Dirichlet Energy Research Articles

Worst-case shape optimization for the Dirichlet energy

We consider the optimization problem for a shape cost functional F(Ω,f) which depends on a domain Ω varying in a suitable admissible class and on a “right-hand side” f. More precisely, the cost functional F is given by an integral which involves the solution u of an elliptic PDE in Ω with right-hand side f; the boundary conditions considered are of the Dirichlet type. When the function f is only known up to some degree of uncertainty, our goal is to obtain the existence of an optimal shape in the worst possible situation. Some numerical simulations are provided, showing the difference in the optimal shape between the case when f is perfectly known and the case when only the worst situation is optimized.

Discrete Connection and Covariant Derivative for Vector Field Analysis and Design

In this article, we introduce a discrete definition of connection on simplicial manifolds, involving closed-form continuous expressions within simplices and finite rotations across simplices. The finite-dimensional parameters of this connection are optimally computed by minimizing a quadratic measure of the deviation to the (discontinuous) Levi-Civita connection induced by the embedding of the input triangle mesh, or to any metric connection with arbitrary cone singularities at vertices. From this discrete connection, a covariant derivative is constructed through exact differentiation, leading to explicit expressions for local integrals of first-order derivatives (such as divergence, curl, and the Cauchy-Riemann operator) and for L 2 -based energies (such as the Dirichlet energy). We finally demonstrate the utility, flexibility, and accuracy of our discrete formulations for the design and analysis of vector, n -vector, and n -direction fields.

The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

We investigate the approximation of the Monge problem (minimizing ∫Ω|T(x)−x|dμ(x) among the vector-valued maps T with prescribed image measure T#μ) by adding a vanishing Dirichlet energy, namely ε∫Ω|DT|2. We study the Γ-convergence as ε→0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H1 map, we study the selected limit map, which is a new “special” Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on ε, where the leading term is of order ε|log⁡ε|.

On uniqueness of heat flow of harmonic maps

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N,h) that have sufficiently small renormalized energies, provided that N is either a unit sphere Sk−1 or a Riemannian homogeneous manifold. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for t ≥ t0 > 0 and the unique limit property at time infinity. As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact Riemannian manifold N whose gradients belong to LqtL l x, for q > 2 and l > n satisfying the Serrin’s condition.

Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy

In this article we present a finite element scheme for solving a nonlocal parabolic problem involving the Dirichlet energy. For time discretization, we use backward Euler method. The nonlocal term causes difficulty while using Newton’s method. Indeed, after applying Newton’s method we get a full Jacobian matrix due to the nonlocal term. In order to avoid this difficulty we use the technique given by Gudi (SIAM J Numer Anal 50(2):657–668, 2012) for elliptic nonlocal problem of Kirchhoff type. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for both semi-discrete and fully discrete formulations. Results based on the usual finite element method are provided to confirm the theoretical estimates.

Asymmetric Domain Walls of Small Angle in Soft Ferromagnetic Films

We focus on a special type of domain walls appearing in the Landau-Lifshitz theory for soft ferromagnetic films. These domain walls are divergence-free $S^2$-valued transition layers that connect two directions in $S^2$ (differing by an angle $2\theta$) and minimize the Dirichlet energy. Our main result is the rigorous derivation of the asymptotic structure and energy of such "asymmetric" domain walls in the limit $\theta \to 0$. As an application, we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model.

Orbifold Tutte embeddings

Injective parameterizations of surface meshes are vital for many applications in Computer Graphics, Geometry Processing and related fields. Tutte's embedding, and its generalization to convex combination maps, are among the most popular approaches for computing parameterizations of surface meshes into the plane, as they guarantee injectivity, and their computation only requires solving a sparse linear system. However, they are only applicable to disk-type and toric surface meshes. In this paper we suggest a generalization of Tutte's embedding to other surface topologies, and in particular the common, yet untreated case, of sphere-type surfaces. The basic idea is to enforce certain boundary conditions on the parameterization so as to achieve a Euclidean orbifold structure. The orbifold-Tutte embedding is a seamless, globally bijective parameterization that, similarly to the classic Tutte embedding, only requires solving a sparse linear system for its computation. In case the cotangent weights are used, the orbifold-Tutte embedding globally minimizes the Dirichlet energy and is shown to approximate conformal and four-point quasiconformal mappings. As far as we are aware, this is the first fully-linear method that produces bijective approximations to conformal mappings. Aside from parameterizations, the orbifold-Tutte embedding can be used to generate bijective inter-surface mappings with three or four landmarks and symmetric patterns on sphere-type surfaces.

Constructing constant mean curvature surfaces with fixed-point half dynamic model

In this paper, we present a method to generate a constant mean curvature (CMC) surface with the given boundary. By assuming the velocity to zero in the dynamic model for every step, a fixed-point iteration is carried out. Either the variation of Dirichlet energy or area energy can be chosen to be the approximation of the mean curvature. Experiments and results show that our method can work for minimal surfaces or CMC surfaces with nonzero mean curvature and with fixed or free boundary. Meanwhile it is robust with poor quality initial mesh as well as non-uniformly distributed vertices.

Moduli spaces of lumps on real projective space

Harmonic maps that minimise the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the behaviour of lumps and their symmetries. An interesting feature is that the moduli space of charge three lumps is a 7-dimensional manifold of cohomogeneity one which can be described as a one-parameter family of symmetry orbits of D2 symmetric maps. In this paper, we discuss the charge three moduli spaces of lumps from two perspectives: discrete symmetries of lumps and the Riemann-Hurwitz formula. We then calculate the metric and find explicit formulas for various geometric quantities. We also discuss the implications for lump decay.

Close-to-conformal deformations of volumes

Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to a conformal deformation is nonlinear and in any case only admits Möbius transformations as solutions. We propose a particular decoupling of scaling and rotation which allows us to find near to conformal deformations as minimizers of a quadratic, convex Dirichlet energy. Applied to tetrahedral meshes we find deformations with low quasiconformal distortion as the principal eigenvector of a (quaternionic) Laplace matrix. The resulting algorithms can be implemented with highly optimized standard linear algebra libraries and yield deformations comparable in quality to far more expensive approaches.

Optimal profiles in a phase-transition model with a saturating flux

It is well known that for the Allen–Cahn equation, the minimizing transition in an infinite cylinder R×ω is one-dimensional and unique up to a translation in the first variable. We analyze in this paper the existence and symmetry of optimal profiles for transitions in a similar phase-separation model with a saturating flux. This amounts to consider transitions in the space of BV functions as we consider the area integral instead of the Dirichlet energy to penalize the creation of wild interfaces.

Robust construction of minimal surface from general initial mesh

We analyze three commonly used energy functions in solving Plateau-Mesh Problem, that is, Dirichlet, area, and the discrete mean curvature(DMC). They all possess unique advantages compared to others, but their drawbacks restrict their usages individually. Our algorithm combines the three steps together to make full use of their features. At first the Dirichlet energy is optimized for faster approximation with better topology. Then the area energy is used to come close to the constrained domain. Finally the DMC energy is engaged to achieve a better converging step. Results show that our method can work under a rather noisy initial mesh, which is even topologically different from the final result.

Multiple valued maps into a separable Hilbert space that almost minimize their $$p$$ p Dirichlet energy or are squeeze and squash stationary

Let \(f : U\subseteq \mathbb {R}^m\rightarrow \fancyscript{Q}_Q(\ell _2)\) be of Sobolev class \(W^{1,p}\), \(1 < p < \infty \). If \(f\) almost minimizes its \(p\) Dirichlet energy then \(f\) is Holder continuous. If \(p=2\) and \(f\) is squeeze and squash stationary then \(f\) is in VMO.

Semicontinuity of eigenvalues under intrinsic flat convergence

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min–max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min–max values of the limit space are larger than or equal to the upper limit of the min–max values of the currents in the sequence. In particular, the infimum of the normalized energy is semicontinuous. On spaces that are infinitesimally Hilbertian, we can define a linear Laplace operator. We can show that semicontinuity under intrinsic flat convergence holds for eigenvalues below the essential spectrum, if the total volume of the spaces converges as well.

Near BPS skyrmions and restricted harmonic maps

Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map φ:(M,g)→(N,h) between Riemannian manifolds is restricted harmonic if it locally extremizes E2 on its SDiff(M) orbit, where SDiff(M) denotes the group of volume preserving diffeomorphisms of (M,g), and E2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that φ is restricted harmonic if and only if φ∗h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R3→SU(2) and R2→S2 are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E2 for φ:Rk→N over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted F-criticality where F is any functional on maps (M,g)→(N,h) which is, in a precise sense, geometrically natural. The case where F is a linear combination of E2 and E4, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R3→S3≡SU(2) is stable restricted F-critical for every such F.

On critical points of the [formula omitted]-energy over a space of measure preserving maps

Let X⊂Rn be a bounded Lipschitz domain and consider the σ2-energy functional Fσ2[u;X]:=∫X|∇u∧∇u|2dx, over the space of admissible maps A(X)={u∈W1,4(X,Rn):u|∂X=x, det∇u=1 for Ln-a.e. in X}. A good measure of how much a map u stretches areas (of 2-dimensional sub-manifolds of the domain X) is the norm of ∇u∧∇u:∧2TX→∧2TX, analogously to |∇u|2 (the Dirichlet energy density) that is a measure of length’s stretching. These kinds of functionals also were arisen as a physical model describing the strong interactions of quantum fields which was introduced by T. Skyrme in 1961. In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler–Lagrange equations associated with Fσ2[⋅;X] over A(X). In particular we present a novel characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions which follows very closely to the ideas that have been introduced by the second author in his recent paper Shahrokhi-Dehkordi and Taheri (2009) [17]. Indeed we show that in even dimensions the latter system of equations admits infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler–Lagrange equations. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.

Dynamics-Adapted Cone Kernels

We present a family of kernels for analysis of data generated by dynamical systems. These so-called cone kernels feature a dependence on the dynamical vector field operating in the phase space manifold, estimated empirically through finite differences of time-ordered data samples. In particular, cone kernels assign strong affinity to pairs of samples whose relative displacement vector lies within a narrow cone aligned with the dynamical vector field. The outcome of this explicit dependence on the dynamics is that, in a suitable asymptotic limit, Laplace--Beltrami operators for data analysis constructed from cone kernels generate diffusions along the integral curves of the dynamical vector field. This property is independent of the observation modality, and endows these operators with invariance under a weakly restrictive class of transformations of the data (including conformal transformations), while it also enhances their capability to extract intrinsic dynamical timescales via eigenfunctions. Here, we study these features by establishing the Riemannian metric tensor induced by cone kernels in the limit of large data. We find that the corresponding Dirichlet energy is governed by the directional derivative of functions along the dynamical vector field, giving rise to a measure of roughness of functions that favors slowly varying observables. We demonstrate the utility of cone kernels in nonlinear flows on the 2-torus and North Pacific sea surface temperature data generated by a comprehensive climate model.

On the Optimality of Shape and Data Representation in the Spectral Domain

A proof of the optimality of the eigenfunctions of the Laplace--Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant--Fischer min-max principle adapted to our case. The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilization of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudometric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can efficiently handle cases that go beyond the scope defined by the observation set that is handled by regular PCA.

Asymptotic Analysis of a Neumann Problem in a Domain with Cusp. Application to the Collision Problem of Rigid Bodies in a Perfect Fluid

We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated with the solution of a Laplace--Neumann problem as the distance $\varepsilon>0$ between the solid and the cavity's bottom tends to zero. Denoting by $\alpha>0$ the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a nonzero velocity for $\alpha <2$ (real shock case), and with null velocity for $\alpha \geqslant 2$ (smooth landing case). Our proof is based on a suitable change of variables sending to infinity the cusp singularity at the contact. More precisely, for every $\varepsilon\geqslant 0$, we transform the Laplace--Neumann problem into a generalized Neumann problem set on a domain containing a horizontal strip $]0,\ell_\varepsilon[\times ]0,1[$, where $\ell_\varepsilon\to +\infty$.

The Hopf-Laplace equation: harmonicity and regularity

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to mappings which are neither harmonic nor homeomorphisms. We prove that such mappings are harmonic outside of a singular set with small image. On the singular set they are locally Lipschitz, but not necessarily differentiable