Wolfgang Bangerth

deal.II—A general-purpose object-oriented finite element library

An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced object-oriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach, deal.II supports a large number of different applications covering a wide range of scientific areas, programming methodologies, and application-specific algorithms, without imposing a rigid framework into which they have to fit. A judicious use of programming techniques allows us to avoid the computational costs frequently associated with abstract object-oriented class libraries. The paper presents a detailed description of the abstractions chosen for defining geometric information of meshes and the handling of degrees of freedom associated with finite element spaces, as well as of linear algebra, input/output capabilities and of interfaces to other software, such as visualization tools. Finally, some results obtained with applications built atop deal.II are shown to demonstrate the powerful capabilities of this toolbox.

Adaptive finite element based tomography for fluorescence optical imaging in tissue

A three-dimensional fluorescence-enhanced optical tomography scheme based upon an adaptive finite element formulation is developed and employed to reconstruct fluorescent targets in turbid media from frequency-domain measurements made in reflectance geometry using area excitation illumination. The algorithm is derived within a Lagrangian framework by treating the photon diffusion model as a constraint to the optimization problem. Adaptively refined meshes are used to separately discretize maps of the forward/adjoint variables and the unknown parameter of fluorescent yield. A truncated Gauss-Newton method with simple bounds is used as the optimization method. Fluorescence yield reconstructions from simulated measurement data with added Gaussian noise are demonstrated for one and two fluorescent targets embedded within a 512ml cubical tissue phantom. We determine the achievable resolution for the area-illumination/area-detection reflectance measurement geometry by reconstructing two 0.4cm diameter spherical targets placed at at a series of decreasing lateral spacings. The results show that adaptive techniques enable the computationally efficient and stable solution of the inverse imaging problem while providing the resolution necessary for imaging the signals from molecularly targeting agents.

The deal.II Library, Version 8.4

Abstract This paper provides an overview of the new features of the finite element library deal.II version 8.5.

Algorithms and data structures for massively parallel generic adaptive finite element codes

Todays largest supercomputers have 100,000s of processor cores and offer the potential to solve partial differential equations discretized by billions of unknowns. However, the complexity of scaling to such large machines and problem sizes has so far prevented the emergence of generic software libraries that support such computations, although these would lower the threshold of entry and enable many more applications to benefit from large-scale computing. We are concerned with providing this functionality for mesh-adaptive finite element computations. We assume the existence of an “oracle” that implements the generation and modification of an adaptive mesh distributed across many processors, and that responds to queries about its structure. Based on querying the oracle, we develop scalable algorithms and data structures for generic finite element methods. Specifically, we consider the parallel distribution of mesh data, global enumeration of degrees of freedom, constraints, and postprocessing. Our algorithms remove the bottlenecks that typically limit large-scale adaptive finite element analyses. We demonstrate scalability of complete finite element workflows on up to 16,384 processors. An implementation of the proposed algorithms, based on the open source software p4est as mesh oracle, is provided under an open source license through the widely used deal.II finite element software library.

Non-contact fluorescence optical tomography with scanning patterned illumination.

This article describes a novel non-contact fluorescence optical tomography scheme which utilizes multiple area illumination patterns, to reduce the ill-posedness of the inverse problem involved in recovering interior fluorescence yield distributions in biological tissue from boundary fluorescence measurements. The image reconstruction is posed as an optimization problem which seeks a tissue optical property distribution minimizing, for all illumination patterns simultaneously, a regularized difference between the observed boundary measurements of light distribution, and the boundary measurements predicted from a physical model. Multiple excitation source illumination patterns are described by line and Gaussian sources scanning the simulated tissue phantom surface and by employing diffractive optics-generated patterns. Multiple measurement data sets generated by scanning excitation sources are processed simultaneously to generate the interior fluorescence distribution in tissue by implementing the fluorescence tomography algorithm in a parallel framework suitable for multiprocessor computers. Image reconstructions for single and multiple fluorescent targets (5mm diameter) embedded in a 512ml simulated tissue phantom are demonstrated, with depths of the fluorescent targets from the illumination plane between 1cm to 2cm. We show both qualitative and quantitative improvements of our algorithm over reconstructions from only a single measurement.

Adaptive finite element methods for the solution of inverse problems in optical tomography

Optical tomography attempts to determine a spatially variable coefficient in the interior of a body from measurements of light fluxes at the boundary. Like in many other applications in biomedical imaging, computing solutions in optical tomography is complicated by the fact that one wants to identify an unknown number of relatively small irregularities in this coefficient at unknown locations, for example corresponding to the presence of tumors. To recover them at the resolution needed in clinical practice, one has to use meshes that, if uniformly fine, would lead to intractably large problems with hundreds of millions of unknowns. Adaptive meshes are therefore an indispensable tool. In this paper, we will describe a framework for the adaptive finite element solution of optical tomography problems. It takes into account all steps starting from the formulation of the problem including constraints on the coefficient, outer Newton-type nonlinear and inner linear iterations, regularization, and in particular the interplay of these algorithms with discretizing the problem on a sequence of adaptively refined meshes. We will demonstrate the efficiency and accuracy of these algorithms on a set of numerical examples of clinical relevance related to locating lymph nodes in tumor diagnosis.

Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection

Using an area-illumination and area-detection scheme, we acquire fluorescence frequency domain measurements from a tissue phantom with an embedded fluorescent target and obtain tomographic reconstructions of the interior fluorescence absorption map with an adaptive finite element based scheme. The tissue phantom consisted of a clear acrylic cubic box (512 ml) filled with 1% Liposyn solution, while the fluorescent targets were 5 mm diameter glass bulbs filled with 1 microM Indocyanine Green dye solution in 1% Liposyn. Frequency domain area illumination and detection employed a planar excitation source using an expanded intensity modulated (100 MHz) 785 nm diode laser light and a gain modulated image intensified charge coupled device camera, respectively. The excitation pattern was characterized by isolating the singly scattered component with cross polarizers and was input into a dual adaptive finite element-based scheme for three dimensional reconstructions of fluorescent targets embedded beneath the phantom surface. Adaptive mesh refinement techniques allowed efficient simulation of the incident excitation light and the reconstruction of fluorescent targets buried at the depths of 1 and 2 cm. The results demonstrate the first clinically relevant noncontact fluorescence tomography with adaptive finite element methods.

Data structures and requirements for hp finite element software

Finite element methods approximate solutions of partial differential equations by restricting the problem to a finite dimensional function space. In hp adaptive finite element methods, one defines these discrete spaces by choosing different polynomial degrees for the shape functions defined on a locally refined mesh. Although this basic idea is quite simple, its implementation in algorithms and data structures is challenging. It has apparently not been documented in the literature in its most general form. Rather, most existing implementations appear to be for special combinations of finite elements, or for discontinuous Galerkin methods. In this article, we discuss generic data structures and algorithms used in the implementation of hp methods for arbitrary elements, and the complications and pitfalls one encounters. As a consequence, we list the information a description of a finite element has to provide to the generic algorithms for it to be used in an hp context. We support our claim that our reference implementation is efficient using numerical examples in two dimensions and three dimensions, and demonstrate that the hp-specific parts of the program do not dominate the total computing time. This reference implementation is also made available as part of the Open Source deal.II finite element library.

An Autonomic Reservoir Framework for the Stochastic Optimization of Well Placement

The adequate location of wells in oil and environmental applications has a significant economic impact on reservoir management. However, the determination of optimal well locations is both challenging and computationally expensive. The overall goal of this research is to use the emerging Grid infrastructure to realize an autonomic self-optimizing reservoir framework. In this paper, we present a policy-driven peer-to-peer Grid middleware substrate to enable the use of the Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm, coupled with the Integrated Parallel Accurate Reservoir Simulator (IPARS) and an economic model to find the optimal solution for the well placement problem.

Application of Grid-enabled technologies for solving optimization problems in data-driven reservoir studies

This paper presents the use of numerical simulations coupled with optimization techniques in oil reservoir modeling and production optimization. We describe three main components of an autonomic oil production management framework. The framework implements a dynamic, data-driven approach and enables Grid-based large scale optimization formulations in reservoir modeling.