Accelerated Algorithm for Splitting a Vector into Two Vectors with Small Uniform Norm

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(NlogN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

We establish estimates from above for the uniform norm of the Chebyshev polynomials associated with a system of continua \(K \subset \mathbb {C}\) by constructing monic polynomials with small norms on K. The estimates are exact (up to a constant factor) in the case where K has a piecewise quasiconformal boundary and its complement \(\varOmega =\overline {\mathbb {C}} \setminus K\) has no outward pointing cusps.

Near minimally normed spline quasi-interpolants on uniform partitions

A trivariate near-best blending quadratic quasi-interpolant