary, the coupled variational problem and its numerical solution are obtained. The unique solvability of the continuous and discrete formulations are discussed, and the error analysis for the problem is also considered. Finally, an a posteriori error estimate for the corresponding problem is provided.

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

Based on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where $n×n$ real symmetric matrices $M$, $C$ and $K$ are constructed so that the quadratic pencil $Q(λ) = λ^{2}M+λC+K$ yields good approximations for the given $k$ eigenpairs. We discuss the case where $M$ is positive definite for $1≤ k≤n$, and a general solution to this problem for $n+1≤k≤2n$. The efficiency of our methods is illustrated by some numerical experiments.

In this short note, we study a relation between the tensor product of matrices and a multilinear map defined by the optimal operator. In this particular case, the linear transform (mediating morphism) hidden in the abstract definition of the general tensor product can be determined explicitly.

In this short note, we present a sharp upper bound for the perturbation of eigenvalues of a singular diagonalizable matrix given by Stanley C. Eisenstat [3].