Linear Algebraic Representation of Big Geometric Data

Transcription

1 Linear Algebraic Representation of Big Geometric Data A. Paoluzzi Università Roma Tre ASI workshop: Big data, nuove tecnologie e futuro dei ground segment Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

2 Outline 1 Motivations 2 New Technology for Computational Modelling and Simulation Linear Algebraic Representation LAR LAR representations: CSR matrices Divide et impera 3 Computational Visualization of Big Geometric Data Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

3 Motivations NASA to Begin New Design Phase For Ground Segment 1 SGSS s importance to the future of NASA s space communications capabilities Space Network Ground Segment Sustainment (SGSS) is updating NASA s Space Network ground communications infrastructure with new, state-of-the-practice technology. 1 Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

4 Motivations NASA to Begin New Design Phase For Ground Segment 1 SGSS s importance to the future of NASA s space communications capabilities Space Network Ground Segment Sustainment (SGSS) is updating NASA s Space Network ground communications infrastructure with new, state-of-the-practice technology. First implemented in the early 1980s and refreshed in the mid-1990s, the ground segment hardware and software is old and increasingly difficult and expensive to sustain. 1 Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

5 Motivations NASA to Begin New Design Phase For Ground Segment 1 SGSS s importance to the future of NASA s space communications capabilities Space Network Ground Segment Sustainment (SGSS) is updating NASA s Space Network ground communications infrastructure with new, state-of-the-practice technology. First implemented in the early 1980s and refreshed in the mid-1990s, the ground segment hardware and software is old and increasingly difficult and expensive to sustain. Installation of an entirely new architecture in each ground terminal Enables easier technology refreshes, simplifies future expansions, and an increase in customer data rate capabilities, while lowering operations and maintenance costs 1 Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

6 Motivations Look at (big) geometric data Similar challenge in CAD/CAE/CAM (PLM): mature technology, hard to parallelise Need: Rethinking the foundations of geometric and topological computing Computational problems in science and technology must deal with increasingly complex geometric information and applications. Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

7 Motivations Look at (big) geometric data Similar challenge in CAD/CAE/CAM (PLM): mature technology, hard to parallelise Need: Rethinking the foundations of geometric and topological computing Computational problems in science and technology must deal with increasingly complex geometric information and applications. Complexity of geometric information stems from dramatic increase in size, diversity, and complexity of geometric data: point clouds, boundary meshes, NURBs representations, finite element meshes, 3D imagery, and so on Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

8 Motivations Rethinking some foundations dealing with Big Data and scalable architectures Google s map-reduce Emerging applications (e.g. space, nano & bio technology, medical 3D) require the convergence of shape synthesis and analysis from: computer imaging computer graphics computer-aided geometric design discrete meshing of domains physical simulations Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

9 Motivations Rethinking some foundations dealing with Big Data and scalable architectures Google s map-reduce Emerging applications (e.g. space, nano & bio technology, medical 3D) require the convergence of shape synthesis and analysis from: computer imaging computer graphics computer-aided geometric design discrete meshing of domains physical simulations The goals of unification, scalability, and massively parallel distributed computing call for rethinking the foundations of geometric and topological computing GOAL: 10 3 times faster and 10 4 times bigger Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

10 Linear Algebraic Representation LAR Example: solid models from 3D images Fast algebraic extraction (GPGPU on Nvidia card) of spongy bone s exact topology Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

11 Linear Algebraic Representation LAR LAR developed in the framework of new medical standard Solid models from 3D images Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

12 Linear Algebraic Representation LAR LAR = algebraic topology + linear algebra A new standard for model topology representation Define a standard for model topology using a very general and simple repr scheme: Models: (co)chain complexes Reprs: sparse binary matrices LAR based on chains, the domains of discrete integration, and cochains, the discrete prototype of differential forms, so naturally integrating the geometric shape with the physical properties SIAM-ACM Geometric and Physical Modeling (GD/SPM), Denver, Nov 11 14, 2013 Computer-Aided Design, Volume 46, January 2014, Pages Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

13 Linear Algebraic Representation LAR Chain and cochain complex C 0, C 1, C 2, C 3 V, E, F, P p Laplacian δ 0, δ 1, δ 2 grad, curl, div cochains (all maps, discrete fields) and coboundary maps (δ d operators) δ d 1 δ d 2 C d C d 1 C 1 C 0 = = = = δ 1 δ 0 d d C d C d 1 C 1 C 0 chains (linear spaces of model subsets) and boundary maps ( d operators) Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

14 Linear Algebraic Representation LAR Chain and cochain complex C 0, C 1, C 2, C 3 V, E, F, P p Laplacian δ 0, δ 1, δ 2 grad, curl, div cochains (all maps, discrete fields) and coboundary maps (δ d operators) δ d 1 δ d 2 C d C d 1 C 1 C 0 = = = = δ 1 δ 0 d d C d C d 1 C 1 C 0 chains (linear spaces of model subsets) and boundary maps ( d operators) δ p = p+1 p = (δ p ) δ p + (δ p 1 ) δ p 1 div curl grad C 3 C 2 C 1 C 0 Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

15 LAR representations: CSR matrices Storage of LAR(X ) CSR(M d ) matrix Amazingly compact storage of a solid model REDUCED LAR (Input and long-term storage) space(lar)= FV = 2 E!!! (Full topology representation) VE + VF = 4 E (Any topological queries) single SpMV multiplication simplicial d-complexes: k = d + 1 cuboidal d-complexes: k = 2 d (Sparse Matrix-Vector Multiplication) is one of the most important computational kernels, for very effective iterative solution methods Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

16 LAR representations: CSR matrices Example: 3D simplicial grid Boundary computation via a single SpMV product Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

17 LAR representations: CSR matrices A random polygon and its boundary Euler characteristic χ = k 0 k 1 + k 2 is pretty 1... chain c 2 chain c 1 = 2 (c 2 ) Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

18 LAR representations: CSR matrices A random polyhedron and its oriented boundary To recover the orientation of 2-cells (unit chains) from that of their coboundary is easy chain c 3 chain c 2 = 3 (c 3 ) Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

19 LAR representations: CSR matrices LAR competitive advantages Several, and very important including mimetic and isogeometric methods Unification computer imaging computer graphics CAD and CAGD discrete meshing of domains physical simulations Compactness amazing long-term storage of solid models Dimensional independence same algorithms on 2D, 3D, 4D representation Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

20 LAR representations: CSR matrices LAR competitive advantages Several, and very important including mimetic and isogeometric methods Unification computer imaging computer graphics CAD and CAGD discrete meshing of domains physical simulations Compactness amazing long-term storage of solid models Dimensional independence same algorithms on 2D, 3D, 4D representation Native parallelism based on linear algebra and GPGPU implementation of SpMV Support of simulations via 3, 2, 1, δ 0, δ 1, δ 2, p Integrated manufacturing e.g.: of composite materials Rapid prototyping tight integration of geometry & CAD/CAE/CAM Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

21 Divide et impera Paradigm: Divide et Impera Bottleneck of GPGPU: moving data from global to local memory Solution: store the (sparse) [ 3] of n 3 voxels in device Constant Memory, and move the (binary) coordinate vectors of chains in Private Memory Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

22 Divide et impera Image serialization & model extraction Serialization is the conversion of a subimage to a stream of bytes (via GPGPU) to be easily moved across the memory hierarchy or streamed across a communication link OpenCL CSR 3 and 2 matrices stored in Constant Memory data cache (read only) single binary (sparse) chains sent to in Local Memory of work groups result vectors from work groups in Local Memory converted to quads in Global Memory each work-group responsible for a chain chunk (subimage) of 16k Input: serialized 3D image Output: stream of 3D model blocks Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

23 Divide et impera Output: Compute with CoChains 3C/LAR Integrated Modeling, Simulation, Manufacturing additive manufacturing (3D printing) Towards the next industrial revolution According to NSF project [Shapiro & Regli, 2013] Cochain Models will encompass computational, logical and information-theoretic models of shape, material, and physical function that apply on all scales (from nano, to meso, to macro) in materials genome, national robotics initiative and advanced manufacturing (3D printing) initiatives Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

24 Divide et impera Output: Compute with CoChains 3C/LAR Integrated Modeling, Simulation, Manufacturing additive manufacturing (3D printing) Towards the next industrial revolution According to NSF project [Shapiro & Regli, 2013] Cochain Models will encompass computational, logical and information-theoretic models of shape, material, and physical function that apply on all scales (from nano, to meso, to macro) in materials genome, national robotics initiative and advanced manufacturing (3D printing) initiatives CochainMachine (U. Wisconsin, Drexel U., RomaTre) cochains as basic objects in a virtually unlimited calculus of physical properties. sound basis for topological, geometrical and physical computations with integrated manufacturing Short term ground management of information from imagery by satellites Long term on-board fabrication (?) for long-range space missions Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

25 Computational Visualization of Big Geometric Data Long-time collaboration with Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

26 Time over... Thank you for attending!! Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20

27 One more thing!!...looking for grants... :o) CCLAR (Compute-with-CoChains over LAR) software in development 1 CAD-PLM Lab, Dip. Matematica e Fisica, Univ Roma Tre 2 Spatial Automation Lab, Univ of Wisconsin at Madison 3 CEDMAV, SCI Institute, Univ of Utah GitHub (social development) Literate programming (Latex + Leo + Nuweb) Haskell (as specification language) Python & PyOpenCL (for rapid Prototyping) Javascript & WebCL (for client-based web applications) C++ & OpenCL (for optimized deployement) Alberto Paoluzzi LAR for Big Geometric Data ASI Rome, Nov 26, / 20