More Like this

1. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

   more » « less
2. Neuromorphic Swarm on RRAM Compute-in-Memory Processor for Solving QUBO Problem

   https://doi.org/10.1109/DAC56929.2023.10247852  

   Lele, Ashwin Sanjay ; Chang, Muya ; Spetalnick, Samuel D. ; Crafton, Brian ; Raychowdhury, Arijit ; Fang, Yan ( July 2023 , IEEE)

   Combinatorial optimization problems prevail in engineering and industry. Some are NP-hard and thus become difficult to solve on edge devices due to limited power and computing resources. Quadratic Unconstrained Binary Optimization (QUBO) problem is a valuable emerging model that can formulate numerous combinatorial problems, such as Max-Cut, traveling salesman problems, and graphic coloring. QUBO model also reconciles with two emerging computation models, quantum computing and neuromorphic computing, which can potentially boost the speed and energy efficiency in solving combinatorial problems. In this work, we design a neuromorphic QUBO solver composed of a swarm of spiking neural networks (SNN) that conduct a population-based meta-heuristic search for solutions. The proposed model can achieve about x20 40 speedup on large QUBO problems in terms of time steps compared to a traditional neural network solver. As a codesign, we evaluate the neuromorphic swarm solver on a 40nm 25mW Resistive RAM (RRAM) Compute-in-Memory (CIM) SoC with a 2.25MB RRAM-based accelerator and an embedded Cortex M3 core. The collaborative SNN swarm can fully exploit the specialty of CIM accelerator in matrix and vector multiplications. Compared to previous works, such an algorithm-hardware synergized solver exhibits advantageous speed and energy efficiency for edge devices. 

   more » « less
3. Estimating the Density of States of Boolean Satisfiability Problems on Classical and Quantum Computing Platforms

   https://doi.org/10.1609/aaai.v34i02.5524  

   Sahai, Tuhin ; Mishra, Anurag ; Pasini, Jose Miguel ; Jha, Susmit ( June 2020 , Proceedings of the AAAI Conference on Artificial Intelligence)

   Given a Boolean formula ϕ(x) in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly e clauses, for all values of e. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the hardness of the problem instance. Consequently, in real-world scenarios, this problem is typically infeasible even when using state-of-the-art algorithms. While finding an exact answer to this problem is a computationally intensive task, we propose a novel approach for estimating density of states based on the concentration of measure inequalities. The methodology results in a quadratic unconstrained binary optimization (QUBO), which is particularly amenable to quantum annealing-based solutions. We present the overall approach and compare results from the D-Wave quantum annealer against the best-known classical algorithms such as the Hamze-de Freitas-Selby (HFS) algorithm and satisfiability modulo theory (SMT) solvers. 

   more » « less
4. Solving Quadratic Unconstrained Binary Optimization with Collaborative Spiking Neural Networks

   https://doi.org/10.1109/ICRC57508.2022.00021  

   Fang, Yan ; Lele, Ashwin Sanjay ( March 2023 , 2022 IEEE International Conference on Rebooting Computing (ICRC))

   Quadratic Unconstrained Binary Optimization (QUBO) problem becomes an attractive and valuable optimization problem formulation in that it can easily transform into a variety of other combinatorial optimization problems such as Graph/number Partition, Max-Cut, SAT, Vertex Coloring, TSP, etc. Some of these problems are NP-hard and widely applied in industry and scientific research. Meanwhile, QUBO has been discovered to be compatible with two emerging computing paradigms, neuromorphic computing, and quantum computing, with tremendous potential to speed up future optimization solvers. In this paper, we propose a novel neuromorphic computing paradigm that employs multiple collaborative spiking neural networks to solve QUBO problems. Each SNN conducts a local stochastic gradient descent search and shares the global best solutions periodically to perform a meta-heuristic search for optima. We simulate our model and compare it to a single SNN solver and a mult-SNN solver without collaboration. Through tests on benchmark problems, the proposed method is demonstrated to be more efficient and effective in searching for QUBO optima. Specifically, it exhibits x10 and x15-20 speedup respectively on the multi-SNN solver without collaboration and the single-SNN solver. 

   more » « less
5. Hybrid Quantum-Classical Algorithms for Solving the Weighted CSP

   Xu, H ; Sun, OK. ; Koenig, S. ; Hen, I. ; Kumar, S. ( January 2020 , Proceedings of the International Symposium on Artificial Intelligence and Mathematics)

   Many kinds of algorithms have been developed for solving the constraint satisfaction problem (WCSP), a combinatorial optimization problem that frequently appears in AI. Unfortunately, its NP-hardness prohibits the existence of an algorithm for solving it that is universally efficient on classical computers. Therefore, a peek into quantum computers may be imperative for solving the WCSP efficiently. In this paper, we focus on a specific type of quantum computer, called the quantum annealer, which approximately solves quadratic unconstrained binary optimization (QUBO) problems. We propose the first three hybrid quantum-classical algorithms (HQCAs) for the WCSP: one specifically for the binary Boolean WCSP and the other two for the general WCSP. We experimentally show that the HQCA based on simple polynomial reformulation works well on the binary Boolean WCSP, but the HQCA based on the constraint composite graph works best on the general WCSP. 

   more » « less