Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Energy Sharing Models for Renewable Energy Integration: Subtransmission Level, Distribution Level, and Community Level

Distributed energy resources (DERs) are being embedded rapidly and widely in the power grid and promoting the transformation of the centralized power industry to a more deregulated mode. However, how to safely and efficiently consume renewable energy is becoming a major concern. In this regard, energy sharing at both grid-scale and community-scale has emerged as a new solution to encourage participants to actively bid instead of acting as price takers and has the potential to accelerate the integration of DERs and decrease energy costs. At the grid level, two risk-averse energy sharing models are developed to safely integrate renewable energy by considering the network constraints and overbidding risk. A risk-averse two-stage stochastic game model is proposed for the regional energy sharing market (ESM). The sample average approximation (SAA) method is used to approximate the stochastic Cournot-Nash equilibrium. In addition, a data-driven joint chance-constrained game is developed for energy sharing in the local energy market (LEM). This model considers the maximum outputs of renewable energy aggregators (REAs) are random variables whose probability distributions are unknown, but the decision-maker has access to finite samples. Case studies show that the proposed game models can effectively increase the profit of reliable players and decrease the overbidding risk. At the community level, a community server enables energy sharing among users based on the Bayesian game-based pricing mechanism. It can also control the community energy storage system (CESS) to smooth the load based on the grid's price signal. A communication-censored ADMM for sharing problems is developed to decrease the communication cost between the community and the grid. Moreover, a co-optimization model for the plan and operation of the shared CESS is developed. By introducing the price uncertainty and degradation cost, the proposed model could more accurately evaluate the performance of the CESS and tap more economic potential. This thesis provides proof of the Nash equilibrium of all game models and the convergence of all market clearing algorithms. The proposed models and methods present performance improvement compared with existing solutions. The work in this thesis indicates that energy sharing is possible to implement at different levels of the power system and could benefit the participants and promote the integration of DERs