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1. Efficient quantum algorithm for dissipative nonlinear differential equations

   https://doi.org/10.1073/pnas.2026805118  

   Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K.; Loureiro, Nuno F.; Trivisa, Konstantina; Childs, Andrew M. (August 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R < 1 , where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T 2 q   poly ( log ⁡ T , log ⁡ n , log ⁡ 1 / ϵ ) / ϵ , where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R ≥ 2 . Finally, we discuss potential applications, showing that the R < 1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. 

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2. Homotopy Continuation for Feedback Linearization of Noncontact Magnetic Manipulators

   https://doi.org/10.23919/ACC45564.2020.9147681  

   Riahi, Nayereh; Tituana, Luis R.; Komaee, Arash (July 2020, 2020 American Control Conference (ACC))

   null (Ed.)

   Magnetic fields render a unique ability to control magnetic objects without a direct mechanical contact. To exploit this potential for a broad range of medical, microrobotics, and microfluidics applications, noncontact magnetic manipulators have been designed using both electromagnets and permanent magnets. By feedback control of these manipulators, magnetic objects can be precisely driven in the directions required by an application of interest. The feedback design process for these manipulators is normally complicated by their highly nonlinear nature, particularly for those utilizing permanent magnets. Yet, feedback linearization techniques can be applied to compensate for the nonlinear nature of most magnetic manipulators. This goal can be achieved by solving an underdetermined system of nonlinear algebraic equations. This paper adopts a homotopy continuation approach to solve this system of equations. It is shown by simulations that the proposed feedback linearization scheme drastically improves the control performance compared to the alternative control design methods used in prior work. 

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3. Modeling of Collective Cell Behaviors Interacting With Extracellular Matrix Using Dual Faceted Linearization

   https://doi.org/10.1115/DSCC2018-9164  

   Mayalu, Michaëlle N.; Kim, Min-Cheol; Asada, H. Harry (September 2018, 2018 ASME Dynamic Systems and Control Conference)

   Cells interacting over an extracellular matrix (ECM) exhibit emergent behaviors, which are often observably different from single-cell dynamics. Fibroblasts embedded in a 3-D ECM, for example, compact the surrounding gel and generate an anisotropic strain field, which cannot be observed in single cellinduced gel compaction. This emergent matrix behavior results from collective intracellular mechanical interaction and is crucial to explain the large deformations and mechanical tensions that occur during embryogenesis, tissue development and wound healing. Prediction of multi-cellular interactions entails nonlinear dynamic simulation, which is prohibitively complex to compute using first principles especially as the number of cells increase. Here, we introduce a new methodology for predicting nonlinear behaviors of multiple cells interacting mechanically through a 3D ECM. In the proposed method, we first apply Dual- Faceted Linearization to nonlinear dynamic systems describing cell/matrix behavior. Using this unique linearization method, the original nonlinear state equations can be expressed with a pair of linear dynamic equations by augmenting the independent state variables with auxiliary variables which are nonlinearly dependent on the original states. Furthermore, we can find a reduced order latent space representation of the dynamic equations by orthogonal projection onto the basis of a lower dimensional linear manifold within the augmented variable space. Once converted to latent variable equations, we superpose multiple dynamic systems to predict their collective behaviors. The method is computationally efficient and accurate as demonstrated through its application for prediction of emergent cell induced ECM compaction. 

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4. A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

   https://doi.org/10.1016/j.camwa.2022.08.032  

   Le, Thuy T.; Nguyen, Loc H.; Tran, Hung V. (October 2022, Computers mathematics with applications)

   We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented. 

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5. Koopman Dynamic Modeling for Global and Unified Representations of Rigid Body Systems Making and Breaking Contact

   O'Neill, Cormac; Asada, Harry (October 2024, IEEE)

   A global modeling methodology based on Koopman operator theory for the dynamics of rigid bodies that make and break contact is presented. Traditionally, robotic systems that contact with their environment are represented as a system comprised of multiple dynamic equations that are switched depending on the contact state. This switching of governing dynamics has been a challenge in both task planning and control. Here, a Koopman lifting linearization approach is presented to subsume multiple dynamics such that no explicit switching is required for examining the dynamic behaviors across diverse contact states. First, it is shown that contact/noncontact transitions are continuous at a microscopic level. This allows for the application of Koopman operator theory to the class of robotic systems that repeat contact/non-contact transitions. Second, an effective method for finding Koopman operator observables for capturing rapid changes to contact forces is presented. The method is applied to the modeling of dynamic peg insertion where a peg collides against and bounces on the chamfer of the hole. Furthermore, the method is applied to the dynamic modeling of a sliding object subject to complex friction and damping properties. Segmented dynamic equations are unified with the Koopman modeling method. 

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