Adaptive Numerical Methods for PDEs

This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing, a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems. Our first target is to develop a boundary strategy for the current PPR algorithm. The current accuracy of PPR near boundaries is not as good as that in the interior of the domain. It might be even worse than without recovery. Some special treatments are needed to improve the accuracy of PPR on the boundary. In this thesis, we present two boundary recovery strategies to resolve the problem caused by boundaries. Numerical experiments indicate that both of the newly proposed strategies made an improvement to the original PPR. Our second target is to generalize PPR to the weak Galerkin methods. Different from the standard finite element methods, the weak Galerkin methods use a different set of degrees of freedom. Instead of the weak gradient information, we are able to obtain the recovered gradient information for the numerical solution in the generalization of PPR. In the PPR process, we are also able to recover the function value at the nodal points which will produce a global continuous solution instead of piecewise continuous function. Our third target is to apply our proposed strategy and WGPPR to interface problems. We treat an interface as a boundary when performing gradient recovery, and the jump condition on the interface can be well captured by the function recovery process. In addition, adaptive methods based on WGPPR recovery type a posteriori error estimator is proposed and numerically tested in this thesis. Application on the elliptic problem and interface problem validate the effectiveness and robustness of our algorithm. Furthermore, WGPPR has been applied to 3D problem and Stokes problem as well. Superconvergent phenomenon is again observed

Fluorescence enhanced optical tomography on breast phantoms with measurements using a gain modulated intensified CCD imaging system

Fluorescence-enhanced optical imaging using near-infrared (NIR) light developed for in-vivo molecular targeting and reporting of cancer provides promising opportunities for diagnostic imaging. However, prior to the administration of unproven contrast agents, the benefits of fluorescence-enhanced optical imaging must be assessed in feasibility phantom studies. A novel intensified charge-coupled device (ICCD) imaging system has been developed to perform 3-D fluorescence tomographic imaging in the frequency-domain using near-infrared contrast agents. This study is unique since it (i) employs a large tissue-mimicking phantom (~1087 cc), which is shaped and sized to resemble a female breast and part of the extended chest wall region, and (ii) enables rapid data acquisition in the frequency-domain by using a gain-modulated ICCD camera. Diagnostic 3-D fluorescence-enhanced optical tomography is demonstrated using 0.5-1 cc single and multiple targets contrasted from their surrounding by ??M concentrations of Indocyanine green (ICG) in the breast-shaped phantom (10 cm diameter), under varying conditions of target-to-background absorption contrast ratios (1:0 and 100:1) and target depths (up to 3 cm deep). Boundary surface fluorescence measurements of referenced amplitude and phase shift were used along with the coupled diffusion equation of light propagation in order to perform 3-D image reconstructions using the approximate extended Kalman filter (AEKF) algorithm, and hence differentiate the target from the background based on fluorescent optical contrast. Detection of single and multiple targets is demonstrated under various conditions of target depths (up to 2 cm deep), absorption optical contrast ratio (1:0 and 100:1), target volumes (0.5-1 cc), and multiple targets (up to three 0.5 cc targets). The feasibility of 3-D image reconstructions from simultaneous multiple point excitation sources are presented. Preliminary lifetime imaging studies with 1:2 and 2:1 optical contrast in fluorescence lifetime of the contrast agents is also demonstrated. The specificity of the optical imager is further assessed from homogeneous phantom studies containing no fluorescently contrasted targets. While nuclear imaging currently provides clinical diagnostic opportunities using radioactive tracers, molecular targeting of tumors using non-ionizing NIR contrast agents tomographically imaged using the frequency-domain ICCD imaging system could possibly become a new method of diagnostic imaging

Traction force microscopy on soft elastic substrates: a guide to recent computational advances

The measurement of cellular traction forces on soft elastic substrates has become a standard tool for many labs working on mechanobiology. Here we review the basic principles and different variants of this approach. In general, the extraction of the substrate displacement field from image data and the reconstruction procedure for the forces are closely linked to each other and limited by the presence of experimental noise. We discuss different strategies to reconstruct cellular forces as they follow from the foundations of elasticity theory, including two- versus three-dimensional, inverse versus direct and linear versus non-linear approaches. We also discuss how biophysical models can improve force reconstruction and comment on practical issues like substrate preparation, image processing and the availability of software for traction force microscopy.Comment: Revtex, 29 pages, 3 PDF figures, 2 tables. BBA - Molecular Cell Research, online since 27 May 2015, special issue on mechanobiolog

Improved fluorescence-enhanced optical imaging and tomography by enhanced excitation light rejection

Fluorescence enhanced optical imaging and tomography studies involve the detection of weak fluorescent signals emanating from nano- to picomolar concentrations of exogenous or endogenously produced fluorophore concurrent with the rejection of an overwhelmingly large component of backscattered excitation light. The elimination of the back-reflected excitation light of the collected signal remains a major and often unrecognized challenge for further reducing the noise floor and increasing sensitivity of small animal fluorescence imaging. In this dissertation, we adapted collimating and gradient index (GRIN) lenses in an existing frequency-domain system to improve excitation light rejection and enhance planar and tomographic imaging. To achieve this goal, we developed planar and tomographic imaging systems based upon ray tracing calculations for improved rejection of excitation light. The “out-of-band (S (λx))” to “in-band (S (λm) - S (λx))” signal ratio assessing excitation leakage was acquired with and without collimating optics. The addition of collimating optics resulted in a 51 to 75% reduction in the transmission ratio of (S (λx))/ (S (λm) - S (λx)) for the phantom studies and an increase of target to background ratio (TBR) from 11% to 31% in animal studies. Additionally, we presented results demonstrating the improvement of model match between experiments and forward simulation models by adaptation of GRIN lens optics to a breast phantom study. In particular, 128 GRIN lenses on the fiber bundle face were employed to align the collected excitation and emission light normal to the filter surface in an existing frequency-domain system. As a result of GRIN lens collimation, we reduced the transmission ratio between 10 and 86 % and improved the model match for tomographic reconstruction of one (1 cm3) and two (0.1 cm3) targets in a 1087 cm3 of breast phantom. Ultimately, this work improves the sensitivity of NIR fluorescence imaging by enhancing the rejection of excitation light and shows that the current sensitivity challenges for translating fluorescence-enhanced optical imaging into the clinic can be overcome

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today