Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions

We address the discretization of optimization problems posed on the cone of convex functions, motivated in particular by the principal agent problem in economics, which models the impact of monopoly on product quality. Consider a two dimensional domain, sampled on a grid of N points. We show that the cone of restrictions to the grid of convex functions is in general characterized by N^2 linear inequalities; a direct computational use of this description therefore has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of discrete convex functions, associated to stencils which can be adaptively, locally, and anisotropically refined. Numerical experiments optimize the accuracy/complexity tradeoff through the use of a-posteriori stencil refinement strategies.Comment: 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2. Modifications are anecdotic.

Tem_357 Harnessing the Power of Digital Transformation, Artificial Intelligence and Big Data Analytics with Parallel Computing

Traditionally, 2D and especially 3D forward modeling and inversion of large geophysical datasets are performed on supercomputing clusters. This was due to the fact computing time taken by using PC was too time consuming. With the introduction of parallel computing, attempts have been made to perform computationally intensive tasks on PC or clusters of personal computers where the computing power was based on Central Processing Unit (CPU). It is further enhanced with Graphical Processing Unit (GPU) as the GPU has become affordable with the launch of GPU based computing devices. Therefore this paper presents a didactic concept in learning and applying parallel computing with the use of General Purpose Graphical Processing Unit (GPGPU) was carried out and perform preliminary testing in migrating existing sequential codes for solving initially 2D forward modeling of geophysical dataset. There are many challenges in performing these tasks mainly due to lack of some necessary development software tools, but the preliminary findings are promising. Traditionally, 2D and especially 3D forward modeling and inversion of large geophysical datasets are performed on supercomputing clusters. This was due to the fact computing time taken by using PC was too time consuming. With the introduction of parallel computing, attempts have been made to perform computationally intensive tasks on PC or clusters of personal computers where the computing power was based on Central Processing Unit (CPU). It is further enhanced with Graphical Processing Unit (GPU) as the GPU has become affordable with the launch of GPU based computing devices. Therefore this paper presents a didactic concept in learning and applying parallel computing with the use of General Purpose Graphical Processing Unit (GPGPU) was carried out and perform preliminary testing in migrating existing sequential codes for solving initially 2D forward modeling of geophysical dataset. There are many challenges in performing these tasks mainly due to lack of some necessary development software tools, but the preliminary findings are promising.Traditionally, 2D and especially 3D forward modeling and inversion of large geophysical datasets are performed on supercomputing clusters. This was due to the fact computing time taken by using PC was too time consuming. With the introduction of parallel computing, attempts have been made to perform computationally intensive tasks on PC or clusters of personal computers where the computing power was based on Central Processing Unit (CPU). It is further enhanced with Graphical Processing Unit (GPU) as the GPU has become affordable with the launch of GPU based computing devices. Therefore this paper presents a didactic concept in learning and applying parallel computing with the use of General Purpose Graphical Processing Unit (GPGPU) was carried out and perform preliminary testing in migrating existing sequential codes for solving initially 2D forward modeling of geophysical dataset. There are many challenges in performing these tasks mainly due to lack of some necessary development software tools, but the preliminary findings are promising

Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries. The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems. Keywords: Anisotropic mesh generation; metric; Nash embedding theorem; isometric; geometry-conforming; restricted Voronoi diagram; constrained centroidal Voronoi tessellation; Steiner vertices; dimension-independen

Multidimensional upwind hydrodynamics on unstructured meshes using graphics processing units - I. Two-dimensional uniform meshes

SJP is supported by a Royal Society University Research Fellowship

Numerical simulation of 2D electrodynamic problems with unstructured triangular meshes

We present a generalization of standard Yee approach for Cauchy problem in electrodynamic simulations on unstructured triangulated mesh. In the paper the whole flow from mesh creation to actual simulation is presented. The proposed computation flow is parallel ready and can be implemented for distributed systems (computation servers, GPU-s, etc.). We studied the influence of non-regular triangulation on stability and dispersion properties of numerical solution

On some interactive mesh deformations

Techniques devoted to deform 3D models are an important research field in Computer Graphics. They can be used in differentstages: the modelling phase, the animation process and also during some special simulations. Additionally, some applications may require the manipulation of 3D models under certain restrictions to preserve the volume of the modified object. Hence, thepresent PhD Dissertation explores new algorithms to perform flexible, robust and efficient 3D deformations. Apart from this, it also researches on a new methodology to restrict these deformations so that the volume of the manipulated model remains constant. Some of the most used methods to achieve smooth deformations are those included in the Cage-Based Deformation paradigm. Cage-based deformations enclose the model to be deformed in a coarse polyhedron, the cage. Then, they usually rely on Generalized Barycentric Coordinates to relate the model with the vertices, and other geometric elements, of this cage, which are the control points or the deformation handles. Finally, every time that one of these handles is dragged, the model is deformed accordingly. Although this paradigm is simple, elegant and performs efficient deformations, some cage-free space deformation techniques have recently appeared. They increase the flexibility of the deformation handles, which do not need to be connected, and define powerful tools that make the deformation process more versatile and intuitive. In this context, the Dissertation introduces new Generalized Barycentric Coordinate systems specially designed to be used in a cage-free environment. Any user who wants to use the presented schemes only needs to locate a set of control points in the vicinity of the model that he or she wants to deform. These handles can be placed wherever he or she considers mode suitable and the only requirement is that the model has to be enclosed in their convex hull. Up to now, there are few techniques to produce volume-preserving space deformations. However, in recent years there has been a growing interest in performing constrained deformations due to their more realistic and physically plausible results. Our contribution to this research line consists in a deformation framework that preserves the volume of the 3D models by means of its gradient and a control surface to restrict the movement of the handles. Moreover, the proposed methodology is not restricted to the cage-based schemes, but it can also be used in a cage-free environment. Finally, our research can be specially useful for spatial deformations of biological and medical models. This kind of models represent real organs and tissues, which are often soft and lack an internal rigid structure. In addition, they are elastic and incompressible. Any application designed to deal with this group of models and to train or assist doctors must be flexible, robust, efficient and user-friendly. The combination of the proposed cage-free systems with the presented volume-preserving deformation framework satisfiesLes deformacions de models 3D s'utilitzen en diverses etapes de la generació de continguts digitals: durant la fase de modelatge, durant el procés d'animació i en alguns tipus de simulacions. A més a més, hi ha aplicacions que necessiten que la manipulació dels models 3D es faci tenint en compte certes restriccions que permeten la conservació del volum de l'objecte modificat. Tot plegat fa que les tècniques de deformació 3D siguin un camp d'estudi molt important dins del món dels Gràfics. Per aquesta raó, aquesta Tesi Doctoral estudia nous algorismes que permetin realitzar deformacions 3D de manera flexible, robusta i eficient i que, a més a més, permetin conservar el volum dels objectes modificats. Un dels paradigmes més utilitzats per tal de realitzar deformacions suaus és el conegut amb el nom de Deformacions Basades en un Poliedre Englobant. Aquesta família de mètodes embolcalla el model que es vol deformar, normalment representat com una malla de triangles, dins d'un poliedre simple, amb poques cares. Un cop fet això, estableix un sistema de Coordenades Baricèntriques Generalitzades per tal de definir els vèrtexs del model a partir dels vèrtexs del poliedre englobant, els quals s'anomenen punts de control o controls de la deformació. D'aquesta manera, cada cop que s'arrossega o es modifica un d'aquests punts de control, el model que es troba dins del poliedre englobant es deforma segons el sistema de coordenades que s'ha definit. Tot i que aquest paradigma és simple, elegant i eficient, des de fa ja uns anys han començat a aparèixer noves tècniques que no necessiten el poliedre englobant per tal de realitzar la deformació. El seu principal objectiu és augmentar la flexibilitat dels controls de la deformació i definir eines que facin que el procés de deformació sigui més versàtil i intuïtiu. Tenint en compte aquest factor, aquesta Tesi també estudia sistemes de Coordenades Baricèntriques Generalitzades dissenyats per realitzar deformacions sense la necessitat de definir el poliedre englobant. D'aquesta manera, qualsevol usuari que vulgui utilitzar els mètodes que es presenten en aquesta Dissertació només s'ha d'encarregar de definir un conjunt de punts de control al voltant del model que vol deformar, podent-los posar allí on consideri més oportú segons la deformació que vulgui obtenir. L'únic requeriment necessari és que el model ha de quedar dins de l'envolupant convexa d'aquests punts de control. Actualment existeixen pocs mètodes que realitzin deformacions 3D amb preservació del volum. No obstant això, d'un temps ençà ha augmentat l'interès per realitzar deformacions subjectes a certes restriccions que fan que el resultat sigui més realista i físicament versemblant. La contribució d'aquesta Tesi dins d'aquesta línia de recerca consisteix en un sistema de deformació que preserva el volum dels objectes 3D gràcies a còmput del seu gradient i a una superfície de control que restringeix el moviment dels punts de control. Aquest mètode es pot aplicar tant als sistemes de deformació que necessiten un poliedre englobant com als que no el necessiten. Finalment, i ja per acabar, la recerca realitzada pot ser especialment útil per tal de realitzar deformacions de models mèdics i biològics. Aquests tipus de models poden representar òrgans i teixits reals, els quals, normalment, són tous, mancats d'una estructura rígida interna, elàstics i incompressibles. Qualsevol aplicació dissenyada per treballar amb aquest tipus de models i per entrenar i donar assistència a usuaris mèdics hauria de ser flexible, robusta, eficient i fàcil d'utilitzar. La combinació dels mètodes de deformació proposats conjuntament amb el sistema de preservació de volum satisfà totes aquestes condicions. Per aquesta raó es creu que les contribucions realitzades poden esdevenir eines importants per produir deformacions mèdiques.Postprint (published version

Deformable Simplicial Complexes

In this dissertation we present a novel method for deformable interface tracking in 2D and 3D|deformable simplicial complexes (DSC). Deformable interfaces are used in several applications, such as fluid simulation, image analysis, reconstruction or structural optimization. In the DSC method, the interface (curve in 2D; surface in 3D) is represented explicitly as a piecewise linear curve or surface. However, the domain is also subject to discretization: triangulation in 2D; tetrahedralization in 3D. This way, the interface can be alternatively represented as a set of edges/triangles separating triangles/tetrahedra marked as outside from those marked as inside. Such an approach allows for robust topological adaptivity. Among other advantages of the deformable simplicial complexes there are: space adaptivity, ability to handle and preserve sharp features, possibility for topology control. We demonstrate those strengths in several applications. In particular, a novel, DSC-based fluid dynamics solver has been developed during the PhD project. A special feature of this solver is that due to the fact that DSC maintains an explicit interface representation, surface tension is more easily dealt with. One particular advantage of DSC is the fact that as an alternative to topology adaptivity, topology control is also possible. This is exploited in the construction of cut loci on tori where a front expands from a single point on a torus and stops when it self-intersects

Development of the VHP-Female Full-Body Computational Model and Its Applications for Biomedical Electromagnetic Modeling

Computational modeling offers better insight into a wide range of bioelectrical and biomechanical problems with improved tools for the design of medical devices and the diagnosis of pathologies. Electromagnetic modeling at low and high frequencies is particularly necessary. Modeling electromagnetic, structural, thermal, and acoustic response of the human body to different internal and external stimuli is limited by the availability of numerically efficient computational human models. This study describes the development to date of a computational full-body human model - Visible Human Project (VHP) - Female Model. Its unique feature is full compatibility both with MATLAB and specialized FEM computational software packages such as ANSYS HFSS/Maxwell 3D. This study also describes progress made to date in using the newly developed tools for segmentation. A visualization tool is implemented within MATLAB and is based on customized version of the constrained 2D Delaunay triangulation method for intersecting objects. This thesis applies a VHP - Female Model to a specific application, transcranial Direct Current Stimulation (tDCS). Transcranial Direct Current Stimulation has been beneficial in the stimulation of cortical activity and treatment of neurological disorders in humans. The placement of electrodes, which is cephalic versus extracephalic montages, is studied for optimal targeting of currents for a given functional area. Given the difficulty of obtaining in vivo measurements of current density, modeling of conventional and alternative electrode montages via the FEM has been utilized to provide insight into the tDCS montage performance. An insight into future work and potential areas of research, such as study of bone quality have been presented too

Development of Human Body CAD Models and Related Mesh Processing Algorithms with Applications in Bioelectromagnetics

Simulation of the electromagnetic response of the human body relies heavily upon efficient computational CAD models or phantoms. The Visible Human Project (VHP)-Female v. 3.1 - a new platform-independent full-body electromagnetic computational model is revealed. This is a part of a significant international initiative to develop powerful computational models representing the human body. This model’s unique feature is full compatibility both with MATLAB and specialized FEM computational software packages such as ANSYS HFSS/Maxwell 3D and CST MWS. Various mesh processing algorithms such as automatic intersection resolver, Boolean operation on meshes, etc. used for the development of the Visible Human Project (VHP)-Female are presented. The VHP - Female CAD Model is applied to two specific low frequency applications: Transcranial Magnetic Stimulation (TMS) and Transcranial Direct Current Stimulation (tDCS). TMS and tDCS are increasingly used as diagnostic and therapeutic tools for numerous neuropsychiatric disorders. The development of a CAD model based on an existing voxel model of a Japanese pregnant woman is also presented. TMS for treatment of depression is an appealing alternative to drugs which are teratogenic for pregnant women. This CAD model was used to study fetal wellbeing during induced peak currents by TMS in two possible scenarios: (i) pregnant woman as a patient; and (ii) pregnant woman as an operator. An insight into future work and potential areas of research such as a deformable phantom, implants, and RF applications will be presented

Numerical simulation of 2D electrodynamic problems with unstructured triangular meshes

We present a generalization of standard leap-frog plus Yee mesh approach for Cauchy problem in electrodynamics simulations on unstructured triangulated mesh. The presented approach still inherits from finite-difference time-domain and do not use techniques developed in finite-volume time-domain approach. In the paper the whole flow from mesh creation to actual simulation is presented.The proposed computation flow is parallel ready and can be implemented for distributed systems (computation servers, graphical processing units, etc.). We studied the influence of nonregular triangulation on stability and dispersion properties of numerical solution