Coffee Time: 15:30 - 16:00. Kernel matrices can be found in computational physics, chemistry, statistics, and machine learning. Fast algorithms for matrix-vector multiplication for kernel matrices have been developed, and is a subject of continuing interest, including here at KAUST. One also often needs fast algorithms to solve systems of equations involving large kernel matrices. Fast direct methods can sometimes be used, for example, when the physical problem is 2-dimensional. In this talk, we address preconditioning for the iterative solution of kernel matrix systems. The spectrum of a kernel matrix significantly depends on the parameters of the kernel function used to define the kernel matrix, e.g., a length scale.

I will discuss the limitations of the current state of the art, and present a proposal for an integrated pipeline, considering data acquisition, meshing, basis design, and numerical optimization as a single challenge, where tradeoffs can be made between different phases to increase automation and efficiency. I will demonstrate that this integrated approach offers many advantages, while opening exciting new geometry processing challenges, and that a fully opaque meshing and analysis solution is already possible for heat transfer and elasticity problems with contact. I will present a set of applications enabled by this approach in reinforcement learning for robotics, force measurements in biology, shape design in mechanical engineering, stress estimation in biomechanics, and simulation of deformable objects in graphics.

Many tasks in the analysis and synthesis of (collections of) 3D shapes boils down to computing a map between two surfaces. Such inter-surface maps can be used to establish correspondences, to transfer information or annotation from one object to another, or to plausibly deform one shape into another. If the two shapes are represented as polygon meshes, a continuous inter-surface map does not only assign the vertices of the source mesh to the target mesh but also maps the interior of the triangles which adds to the complexity of the task.

We will discuss several settings in which smooth developable strips are attached to each other or assembled in some other way to obtain surfaces with interesting geometric properties. Driven by the view towards applications we will investigate collections of strips which lie orthogonal or tangential to a reference surface and assume particular shapes. Such strips can serve as support structures or cladding panels of free-form shapes in architectural contexts. Our focus will lie on surfaces with a constant ratio of principal curvatures, cone nets, geodesic grid shells, and others.

We present a topological analysis of the vorticity formulation in describing fluid dynamics.  Despite its widespread use in fluid mechanics, this formulation is insufficient at describing fluid dynamics on a non-simply-connected domain.  What is missing is an equation of motion for fluid's cohomology component, which exhibits fascinating dynamics previously under explored.  Using geometric language, we derive the new equation of motion and establish new conservation laws, as Casimir invariants in Hamiltonian mechanics, for fluids on domains with general topology.  Significantly, we present the first physically correct vortex method on curved surfaces with genus and boundaries.

Being able to accurately solve PDEs on arbitrary polygonal/polyhedral meshes is a central goal and has been considered for various differential operators over the last years. In this talk I will present a simple approach for computing (piecewise) linear and quadratic basis functions for general polygons and polyhedra, from which discrete operators for gradient, divergence and Laplacian can be derived.

In this talk, I will discuss our recent advances in approximation of free-form surfaces by motions of curvature varying tools in the context of 5-axis flank CNC machining. In particular, I will discuss path-planning strategies using fixed tools, or custom-shaped ones, and on an example of spiral bevel gears will demonstrate even more efficient variant of flank machining, called double-flank.

The “KAUST Workshop on Applied Geometry and Visual Computing” brings together leading scientists from Europe and the United States, presenting their latest results in - Applied and Discrete Differential Geometry - Geometry Processing - Computational Fabrication The talks are related to various problems in Applied Mathematics in general and to further areas of Visual Computing such as Computer Graphics, Physical Simulation and Scientific Visualization. The workshop provides a great opportunity to learn about latest developments and to discuss ongoing work with top researchers in the field.

The Visual Computing Center at KAUST offers a unique opportunity to strengthen this link between architecture and science. We invite interested students of architecture to a week-long workshop on computational architecture. We want to bridge the gap between the latest research in geometry, computer graphics, simulation and modeling, and machine learning and design practice. Participants will have the opportunity to learn directly from our professors about their latest results in their fields.