266 research outputs found
Interactive Extraction of High-Frequency Aesthetically-Coherent Colormaps
Color transfer functions (i.e. colormaps) exhibiting a high frequency luminosity component have proven to be useful in the visualization of data where feature detection or iso-contours recognition is essential. Having these colormaps also display a wide range of color and an aesthetically pleasing composition holds the potential to further aid image understanding and analysis. However producing such colormaps in an efficient manner with current colormap creation tools is difficult. We hereby demonstrate an interactive technique for extracting colormaps from artwork and pictures. We show how the rich and careful color design and dynamic luminance range of an existing image can be gracefully captured in a colormap and be utilized effectively in the exploration of complex datasets
R-adaptive multisymplectic and variational integrators
Moving mesh methods (also called r-adaptive methods) are space-adaptive
strategies used for the numerical simulation of time-dependent partial
differential equations. These methods keep the total number of mesh points
fixed during the simulation, but redistribute them over time to follow the
areas where a higher mesh point density is required. There are a very limited
number of moving mesh methods designed for solving field-theoretic partial
differential equations, and the numerical analysis of the resulting schemes is
challenging. In this paper we present two ways to construct r-adaptive
variational and multisymplectic integrators for (1+1)-dimensional Lagrangian
field theories. The first method uses a variational discretization of the
physical equations and the mesh equations are then coupled in a way typical of
the existing r-adaptive schemes. The second method treats the mesh points as
pseudo-particles and incorporates their dynamics directly into the variational
principle. A user-specified adaptation strategy is then enforced through
Lagrange multipliers as a constraint on the dynamics of both the physical field
and the mesh points. We discuss the advantages and limitations of our methods.
Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure
Discrete Poincaré Lemma
This paper proves a discrete analogue of the PoincarÂŽe lemma in the context of a discrete exterior calculus based on simplicial cochains. The proof requires the construction of a generalized cone operator, p : Ck(K) -> Ck+1(K), as the geometric cone of a simplex cannot, in general, be interpreted as a chain in the simplicial complex. The corresponding cocone operator H : Ck(K) -> Ckâ1(K) can be shown to be a homotopy operator, and this yields the discrete PoincarÂŽe lemma.
The generalized cone operator is a combinatorial operator that can be constructed for any simplicial complex that can be grown by a process of local augmentation. In particular, regular triangulations and tetrahedralizations of R2 and R3 are presented, for which the discrete PoincarÂŽe lemma is globally valid
Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT
We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions over the domain . Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to -convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem
On the coupling between an ideal fluid and immersed particles
In this paper we use Lagrange-Poincare reduction to understand the coupling
between a fluid and a set of Lagrangian particles that are supposed to simulate
it. In particular, we reinterpret the work of Cendra et al. by substituting
velocity interpolation from particle velocities for their principal connection.
The consequence of writing evolution equations in terms of interpolation is
two-fold. First, it gives estimates on the error incurred when interpolation is
used to derive the evolution of the system. Second, this form of the equations
of motion can inspire a family of particle and hybrid particle-spectral methods
where the error analysis is "built-in". We also discuss the influence of other
parameters attached to the particles, such as shape, orientation, or
higher-order deformations, and how they can help with conservation of momenta
in the sense of Kelvin's circulation theorem.Comment: to appear in Physica D, comments and questions welcom
Variational Partitioned RungeâKutta Methods for Lagrangians Linear in Velocities
In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the âHamiltonianâ equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational RungeâKutta methods and analyze their properties. The general properties of RungeâKutta methods depend on the âvelocityâ part of the Lagrangian. If the âvelocityâ part is also linear in the position coordinate, then we show that non-partitioned variational RungeâKutta methods are equivalent to integration of the corresponding first-order EulerâLagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the RungeâKutta method are retained. If the âvelocityâ part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We verified our results through numerical experiments for various dynamical systems
A Discrete Geometric Optimal Control Framework for Systems with Symmetries
This paper studies the optimal motion control of
mechanical systems through a discrete geometric approach. At
the core of our formulation is a discrete Lagrange-dâAlembert-
Pontryagin variational principle, from which are derived discrete
equations of motion that serve as constraints in our optimization
framework. We apply this discrete mechanical approach to
holonomic systems with symmetries and, as a result, geometric
structure and motion invariants are preserved. We illustrate our
method by computing optimal trajectories for a simple model of
an air vehicle flying through a digital terrain elevation map, and
point out some of the numerical benefits that ensue
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