Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities that depend quadratically on the departure of the coefficient matrix from normality: these bounds suggest that the larger the departure from normality, the ...

How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To il...

Constrained diffusions in convex polyhedral domains with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter ε > 0, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded domains from [13], certain uniform in ε lower bounds on transition densities of such constrained diffusions are ...

We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for Γ0(2). Numerical evidence indicates that a sharper bound holds for the weights h ≡ 2 ( mod 4). We derive upper bounds for the minimum positive integer represented by level two even positive-definite quadratic forms. Our data suggest that, ...

We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for Γ0(2). Numerical evidence indicates that a sharper bound holds for the weights h ≡ 2 ( mod 4). We derive upper bounds for the minimum positive integer represented by level two even positive-definite quadratic forms. Our data suggest that, ...

The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the first-named author is used to develop condition number bounds for conforming linear ele...

We derive some bounds which can be viewed as an evidence of increasing stability in the Cauchy Problem for the Helmholtz equation with lower order terms when frequency is growing. These bounds hold under certain (pseudo)convexity properties of the surface where the Cauchy data are given and of variable zero order coefficient of the Helmholtz equation. Proofs use Carleman estimates, the theory o...

How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To il...

A boundary value problem for a 4-th order self-adjoint ordinary differential equation is considered in the case where the coefficients of the equation and its right-hand side can be nonsmooth (discontinuous, concentrated or rapidly oscillating functions). Generalized cubic splines of deficiency 1 depending on the major coefficient of the equation are applied. An error analysis of finite element...

Stone's dimensionality reduction principle has been confirmed on several occasions for independent observations. When dependence is expressed with C-mixing, a minimum distance es t imate ~n is proposed for a smooth projection pursuit regression-type function t~ C e, that is either additive or multiplicative, in the presence of or without interactions. Upper bounds on the Ll-risk and the Ll-erro...