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We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an auto-encoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train a neural network to map the latent variables representing the boundary data to the solution of the PDE. Finally, we solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method in the linear setting and establish optimal error estimates in the $H^1$ and $L^2$-norms. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. Tikhonov regularisation is necessary for high order polynomial approximation, , but its weak consistency may be tuned to allow for optimal convergence. For low order elements a simple numerical scheme with optimal convergence is obtained without stabilization. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability and consistency assumptions. Numerical experiments are provided to illustrate the theoretical results.

In arXiv:2312.14108, we argued that a sliver at the black hole mass in the Hilbert space of a quantum field theory on an AdS black hole with a stretched horizon, has many desirable features of black hole microstates. A key observation is that the stretched horizon requires a finite Planck length, i.e., finite-$N$. Therefore it is best viewed as a ``quantum horizon" -- a proxy for the UV-complete bulk description, and not directly an element in the bulk EFT. It was shown in arXiv:2312.14108 that despite the manifest absence of the interior, the 2-point function in the sliver is indistinguishable from the smooth horizon correlator, up to the Page time. In this paper, instead of boundary correlators, we work directly in the bulk, and demonstrate the appearance of the bulk Hartle-Hawking correlator in the large-$N$ limit. This is instructive because it shows that the analytic continuation across the horizon is emergent. It is a $bulk$ transition to a Type III algebra and provides a structural distinction between black holes and weakly coupled thermal systems. We also identify a mechanism for universal code subspaces and interior tensor factors to appear via a quantum horizon version of thermal factorization. Our claims apply directly only within a ``Page window", so they are not in immediate tension with firewall arguments. During a Page window, typical heavy microstates probed by light single trace operators respond with effectively smooth horizons in low-point correlators. We work in 2+1 dimensions to be concrete, but expect our results to hold in all higher dimensions. We discuss some important differences between our approach and the conventional fuzzball program, and also argue that probe-notions (like infalling boundary conditions) must be distinguished from microstate-notions (like size of the Einstein-Rosen bridge) to make meaningful statements about post-Page smoothness., Comment: 70 pages, 3 contractions, 1 wormhole

Event-based vision sensors produce an asynchronous stream of events which are triggered when the pixel intensity variation exceeds a predefined threshold. Such sensors offer significant advantages, including reduced data redundancy, micro-second temporal resolution, and low power consumption, making them valuable for applications in robotics and computer vision. In this work, we consider the problem of video reconstruction from events, and propose an approach for dynamic feedback control of activation thresholds, in which a controller network analyzes the past emitted events and predicts the optimal distribution of activation thresholds for the following time segment. Additionally, we allow a user-defined target peak-event-rate for which the control network is conditioned and optimized to predict per-column activation thresholds that would eventually produce the best possible video reconstruction. The proposed OnTheFly control scheme is data-driven and trained in an end-to-end fashion using probabilistic relaxation of the discrete event representation. We demonstrate that our approach outperforms both fixed and randomly-varying threshold schemes by 6-12% in terms of LPIPS perceptual image dissimilarity metric, and by 49% in terms of event rate, achieving superior reconstruction quality while enabling a fine-tuned balance between performance accuracy and the event rate. Additionally, we show that sampling strategies provided by our OnTheFly control are interpretable and reflect the characteristics of the scene. Our results, derived from a physically-accurate simulator, underline the promise of the proposed methodology in enhancing the utility of event cameras for image reconstruction and other downstream tasks, paving the way for hardware implementation of dynamic feedback EVS control in silicon., Comment: 17 pages, 5 figures, ECCV 2024, NEVI workshop

In this article, we first obtain, for the Kolmogorov distance, an error bound between a tempered stable and a compound Poisson distribution and also an error bound between a tempered stable and an alpha stable distribution via Stein method. For the smooth Wasserstein distance, an error bound between two tempered stable distributions is also derived. As examples, we discuss the approximation of a tempered stable to normal and variance gamma distributions. As corollaries, the corresponding limit theorem follows, Comment: 19 pages

The M6 Competition assessed the performance of competitors using a ranked probability score and an information ratio (IR). While these metrics do well at picking the winners in the competition, crucial questions remain for investors with longer-term incentives. To address these questions, we compare the competitors' performance to a number of conventional (long-only) and alternative indices using standard industry metrics. We apply factor models to measure the competitors' value-adds above industry-standard benchmarks and find that competitors with more extreme performance are less dependent on the benchmarks. We also uncover that most competitors could not generate significant out-performance compared to randomly selected long-only and long-short portfolios but did generate out-performance compared to short-only portfolios. We further introduce two new strategies by picking the competitors with the best (Superstars) and worst (Superlosers) recent performance and show that it is challenging to identify skill amongst investment managers. We also discuss the incentives of winning the competition compared to professional investors, where investors wish to maximize fees over an extended period of time., Comment: Forecasting Competitions, M Competitions, Financial Analysis, Investment Management, Hedge Fund, Portfolio Optimization

In this paper we continue the work on implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations that we started in \cite{BGG23} (E. Burman, D. Garg, J. Guzm\`an, {\emph{Implicit-explicit time discretization for Oseen's equation at high Reynolds number with application to fractional step methods}}, SIAM J. Numer. Anal., 61, 2859--2886, 2023). The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein we focus on the implicit-explicit Crank-Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions we prove stability of the implicit-explicit Crank-Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order $O(h^{k+\frac12} + \tau^2)$. Here $h$ is the mesh parameter, $k$ the polynomial order and $\tau$ the time step. Finally we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier-Stokes' equations., Comment: 33 pages

Probing magnetic fields in high-redshift galactic systems is crucial to investigate galactic dynamics and evolution. Utilizing the rotation measure of the background quasars, we have developed a radial profile of the magnetic field in a typical high-$z$ galaxy. We have compiled a catalog of 59 confirmed quasar sightlines, having one intervening Mg \rom{2} absorber in the redshift range $0.372\leq z_{\text{abs}} \leq 0.8$. The presence of the foreground galaxy is ensured by comparing the photometric and spectroscopic redshifts within $3 \sigma_{z-\text{photo}}$ and visual checks. These quasar line-of-sights (LoS) pass through various impact parameters (D) up to $160$ kpc, covering the circumgalactic medium of a typical Milky-Way type galaxy. Utilizing the residual rotation measure (RRM) of these sightlines, we estimated the excess in RRM dispersion, $\sigma_{\text{ex}}^{\text{RRM}}$. We found $\sigma_{\text{ex}}^{\text{RRM}}$ decreases with increasing D. We translated $\sigma_{\text{ex}}^{\text{RRM}}$ to average LoS magnetic field strength, $\langle B_{\|}\rangle$ by considering a typical electron column density. Consequently, the decreasing trend is sustained in the magnetic field. In particular for sightlines with $\text{D} \leq 50$ kpc and $\text{D} > 50$ kpc, $\langle B_{\|}\rangle$ is found to be $2.39 \pm 0.7 \ \mu$G and $1.67 \pm 0.38 \ \mu$G, respectively. This suggests a clear indication of varying magnetic field from the disk to the circumgalactic medium. This work provides a methodology that, when applied to ongoing and future radio polarisation surveys such as LOFAR and SKA, promises to significantly enhance our understanding of magnetic field mapping in galactic systems., Comment: Published in JCAP

We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time methods have been used in previous work to obtain stable and accurate solutions to this reconstruction problem. Here we propose to solve the problem using a standard discontinuous Galerkin method for the temporal discretization and continuous finite elements for the space discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies which can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.

We consider a parabolic-parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin-Robin splitting method analyzed in [J. Numer. Math., 31(1):59--77, 2023]. We show that the errors of the correction step converge at $\mathcal O((\Delta t)^2)$, under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where $\Delta t$ stands for the time-step length. Numerical results are shown to support our analysis and the assumptions.

We consider a loosely coupled, non-iterative Robin-Robin coupling method proposed and analyzed in [J. Numer. Math., 31(1):59--77, 2023] for a parabolic-parabolic interface problem and prove estimates for the discrete time derivatives of the scalar field in different norms. When the interface is flat and perpendicular to two of the edges of the domain we prove error estimates in the $H^2$-norm. Such estimates are key ingredients to analyze a defect correction method for the parabolic-parabolic interface problem. Numerical results are shown to support our findings.

In this paper, we consider the unique continuation problem for the Schr\"odinger equations. We prove a H\"older type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a priori} knowledge of the solution, achieving error estimates in interior domains with convergence up to continuous stability. The approximability of the scheme to solutions with only $H^1$-regularity is studied and the convergence rate for solutions with regularity higher than $H^1$ is also shown. Comparisons in terms of different parameterization for different regularities will be illustrated with respect to the convergence and condition numbers of the linear systems. Finally, numerical experiments will be given to illustrate the theory.

We provide a direct correspondence between the $b$-Hurwitz numbers with $b=1$ from \cite{ChapuyDolega}, and twisted Hurwtiz numbers from \cite{TwistedHurwitz}. This provides a description of real coverings of the sphere with ramification on the real line in terms of monodromy., Comment: 15 pages, 6 figures

This paper discusses the effect of rotation on the boundary layer in high Reynolds number flow over a ridge using a numerical method based on stabilised finite elements that captures steady solutions up to Reynolds number of order $10^6$. The results are validated against boundary layer computations in shallow flows and for deep flows against experimental observations reported in Machicoane et al. (Phys. Rev. Fluids, 2018). In all cases considered the boundary layer remains attached, even at large Reynolds numbers, provided the Rossby number of the flow is sufficiently small. At any fixed Rossby number the flow detaches at sufficiently high Reynolds number to form a steady recirculating region in the lee of the ridge. At even higher Reynolds numbers no steady flow is found. This disappearance of steady solutions closely reproduces the transition to unsteadiness seen in the laboratory.

We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.

Reinforcement Learning (RL) has made significant strides in enabling artificial agents to learn diverse behaviors. However, learning an effective policy often requires a large number of environment interactions. To mitigate sample complexity issues, recent approaches have used high-level task specifications, such as Linear Temporal Logic (LTL$_f$) formulas or Reward Machines (RM), to guide the learning progress of the agent. In this work, we propose a novel approach, called Logical Specifications-guided Dynamic Task Sampling (LSTS), that learns a set of RL policies to guide an agent from an initial state to a goal state based on a high-level task specification, while minimizing the number of environmental interactions. Unlike previous work, LSTS does not assume information about the environment dynamics or the Reward Machine, and dynamically samples promising tasks that lead to successful goal policies. We evaluate LSTS on a gridworld and show that it achieves improved time-to-threshold performance on complex sequential decision-making problems compared to state-of-the-art RM and Automaton-guided RL baselines, such as Q-Learning for Reward Machines and Compositional RL from logical Specifications (DIRL). Moreover, we demonstrate that our method outperforms RM and Automaton-guided RL baselines in terms of sample-efficiency, both in a partially observable robotic task and in a continuous control robotic manipulation task.

Biopolymer topology is critical for determining interactions inside cell environments, exemplified by DNA where its response to mechanical perturbation is as important as biochemical properties to its cellular roles. The dynamic structures of chiral biopolymers exhibit complex dependence with extension and torsion, however the physical mechanisms underpinning the emergence of structural motifs upon physiological twisting and stretching are poorly understood due to technological limitations in correlating force, torque and spatial localization information. We present COMBI-Tweez (Combined Optical and Magnetic BIomolecule TWEEZers), a transformative tool that overcomes these challenges by integrating optical trapping, time-resolved electromagnetic tweezers, and fluorescence microscopy, demonstrated on single DNA molecules, that can controllably form and visualise higher order structural motifs including plectonemes. This technology combined with cutting-edge MD simulations provides quantitative insight into complex dynamic structures relevant to DNA cellular processes and can be adapted to study a range of filamentous biopolymers.

Recent observations on type III algebras in AdS/CFT raise the possibility that smoothness of the black hole horizon is an emergent feature of the large-$N$ limit. In this paper, we present a $bulk$ model for the finite-$N$ mechanism underlying this transition. We quantize a free scalar field on a BTZ black hole with a Planckian stretched horizon placed as a Dirichlet boundary for the field. This is a tractable model for the stretched horizon that does not ignore the angular directions, and it defines a black hole vacuum which has similarities to (but is distinct from) the Boulware state. Using analytic approximations for the normal modes, we first improve upon 't Hooft's brick wall calculation: we are able to match $both$ the entropy and the temperature, $exactly$. Emboldened by this, we compute the boundary Wightman function of the scalar field in a typical pure state built on our stretched horizon vacuum, at an energy sliver at the mass of the black hole. A key result is that despite the manifest lack of smoothness, this single-sided pure state calculation yields precisely the Hartle-Hawking thermal correlator associated to the smooth horizon, in the small-$G_N$ limit. At finite $G_N$, there are variance corrections that are suppressed as $\mathcal{O}(e^{-S_{BH}/2})$. They become important at late times and resolve Maldacena's information paradox. Highly excited typical pure states on the stretched horizon vacuum are therefore models for black hole microstates, while the smooth horizon describes the thermal state. We note that heavy excited states on the stretched horizon are better defined than the vacuum itself. These results suggest that complementarity in the bulk EFT could arise from a UV complete bulk description in which the black hole interior is not manifest., Comment: V2: JHEP version + very minor improvements, v3: Clearer discussion about (averages over) eigenstates vs microcanonical typical states. Results and conclusions unchanged

Introducing a coupling framework reminiscent of FETI methods, but here on abstract form, we establish conditions for stability and minimal requirements for well-posedness on the continuous level, as well as conditions on local solvers for the approximation of subproblems. We then discuss stability of the resulting Lagrange multiplier methods and show stability under a mesh conditions between the local discretizations and the mortar space. If this condition is not satisfied we show how a stabilization, acting only on the multiplier can be used to achieve stability. The design of preconditioners of the Schur complement system is discussed in the unstabilized case. Finally we discuss some applications that enter the framework.

Patients being admitted to a hospital will most often be associated with a certain clinical development during their stay. However, there is always a risk of patients being subject to the wrong diagnosis or to a certain treatment not pertaining to the desired effect, potentially leading to adverse events. Our research aims to develop an anomaly detection system for identifying deviations from expected clinical trajectories. To address this goal we analyzed 16 months of vital sign recordings obtained from the Nordland Hospital Trust (NHT). We employed an self-supervised framework based on the STraTS transformer architecture to represent the time series data in a latent space. These representations were then subjected to various clustering techniques to explore potential patient phenotypes based on their clinical progress. While our preliminary results from this ongoing research are promising, they underscore the importance of enhancing the dataset with additional demographic information from patients. This additional data will be crucial for a more comprehensive evaluation of the method's performance., Comment: 10 pages, 6 figures

We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition.