267 research outputs found
Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions
In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces
Almost- splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems
Isogeometric Analysis generalizes classical finite element analysis and
intends to integrate it with the field of Computer-Aided Design. A central
problem in achieving this objective is the reconstruction of analysis-suitable
models from Computer-Aided Design models, which is in general a non-trivial and
time-consuming task. In this article, we present a novel spline construction,
that enables model reconstruction as well as simulation of high-order PDEs on
the reconstructed models. The proposed almost- are biquadratic splines on
fully unstructured quadrilateral meshes (without restrictions on placements or
number of extraordinary vertices). They are smooth almost everywhere,
that is, at all vertices and across most edges, and in addition almost (i.e.
approximately) smooth across all other edges. Thus, the splines form
-nonconforming analysis-suitable discretization spaces. This is the
lowest-degree unstructured spline construction that can be used to solve
fourth-order problems. The associated spline basis is non-singular and has
several B-spline-like properties (e.g., partition of unity, non-negativity,
local support), the almost- splines are described in an explicit
B\'ezier-extraction-based framework that can be easily implemented. Numerical
tests suggest that the basis is well-conditioned and exhibits optimal
approximation behavior
Fractional Stochastic Partial Differential Equation for Random Tangent Fields on the Sphere
This paper develops a fractional stochastic partial differential equation
(SPDE) to model the evolution of a random tangent vector field on the unit
sphere. The SPDE is governed by a fractional diffusion operator to model the
L\'{e}vy-type behaviour of the spatial solution, a fractional derivative in
time to depict the intermittency of its temporal solution, and is driven by
vector-valued fractional Brownian motion on the unit sphere to characterize its
temporal long-range dependence. The solution to the SPDE is presented in the
form of the Karhunen-Lo\`{e}ve expansion in terms of vector spherical
harmonics. Its covariance matrix function is established as a tensor field on
the unit sphere that is an expansion of Legendre tensor kernels. The variance
of the increments and approximations to the solutions are studied and
convergence rates of the approximation errors are given. It is demonstrated how
these convergence rates depend on the decay of the power spectrum and variances
of the fractional Brownian motion.Comment: 20 page
On fusogenicity of positively charged phased-separated lipid vesicles: experiments and computational simulations
This paper studies the fusogenicity of cationic liposomes in relation to
their surface distribution of cationic lipids and utilizes membrane phase
separation to control this surface distribution. It is found that concentrating
the cationic lipids into small surface patches on liposomes, through
phase-separation, can enhance liposome's fusogenicity. Further concentrating
these lipids into smaller patches on the surface of liposomes led to an
increased level of fusogenicity. These experimental findings are supported by
numerical simulations using a mathematical model for phase-separated charged
liposomes. Findings of this study may be used for design and development of
highly fusogenic liposomes with minimal level of toxicity
Differential operators on sketches via alpha contours
A vector sketch is a popular and natural geometry representation depicting
a 2D shape. When viewed from afar, the disconnected vector strokes of a
sketch and the empty space around them visually merge into positive space
and negative space, respectively. Positive and negative spaces are the key
elements in the composition of a sketch and define what we perceive as the
shape. Nevertheless, the notion of positive or negative space is mathematically ambiguous: While the strokes unambiguously indicate the interior
or boundary of a 2D shape, the empty space may or may not belong to the
shape’s exterior.
For standard discrete geometry representations, such as meshes or point
clouds, some of the most robust pipelines rely on discretizations of differential operators, such as Laplace-Beltrami. Such discretizations are not
available for vector sketches; defining them may enable numerous applications of classical methods on vector sketches. However, to do so, one needs
to define the positive space of a vector sketch, or the sketch shape.
Even though extracting this 2D sketch shape is mathematically ambiguous,
we propose a robust algorithm, Alpha Contours, constructing its conservative
estimate: a 2D shape containing all the input strokes, which lie in its interior
or on its boundary, and aligning tightly to a sketch. This allows us to define
popular differential operators on vector sketches, such as Laplacian and
Steklov operators.
We demonstrate that our construction enables robust tools for vector
sketches, such as As-Rigid-As-Possible sketch deformation and functional
maps between sketches, as well as solving partial differential equations on a
vector sketch
Shape analysis of the human brain.
Autism is a complex developmental disability that has dramatically increased in prevalence, having a decisive impact on the health and behavior of children. Methods used to detect and recommend therapies have been much debated in the medical community because of the subjective nature of diagnosing autism. In order to provide an alternative method for understanding autism, the current work has developed a 3-dimensional state-of-the-art shape based analysis of the human brain to aid in creating more accurate diagnostic assessments and guided risk analyses for individuals with neurological conditions, such as autism. Methods: The aim of this work was to assess whether the shape of the human brain can be used as a reliable source of information for determining whether an individual will be diagnosed with autism. The study was conducted using multi-center databases of magnetic resonance images of the human brain. The subjects in the databases were analyzed using a series of algorithms consisting of bias correction, skull stripping, multi-label brain segmentation, 3-dimensional mesh construction, spherical harmonic decomposition, registration, and classification. The software algorithms were developed as an original contribution of this dissertation in collaboration with the BioImaging Laboratory at the University of Louisville Speed School of Engineering. The classification of each subject was used to construct diagnoses and therapeutic risk assessments for each patient. Results: A reliable metric for making neurological diagnoses and constructing therapeutic risk assessment for individuals has been identified. The metric was explored in populations of individuals having autism spectrum disorders, dyslexia, Alzheimers disease, and lung cancer. Conclusion: Currently, the clinical applicability and benefits of the proposed software approach are being discussed by the broader community of doctors, therapists, and parents for use in improving current methods by which autism spectrum disorders are diagnosed and understood
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