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1. Optimal order multigrid preconditioners for the distributed control of parabolic equations with coarsening in space and time

   https://doi.org/10.1080/10556788.2021.2022145  

   Drăgănescu, Andrei; Hajghassem, Mona (February 2022, Optimization Methods and Software)

   We devise multigrid preconditioners for linear-quadratic space-time distributed parabolic optimal control problems. While our method is rooted in earlier work on elliptic control, the temporal dimension presents new challenges in terms of algorithm design and quality. Our primary focus is on the cG(s)dG(r) discretizations which are based on functions that are continuous in space and discontinuous in time, but our technique is applicable to various other space-time finite element discretizations. We construct and analyse two kinds of multigrid preconditioners: the first is based on full coarsening in space and time, while the second is based on semi-coarsening in space only. Our analysis, in conjunction with numerical experiments, shows that both preconditioners are of optimal order with respect to the discretization in case of cG(1)dG(r) for r = 0, 1 and exhibit a suboptimal behaviour in time for Crank–Nicolson. We also show that, under certain conditions, the preconditioner using full space-time coarsening is more efficient than the one involving semi-coarsening in space, a phenomenon that has not been observed previously. Our numerical results confirm the theoretical findings. 

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2. Non-euclidean virtual reality II: explorations of H²×E

   Hart, Vi; Hawksley, Andrea; Matsumoto, Elisabetta A; Segerman, Henry (July 2017, Bridges 2017 Conference Proceedings)

   We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at h2xe.hypernom.com. 

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3. Control theory on Wasserstein space: a new approach to optimality conditions

   https://doi.org/10.4310/AMSA.2023.v8.n3.a6  

   Bensoussan, Alain; Huang, Ziyu; Yam, Sheung_Chi Phillip (January 2023, Annals of Mathematical Sciences and Applications)

   We study the deterministic control problem in the Wasserstein space, following the recent works of Bonnet and Frankowska, but with a new approach. One of the major advantages of our approach is that it reconciles the closed loop and the open loop approaches, without the technicalities of the traditional feedback control methodology. It allows also to embed the control problem in the Wasserstein space into a control problem in a Hilbert space, similar to the lifting method introduced by P. L. Lions, used already in our previous works. The Hilbert space is different from that proposed by P. L. Lions, and it allows to recover the control problem in the Wasserstein space as a particular case. 

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4. Semiexplicit symplectic integrators for non-separable Hamiltonian systems

   https://doi.org/10.1090/mcom/3778  

   Jayawardana, Buddhika; Ohsawa, Tomoki (January 2023, Mathematics of Computation)

   We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao’s explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems. 

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5. Temporal Supervised Contrastive Learning for Modeling Patient Risk Progression

   Noroozizadeh, Shahriar; Weiss, Jeremy C; Chen, George H (December 2023, Proceedings of the 3rd Machine Learning for Health Symposium)

   We consider the problem of predicting how the likelihood of an outcome of interest for a patient changes over time as we observe more of the patient’s data. To solve this problem, we propose a supervised contrastive learning framework that learns an embedding representation for each time step of a patient time series. Our framework learns the embedding space to have the following properties: (1) nearby points in the embedding space have similar predicted class probabilities, (2) adjacent time steps of the same time series map to nearby points in the embedding space, and (3) time steps with very different raw feature vectors map to far apart regions of the embedding space. To achieve property (3), we employ a nearest neighbor pairing mechanism in the raw feature space. This mechanism also serves as an alternative to "data augmentation", a key ingredient of contrastive learning, which lacks a standard procedure that is adequately realistic for clinical tabular data, to our knowledge. We demonstrate that our approach outperforms state-of-the-art baselines in predicting mortality of septic patients (MIMIC-III dataset) and tracking progression of cognitive impairment (ADNI dataset). Our method also consistently recovers the correct synthetic dataset embedding structure across experiments, a feat not achieved by baselines. Our ablation experiments show the pivotal role of our nearest neighbor pairing. 

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