Ivan P. Gavrilyuk

H-matrix approximation for the operator exponential with applications

Summary. We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4].In the preceding papers [12]–[17], a class of matrices (

Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems

\mathcal{H}

Data-sparse approximation to a class of operator-valued functions

-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. In particular, the matrix-vector/matrix-matrix product with such matrices as well as the computation of the inverse have linear-logarithmic cost. In the present paper, we apply the

Data-sparse approximation to the operator-valued functions of elliptic operator

\mathcal{H}

Exponentially Convergent Algorithms for the Operator Exponential with Applications to Inhomogeneous Problems in Banach Spaces

-matrix techniques to approximate the exponent of an elliptic operator.Starting with the Dunford-Cauchy representation for the operator exponent, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the

Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations

\mathcal{H}

Natural sloshing frequencies in rigid truncated conical tanks

-matrices. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different time values. In the case of smooth data (coefficients, boundaries), we prove the linear-logarithmic complexity of the method.

Stability and Regularization of Three-Level Difference Schemes with Unbounded Operator Coefficients in Banach Spaces

Abstract The class of -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the -matrix technique combined with the Kronecker tensor-product approximation (cf. [2, 20]) to represent the inverse of a discrete elliptic operator in a hypercube (0, 1)d∈ℝd in the case of a high spatial dimension d. In this data-sparse format, we also represent the operator exponential, the fractional power of an elliptic operator as well as the solution operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by (dn log qn), where N=nd is the discrete problem size.

ℋ-matrix approximation for elliptic solution operators in cylinder domains

In earlier papers we developed a method for the data-sparse approximation of the solution operators for elliptic, parabolic, and hyperbolic PDEs based on the Dunford-Cauchy representation to the operator-valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents. In the present paper, we discuss how these techniques can be applied to approximate a hierarchy of the operator-valued functions generated by an elliptic operator £.

Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces

llIn previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator L. The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent (zI - L) -1 , z ∈ C. In the present paper, we consider various operator functions, the operator exponential e -tL , negative fractional powers L -α , the cosine operator function cos(t√L) L-k and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents (z k I - L) -1 mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.