We present algorithms for the solution of a class of source identification problems for systems governed by elliptic partial differential equations (PDEs) on two-dimensional regular geometries. The state is the solution of the PDE, which is driven by an unknown source field. Given observations of the state, we seek to reconstruct the source field. We consider the cases of full domain observations and boundary observations. The problem is formulated as a least-squares PDE-constrained optimization problem. We use a reduced space approach in which we “invert” the associated Hessian using a preconditioned conjugate gradient (PCG) algorithm. Using standard Fourier analysis, we derive analytical solutions for the case in which the governing PDE has constant coefficients. Based on these solutions, we construct preconditioners that accelerate the convergence of PCG in the case of variable-coefficient elliptic PDE constraints. We performed numerical experiments to show the effectiveness of the preconditioners for variable coefficients with different contrasts and smoothness properties. We observed mesh-independent and $\beta$-independent convergence for different cases of the variable coefficients. The computational complexity of solving the source identification problem is $\mathcal{O}(N\log N)$. The construction of the preconditioner costs $\mathcal{O}(N^{3/2})$, where N is the discretization size for the source.

In this article, we present Dendro, a suite of parallel algorithms for the discretization and solution of partial differential equations (PDEs) involving second-order elliptic operators. Dendro uses trilinear finite element discretizations constructed using octrees. Dendro, comprises four main modules: a bottom-up octree generation and 2:1 balancing module, a meshing module, a geometric multiplicative multigrid module, and a module for adaptive mesh refinement (AMR). Here, we focus on the multigrid and AMR modules. The key features of Dendro are coarsening/refinement, inter-octree transfers of scalar and vector fields, and parallel partition of multilevel octree forests. We describe a bottom-up algorithm for constructing the coarser multigrid levels. The input is an arbitrary 2:1 balanced octree-based mesh, representing the fine level mesh. The output is a set of octrees and meshes that are used in the multigrid sweeps. Also, we describe matrix-free implementations for the discretized PDE operators and the intergrid transfer operations. We present results on up to 4096 CPUs on the Cray XT3 (ldquoBigBenrdquo), the Intel 64 system (ldquoAberdquo), and the Sun Constellation Linux cluster (ldquoRangerrdquo).

In this article we propose parallel algorithms for the construction of conforming finite-element discretization on linear octrees. Existing octree-based discretizations scale to billions of elements, but the complexity constants can be high. In our approach we use several techniques to minimize overhead: a novel bottom-up tree-construction and 2:1 balance constraint enforcement; a Golomb-Rice encoding for compression by representing the octree and element connectivity as an Uniquely Decodable Code (UDC); overlapping communication and computation; and byte alignment for cache efficiency. The cost of applying the Laplacian is comparable to that of applying it using a direct indexing regular grid discretization with the same number of elements. Our algorithm has scaled up to four billion octants on 4096 processors on a Cray XT3 at the Pittsburgh Supercomputing Center. The overall tree construction time is under a minute in contrast to previous implementations that required several minutes; the evaluation of the discretization of a variable-coefficient Laplacian takes only a few seconds.