Preconditioning for hyperelasticity-based mesh optimisation

A robust mesh optimisation method is presented that directly enforces the resulting deformation to be orientation preserving. Motivated by aspects from mathematical elasticity, the energy functional of the mesh deformation can be related to a stored energy functional of a hyperelastic material. Formulating the functional in the principal invariants of the deformation gradient allows fine grained control over the resulting deformation. Solution techniques for the arising nonconvex and highly nonlinear system are presented. As existing preconditioners are not sufficient, a PDE-based preconditioner is developed

Diamond-based models for scientific visualization

Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains including scientific visualization, finite element analysis and shape modeling and analysis. A popular class of such approaches is based on the regular simplex bisection operator, which bisects simplices (e.g. line segments, triangles, tetrahedra) along the midpoint of a predetermined edge. Regular simplex bisection produces adaptive simplicial meshes of high geometric quality, while simplifying the extraction of crack-free, or conforming, approximations to the original dataset. Efficient multiresolution representations for such models have been achieved in 2D and 3D by clustering sets of simplices sharing the same bisection edge into structures called diamonds. In this thesis, we introduce several diamond-based approaches for scientific visualization. We first formalize the notion of diamonds in arbitrary dimensions in terms of two related simplicial decompositions of hypercubes. This enables us to enumerate the vertices, simplices, parents and children of a diamond. In particular, we identify the number of simplices involved in conforming updates to be factorial in the dimension and group these into a linear number of subclusters of simplices that are generated simultaneously. The latter form the basis for a compact pointerless representation for conforming meshes generated by regular simplex bisection and for efficiently navigating the topological connectivity of these meshes. Secondly, we introduce the supercube as a high-level primitive on such nested meshes based on the atomic units within the underlying triangulation grid. We propose the use of supercubes to associate information with coherent subsets of the full hierarchy and demonstrate the effectiveness of such a representation for modeling multiresolution terrain and volumetric datasets. Next, we introduce Isodiamond Hierarchies, a general framework for spatial access structures on a hierarchy of diamonds that exploits the implicit hierarchical and geometric relationships of the diamond model. We use an isodiamond hierarchy to encode irregular updates to a multiresolution isosurface or interval volume in terms of regular updates to diamonds. Finally, we consider nested hypercubic meshes, such as quadtrees, octrees and their higher dimensional analogues, through the lens of diamond hierarchies. This allows us to determine the relationships involved in generating balanced hypercubic meshes and to propose a compact pointerless representation of such meshes. We also provide a local diamond-based triangulation algorithm to generate high-quality conforming simplicial meshes

AMM: Adaptive Multilinear Meshes

We present Adaptive Multilinear Meshes (AMM), a new framework that significantly reduces the memory footprint compared to existing data structures. AMM uses a hierarchy of cuboidal cells to create continuous, piecewise multilinear representation of uniformly sampled data. Furthermore, AMM can selectively relax or enforce constraints on conformity, continuity, and coverage, creating a highly adaptive and flexible representation to support a wide range of use cases. AMM supports incremental updates in both spatial resolution and numerical precision establishing the first practical data structure that can seamlessly explore the tradeoff between resolution and precision. We use tensor products of linear B-spline wavelets to create an adaptive representation and illustrate the advantages of our framework. AMM provides a simple interface for evaluating the function defined on the adaptive mesh, efficiently traversing the mesh, and manipulating the mesh, including incremental, partial updates. Our framework is easy to adopt for standard visualization and analysis tasks. As an example, we provide a VTK interface, through efficient on-demand conversion, which can be used directly by corresponding tools, such as VisIt, disseminating the advantages of faster processing and a smaller memory footprint to a wider audience. We demonstrate the advantages of our approach for simplifying scalar-valued data for commonly used visualization and analysis tasks using incremental construction, according to mixed resolution and precision data streams

Approximation methods in geometry and topology: learning, coarsening, and sampling

Data materialize in many different forms and formats. These can be continuous or discrete, from algebraic expressions to unstructured pointclouds and highly structured graphs and simplicial complexes. Their sheer volume and plethora of different modalities used to manipulate and understand them highlight the need for expressive abstractions and approximations, enabling novel insights and efficiency. Geometry and topology provide powerful and intuitive frameworks for modelling structure, form, and connectivity. Acting as a multi-focal lens, they enable inspection and manipulation at different levels of detail, from global discriminant features to local intricate characteristics. However, these fundamentally algebraic theories do not scale well in the digital world. Adjusting topology and geometry to the computational setting is a non-trivial task, adhering to the “no free lunch” adage. The necessary discretizations can be inaccurate, the underlying combinatorial structures can grow unmanageably in size, and computing salient topological and geometric features can become computationally taxing. Approximations are a necessity when theory cannot accommodate for efficient algorithms. This thesis explores different approaches to simplifying computations pertaining to geometry and topology via approximations. Our methods contribute to the approximation of topological features on discrete domains, and employ geometry and topology to efficiently guide discretizations and approximations. This line of work fits un der the umbrella of Topological Data Analysis (TDA) and Discrete Geometry, which aim to bridge the continuous algebraic mindset with the discrete. We construct topological and geometric approximation methods operating on three different levels. We approximate topological features on discrete combinatorial spaces; we approximate the combinatorial spaces themselves; and we guide processes that allow us to discretize domains via sampling. With our Dist2Cycle model we learn geometric manifestations of topological features, the “optimal” homology generating cycles. This is achieved by a novel simplicial complex neural network that exploits the kernel of Hodge Laplacian operators to localize concise homology generators. Compression of meshes and arbitrary simplicial complexes is made possible by our general spectral coarsening strategy. Functional and structural properties are preserved by optimizing for important eigenspaces of general differential operators, the Hodge Laplacians, at multiple dimensions. Finally, we offer a geometry-driven sampling strategy for data accumulation and stochastic integration. By employing the kd-tree geometric partitioning algorithm we construct a sample set with provable equidistribution guarantees. Our findings are contextualized within prior and recent work, and our methods are thoroughly discussed and evaluated on diverse settings. Ultimately, we are making a claim towards the usefulness of examining the ever-present topological and geometric properties of data, not only in terms of feature discovery, but also as informed generation, manipulation, and simplification tools

Nonlinear hyperelasticity-based mesh optimisation

In this work, various aspects of PDE-based mesh optimisation are treated. Different existing methods are presented, with the focus on a class of nonlinear mesh quality functionals that can guarantee the orientation preserving property. This class is extended from simplex to hypercube meshes in 2d and 3d. The robustness of the resulting mesh optimisation method allows the incorporation of unilateral boundary conditions of place and r-adaptivity with direct control over the resulting cell sizes. Also, alignment to (implicit) surfaces is possible, but in all cases, the resulting functional is hard to treat analytically and numerically. Using theoretical results from mathematical elasticity for hyperelastic materials, the existence and non-uniqueness of minimisers can be established. This carries over to the discrete case, for the solution of which tools from nonlinear optimisation are used. Because of the considerable numerical effort, a class of linear preconditioners is developed that helps to speed up the solution process

Harnessing the Power of Distributed Computing: Advancements in Scientific Applications, Homomorphic Encryption, and Federated Learning Security

Data explosion poses lot of challenges to the state-of-the art systems, applications, and methodologies. It has been reported that 181 zettabytes of data are expected to be generated in 2025 which is over 150\% increase compared to the data that is expected to be generated in 2023. However, while system manufacturers are consistently developing devices with larger storage spaces and providing alternative storage capacities in the cloud at affordable rates, another key challenge experienced is how to effectively process the fraction of large scale of stored data in time-critical conventional systems. One transformative paradigm revolutionizing the processing and management of these large data is distributed computing whose application requires deep understanding. This dissertation focuses on exploring the potential impact of applying efficient distributed computing concepts to long existing challenges or issues in (i) a widely data-intensive scientific application (ii) applying homomorphic encryption to data intensive workloads found in outsourced databases and (iii) security of tokenized incentive mechanism for Federated learning (FL) systems.The first part of the dissertation tackles the Microelectrode arrays (MEAs) parameterization problem from an orthogonal viewpoint enlightened by algebraic topology, which allows us to algebraically parametrize MEAs whose structure and intrinsic parallelism are hard to identify otherwise. We implement a new paradigm, namely Parma, to demonstrate the effectiveness of the proposed approach and report how it outperforms the state-of-the-practice in time, scalability, and memory usage.The second part discusses our work on introducing the concept of parallel caching of secure aggregation to mitigate the performance overhead incurred by the HE module in outsourced databases. The key idea of this optimization approach is caching selected radix-ciphertexts in parallel without violating existing security guarantees of the primitive/base HE scheme. A new radix HE algorithm was designed and applied to both batch and incremental HE schemes, and experiments carried out on six workloads show that the proposed caching boost state-of-the-art HE schemes by high orders of magnitudes.In the third part, I will discuss our work on leveraging the security benefit of blockchains to enhance or protect the fairness and reliability of tokenized incentive mechanism for FL systems. We designed a blockchain-based auditing protocol to mitigate Gaussian attacks and carried out experiments with multiple FL aggregation algorithms, popular data sets and a variety of scales to validate its effectiveness