Approximating inverse FEM matrices on non-uniform meshes with $${\mathcal{H}}$$-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $${\mathcal{H}}$$ H -matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $${\mathcal{H}}$$ H -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

3D general-measure inversion of crosswell EM data using a direct solver

We present a three-dimensional (3D) general-measure inversion scheme of crosswell electromagnetic (EM) data in the frequency domain with a direct forward solver. In the forward problem, we discretised the EM Helmholtz equation by the staggered-grid finite difference (SGFD) scheme and solved it using the Intel MKL PARDISO direct solver. By applying a direct solver, we simultaneously solved the multisource forward problems at a given frequency. In the inversion, we integrated a general measure of data misfit and model constraints with linearised least-squares inversion. We reconstructed a model with blocky features by selecting the appropriate measure parameters and model constraints. We used the adjoint equation method to explicitly calculate the Jacobian matrix, which facilitated the determination of an appropriate initial value for the regularisation coefficient in the objective function. We illustrated the inversion scheme using synthetic crosswell EM data with a general measure, the L2 norm, and, specifically, two mixed norms.

Computational Costs of Multi-Frontal Direct Solvers with Analysis-Suitable T-Splines

In this paper, we consider the computational cost of a multi-frontal direct solver used for the factorization of matrices resulting from a discretization of isogeometric analysis with T-splines, and analysis-suitable T-splines. We start from model projection or model heat transfer problems discretized over two-dimensional meshes, either uniformly refined or refined towards a point or an edge. These grids preserve several symmetries and they are the building blocks of more complicated grids constructed during adaptive isotropic refinement procedures. A large class of computational problems construct meshes refined towards point or edge singularities. We propose an ordering that permutes the matrix in a way that the computational cost of a multi-frontal solver executed on adaptive grids is linear. We show that analysis-suitable T-splines with our ordering, besides having other well-known advantages, also significantly reduce the computational cost of factorization with the multi-frontal direct solver. Namely, the factorization with N T-splines of order p over meshes refined to a point has a linear O(Np4) cost, and the factorization with T-splines on meshes refined to an edge has O(N2pp2) cost. We compare the execution time of the multi-frontal solver with our ordering to the Approximate Minimum Degree (AMD) and Cuthill–McKee orderings available in Octave.