Optimal Decomposition Meera Sitharam Efficient Realization of Geometric Constraint Introduction Systems via Optimal Recursive Decomposition Recursive De- composition and Cayley Convexification Main Result: Optimal DR-Plan Algorithm Main Result: Meera Sitharam Solving Inde- composables via Cayley University of Florida Convexifica- tion July 29, 2016
Talk Based On.. I Optimal Papers on Decomposition [1] [3] [4] Convex Cayley Spaces, [7] [9] [11] [10] Applications [5] [6] [12] [2] [8] Decomposition Supported in part by NSF/DMS grants 2007,2011,2016 Meera T. Baker, M. Sitharam, M. Wang, and J. Willoughby. Sitharam Optimal decomposition and recombination of isostatic geometric constraint systems for designing layered materials. Computer Aided Geometric Design , 40:1 – 25, 2015. Introduction Mikl´ os B´ ona, Meera Sitharam, and Andrew Vince. Recursive De- composition Enumeration of viral capsid assembly pathways: Treeorbits under permutation group action. Bulletin of Mathematical Biology , 73(4):726–753, 2011. Main Result: Optimal Christoph M. Hoffman, Andrew Lomonosov, and Meera Sitharam. DR-Plan Decomposition plans for geometric constraint systems, part I: Performance measures for CAD. Algorithm Journal of Symbolic Computation , 31(4):367–408, 2001. Main Result: Christoph M Hoffman, Andrew Lomonosov, and Meera Sitharam. Solving Inde- Decomposition plans for geometric constraint systems, part ii: Algorithms. composables Journal of Symbolic Computation , 31(4), 2001. via Cayley Convexifica- Aysegul Ozkan and Meera Sitharam. tion EASAL (efficient atlasing, analysis and search of molecular assembly landscapes). In Proceedings of the ISCA 3rd International Conference on Bioinformatics and Computational Biology, BICoB-2011 , pages 233–238, 2011. Meera Sitharam and Mavis Agbandje-mckenna. Modelling virus self-assembly pathways: Avoiding dynamics using geometric constraint decomposition. J. Comp. Biol , page 65, 2006.
Talk Based On.. II Optimal Decomposition Meera Sitharam and Heping Gao. Characterizing graphs with convex and connected cayley configuration spaces. Meera Discrete & Computational Geometry , 43(3):594–625, 2010. Sitharam Meera Sitharam, Andrew Vince, Menghan Wang, and Mikls Bna. Introduction Symmetry in sphere-based assembly configuration spaces. Symmetry , 8(1):5, 2016. Recursive De- composition Meera Sitharam and Menghan Wang. How the beast really moves: Cayley analysis of mechanism realization spaces using caymos. Main Result: Computer-Aided Design SIAM SPM 2013 issue , 46:205 – 210, 2014. Optimal 2013 { SIAM } Conference on Geometric and Physical Modeling. DR-Plan Algorithm Meera Sitharam and Joel Willoughby. Main Result: On Flattenability of Graphs , pages 129–148. ADG Springer Lecture Notes, 2015. Solving Inde- composables Menghan Wang and Meera Sitharam. via Cayley Convexifica- Algorithm 951: Cayley analysis of mechanism configuration spaces using caymos: Software tion functionalities and architecture. ACM Trans. Math. Softw. , 41(4):27:1–27:8, October 2015. Ruijin Wu, Aysegul Ozkan, Antonette Bennett, Mavis Agbandje-Mckenna, and Meera Sitharam. Robustness measure for an adeno-associated viral shell self-assembly is accurately predicted by configuration space atlasing using easal. In Proceedings of the ACM Conference on Bioinformatics, Computational Biology and Biomedicine , BCB ’12, pages 690–695, New York, NY, USA, 2012. ACM.
Outline Optimal Decomposition Meera Sitharam 1 Introduction Introduction Recursive De- composition 2 Recursive Decomposition Main Result: Optimal DR-Plan Algorithm 3 Main Result: Optimal DR-Plan Algorithm Main Result: Solving Inde- composables via Cayley Convexifica- 4 Main Result: Solving Indecomposables via Cayley tion Convexification
Realizing Linkages given dimension Optimal Decomposition Meera Sitharam EDM Completion given rank = PSD Completion given rank Introduction Recursive De- Definition (Realizing a Linkage) composition Given graph G = ( V , E , δ ) with δ : E → Q , Main Result: Optimal • find/describe the set of all p : V → R d with DR-Plan Algorithm || p u − p v || = δ ( u , v ), modulo trivial transformations. Main Result: Solving Inde- • equivalently, find/describe the set of all completions of composables via Cayley δ ( u , v ) = || p u − p v || from E to V × V . Convexifica- tion
Realization = Solution of GCS Optimal Decomposition Meera Sitharam • Problem: Finding/Roadmapping the real solution set of Introduction the corresponding polynomial (typically quadratic) system. Recursive De- composition • Extends to other Geometric Constraint Systems with Main Result: underlying constraint (hyper)graphs (other distance Optimal DR-Plan metrics, types of constraints), with corresponding trivial Algorithm transformation groups. Main Result: Solving Inde- • Numerous applications: Computer Aided composables via Cayley Mechanical/Structural design, Robotics, Graphics and Convexifica- tion Computer Vision, Molecular Configuration Spaces.
A linkage Optimal Decomposition Meera Sitharam Introduction Recursive De- composition Main Result: Optimal DR-Plan Algorithm Main Result: Solving Inde- composables via Cayley Convexifica- tion
Connected Components Optimal Decomposition Meera Sitharam Introduction Recursive De- composition Main Result: Optimal DR-Plan Algorithm Main Result: Solving Inde- composables via Cayley Convexifica- tion
Problems ● Configuration Space Atlasing & ● Configurational Entropy Computation for – Assemblies of upto 5 rigid molecular motifs given ● pair potentials and sterics ● global constraints
Problems ● Configuration Space Atlasing & ● Configurational Entropy Computation for – Assemblies of upto 5 rigid molecular motifs given ● pair potentials and sterics ● global constraints Courtesy: Atoms in Motion
Problems ● Configuration Space Atlasing & ● Configurational Entropy Computation for – Assemblies of upto 5 rigid molecular motifs – Small molecules with loop closure, pair potentials/sterics Courtesy: CUIK project, Barcelona
Problems ● Configuration Space Atlasing & ● Configurational Entropy Computation for Assemblies of upto 5 rigid molecular motifs – Small molecules with loop closure – – Sticky sphere systems (sterics) Courtesy: “A geometrical approach to computing free-energy landscapes from short-ranged potentials” Miranda Holmes-Cerfon, Steven J. Gortler, Michael P. Brenner PNAS v110(1)
Problems ● Configuration Space Atlasing & ● Computation of Free Energy & Formation Rate (kinetics) for Assemblies of upto 5 rigid molecular motifs – Small molecules with loop closure – Sticky sphere systems – ● Prediction of Crucial Interactions for Larger Assemblies e.g. Viral Shells
Motivating Decomposition Optimal Decomposition Meera –Complexity of solving a quadratic system prohibitively high. Sitharam – Easy Case: Triangularizable System (maintaining degree 2) - Introduction QRS (quadratically radically solvable, or ”ruler and compass” Recursive De- composition systems. Main Result: – A corresponding natural class of graphs: Optimal DR-Plan Algorithm Definition Main Result: For dimension 2, G is △ -decomposable if it is a single edge, or Solving Inde- composables can be divided into 3 △ -decomposable subgraphs s.t. every via Cayley Convexifica- two of them share a single vertex. tion Note: △ -decomposable implies minimally rigid
Optimal Decomposition Meera Sitharam • There is a base edge f with a graph Introduction construction from f : each step Recursive De- v3 appends a new vertex shared by 2 a composition v1 △ -decomposable subgraphs Main Result: Optimal (clusters) v0 v0' DR-Plan Algorithm f • Corresponding linkages have a ruler Main Result: Solving Inde- and compass realization parallel to v2 composables b v4 the graph theoretical construction via Cayley Convexifica- tion • Extends to arbitrary dimension d .
Outline Optimal Decomposition Meera Sitharam 1 Introduction Introduction Recursive De- composition 2 Recursive Decomposition Main Result: Optimal DR-Plan Algorithm 3 Main Result: Optimal DR-Plan Algorithm Main Result: Solving Inde- composables via Cayley Convexifica- 4 Main Result: Solving Indecomposables via Cayley tion Convexification
Decomposition for Recombination Optimal Decomposition Meera Sitharam Introduction Definition (Decomposition-recombination (DR-) plan) Recursive De- composition A DR-plan of constraint graph G is a forest where: Main Result: • Each node is a rigid subgraph of G . Optimal DR-Plan • A root node is a vertex-maximal rigid subgraph. Algorithm Main Result: • An internal node is the union of its children. Solving Inde- composables via Cayley • A leaf node is a single edge Convexifica- tion
Example DR-Plans: C 2 × C 3 Optimal Decomposition Meera Sitharam Introduction Recursive De- composition Main Result: Optimal DR-Plan Algorithm Main Result: Solving Inde- composables via Cayley Convexifica- tion
Example DR-Plans: C 2 × C 3 Optimal Decomposition Meera Sitharam Introduction Recursive De- composition Main Result: Optimal DR-Plan Algorithm Main Result: Solving Inde- composables via Cayley Convexifica- tion
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