Geometric PDEs Research Articles

Mixed Finite Elements for Variational Surface Modeling

AbstractMany problems in geometric modeling can be described using variational formulations that define the smoothness of the shape and its behavior w.r.t. the posed modeling constraints. For example, high‐quality C2 surfaces that obey boundary conditions on positions, tangents and curvatures can be conveniently defined as solutions of high‐order geometric PDEs; the advantage of such a formulation is its conceptual representation‐independence. In practice, solving high‐order problems efficiently and accurately for surfaces approximated by meshes is notoriously difficult. For modeling applications, the preferred approach is to use discrete geometric schemes which are efficient and robust, but their convergence properties are less well understood compared to higher‐order FEM. In this paper, we explore discretizations of common geometric PDEs on meshes using mixed finite elements, where additional variables for the derivatives in the problem are introduced. Such formulations use first‐order derivatives only, allowing a discretization with simple linear elements. Various boundary conditions can be naturally discretized in this setting. We formalize continuous region constraints commonly used in modeling applications, and show that these seamlessly fit into the mixed framework. We demonstrate that some of the commonly used discrete geometric discretizations can be regarded as a particular case of mixed finite elements. We study the convergence behavior of our discretizations, and how they can be applied to implement common modeling tasks.

Affine harmonic maps

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemannian geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic maps in a given homotopy class when the target has nonpositive sectional curvature and some global nontriviality condition is met. An example shows that such a condition is necessary. The analytical part is made difficult by the absence of a variational structure underlying affine harmonic maps. We therefore need to combine estimation techniques from geometric analysis and PDE theory with global geometric considerations.

Mini-Workshop: Control of Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

Regularization of diffusion tensor MRI via local coordinates

AbstractWe propose a novel framework for regularization of symmetric positive‐definite (SPD) tensors (e.g., diffusion tensors). This framework is based on differential geometry. The space of SPD matrices, Pn, is described as a Riemannian manifold that is parameterized via the Iwasawa coordinate system. Then, distances on Pn are measured in terms of a natural GL (n)‐invariant Riemannian metric. Using the Beltrami framework we construct a set of coupled geometric PDEs with respect to the Iwasawa coordinates. Then, by means of the gradient descent method these equations define the regularization flow over Pn. It appears to be that the local coordinate approach via that coordinate system results in very simple numerics that leads to fast convergence of the algorithm. We demonstrate the efficiency of this algorithm on real volumetric DTI datasets. Results of fibers tractography before and afterthe regularization process arepresented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Special Lagrangian cones with higher genus links

For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.

Bi-Hamiltonian operators, integrable flows of curves using moving frames and geometric map equations

Moving frames of various kinds are used to derive bi-Hamiltonian operators and associated hierarchies of multi-component soliton equations from group-invariant flows of non-stretching curves in constant curvature manifolds and Lie group manifolds. The hierarchy in the constant-curvature case consists of a vector mKdV equation coming from a parallel frame, a vector potential mKdV equation coming from a covariantly-constant frame, and higher order counterparts generated by an underlying vector mKdV recursion operator. In the Lie-group case the hierarchy comprises a group-invariant analog of the vector NLS equation coming from a left-invariant frame, along with higher order counterparts generated by a recursion operator that is like a square-root of the mKdV one. The corresponding respective curve flows are found to be given by geometric nonlinear PDEs, specifically mKdV and group-invariant analogs of Schrodinger maps. In all cases the hierarchies also contain variants of vector sine-Gordon equations arising from the kernel of the respective recursion operators. The geometric PDEs that describe the corresponding curve flows are shown to be wave maps. Full details of these results are presented for two main cases: $S^2,S^3\simeq SU(2)$.

Computation of geometric partial differential equations and mean curvature flow

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Regularity of generalized sphere valued p -harmonic maps with small mean oscillations

The results of this note belong to the regularity theory of nonlinear elliptic systems with the right hand side growing critically with the gradient of solution. The particular system we have in mind may appear special but it is closely linked with vast parts of geometric analysis and PDE. In case m = n = 2 its solutions agree with asymptotic limits of solutions of a complex valued Ginzburg–Landau equation involving a small parameter, see Bethuel, Brezis and Helein [4] and Hardt and Lin [22]. We consider solutions with weak integrability assumptions, weaker than those which are needed for a variational interpretration of the system. The proofs are based on a mixture of (well known) subtle hard analytic methods.

Some Applications of Integral Formulas in Riemannian Geometry and PDE?s

We show some applications of integral formulas, most notably the Hsiung-Minkowski formulas, the Rellich-Pohozaev identities, and variations thereof, to the study of geometric problems and PDE's. Our presentation aims at miderlining the common geometrical and analytical features of such formula.

Anisotropic diffusion of surfaces and functions on surfaces

We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated two-manifold surface meshes inIR3and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine theC1limit representation of Loop's subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively solving the sparse linear system yields a sequence of faired (smoothed) meshes as well as faired functions.

Segmentation for radiotherapy treatment planning using geometric PDEs

Segmentation for radiotherapy treatment planning using geometric PDEs