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Regardless of the development of various finite element methods for fluid-structure interaction (FSI) problems, optimal-order convergence of finite element discretizations of the FSI problems in the $L^\infty(0,T;L^2)$ norm has not been proved due to the incompatibility between standard Ritz projections and the interface conditions in the FSI problems. To address this issue, we define a dynamic Ritz projection (which satisfies a dynamic interface condition) associated to the FSI problem and study its approximation properties in the $L^\infty(0,T;H^1)$ and $L^\infty(0,T;L^2)$ norms. Existence and uniqueness of the dynamic Ritz projection of the solution, as well as estimates of the error between the solution and its dynamic Ritz projection, are established. By utilizing the established results, we prove optimal-order convergence of finite element methods for the FSI problem in the $L^\infty(0,T;L^2)$ norm.

This paper addresses the problem of friction-free contact between two elastic bodies. We develop an augmented Lagrangian method that provides computational convenience by reformulating the contact problem as a nonlinear variational equality. To achieve this, we propose a Nitsche-based method incorporating a hybrid displacement variable defined on an interstitial layer. This approach enables complete decoupling of the contact domains, with interaction occurring exclusively through the interstitial layer. The layer is independently approximated, eliminating the need to handle intersections between unrelated meshes. Additionally, the method supports introducing an independent model on the interface, which we leverage to represent a membrane covering one of the bodies. We present the formulation of the method, establish stability and error estimates, and demonstrate its practical utility through illustrative numerical examples., Comment: 33 pages, 14 figures

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 autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to 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 an optimal error estimate in the $H^1$-norm. 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.

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. The necessity and effectiveness of the regularization is illustrated in numerical experiments. We also explore numerically the effect of the heterogeneity in the coefficients on the ability to reconstruct the solution outside the data domain. For Helmholtz equations we find that a jump in the flux impacts the stability of the problem significantly less than the size of the wavenumber., Comment: 39 pages, 9 figures

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network., Comment: 29 pages, 8 figures

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the Poisson--Boltzmann equation, where the finite element method is applied in a confined {\it solute} region, and the boundary element method in the external {\it solvent} region. As a proof-of-concept exercise, we use the simplest methods available: Johnson--N\'ed\'elec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. We validate our implementation against the finite difference code APBS (to 0.5\%), and show scaling from protein G B1 (955 atoms) up to immunoglobulin G (20\,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large sized problems on a single workstation. Development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments for molecular electrostatics with implicit solvent models.

We revisit the celebrated Kohn-Vogelius penalty method and discuss how to use it for the unique continuation problem where data is given in the bulk of the domain. We then show that the primal-dual mixed finite element methods for the elliptic Cauchy problem introduced in \cite{BLO18} (\emph{E. Burman, M. Larson, L. Oksanen, Primal-dual mixed finite element methods for the elliptic Cauchy problem, SIAM J. Num. Anal., 56 (6), 2018}) can be interpreted as a Kohn-Vogelius penalty method and modify it to allow for unique continuation using data in the bulk. We prove that the resulting linear system is invertible for all data. Then we show that by introducing a singularly perturbed Robin condition on the discrete level sufficient regularization is obtained so that error estimates can be shown using conditional stability. Finally we show how the method can be used for the identification of the diffusivity coefficient in a second order elliptic operator with partial data. Some numerical examples are presented showing the performance of the method for unique continuation and for impedance computed tomography with partial data., Comment: Submitted to the conference proceedings of the Peruvian conference on Scientific Computing, October 2023

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier--Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system., Comment: 26 PAGES

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem., Comment: 36 pages, 10 figures

In this note we design a cut finite element method for a low order divergence free element applied to a boundary value problem subject to Stokes' equations. For the imposition of Dirichlet boundary conditions we consider either Nitsche's method or a stabilized Lagrange multiplier method. In both cases the normal component of the velocity is constrained using a multiplier, different from the standard pressure approximation. The divergence of the approximate velocities is pointwise zero over the whole mesh domain, and we derive optimal error estimates for the velocity and pressures, where the error constant is independent of how the physical domain intersects the computational mesh, and of the regularity of the pressure multiplier imposing the divergence free condition.

In this paper we will review recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier--free stabilised methods. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree $p$ with $k$ continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.

In this paper we propose, analyze, and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided., Comment: 3 figures, 23 pages

In this note, we develop a new stabilization mechanism for cut finite element methods that generalizes previous approaches of ghost penalty type in two ways: (1) The quantity that is stabilized and (2) The choice of elements that are connected in the stabilization. In particular, we can stabilize functionals of the discrete function such as finite element degrees of freedom. We subsequently show that the kernel of our ghost penalty operator defines a finite element space based on discrete extensions in the spirit of those introduced in Burman, E.; Hansbo, P. and Larson, M. G., CutFEM Based on Extended Finite Element Spaces, arXiv2101.10052, 2021., Comment: 31 pages, 12 figures

Coronary artery disease(CAD) is the most common type of heart disease and the leading cause of death worldwide[1]. A progressive state of this disease marked by plaque rupture and clot formation in the coronary arteries, also known as an acute coronary syndrome (ACS), is a condition of the heart associated with sudden, reduced blood flow caused due to partial or full occlusion of coronary vasculature that normally perfuses the myocardium and nerve bundles, compromising the proper functioning of the heart. Often manifesting with pain or tightness in the chest as the second most common cause of emergency department visits in the United States, it is imperative to detect ACS at the earliest. This is particularly relevant to diabetic patients at home, that may not feel classic chest pain symptoms, and are susceptible to silent myocardial injury. In this study, we developed the RCE- ECG-Detect algorithm, a machine learning model to detect the morphological patterns in significant ST change associated with myocardial ischemia. We developed the RCE- ECG-Detect using data from the LTST database which has a sufficiently large sample set to train a reliable model. We validated the predictive performance of the machine learning model on a holdout test set collected using RCE's ECG wearable. Our deep neural network model, equipped with convolution layers, achieves 90.31% ROC-AUC, 89.34% sensitivity, 87.81% specificity.

Platform trials can evaluate the efficacy of several treatments compared to a control. The number of treatments is not fixed, as arms may be added or removed as the trial progresses. Platform trials are more efficient than independent parallel-group trials because of using shared control groups. For arms entering the trial later, not all patients in the control group are randomised concurrently. The control group is then divided into concurrent and non-concurrent controls. Using non-concurrent controls (NCC) can improve the trial's efficiency, but can introduce bias due to time trends. We focus on a platform trial with two treatment arms and a common control arm. Assuming that the second treatment arm is added later, we assess the robustness of model-based approaches to adjust for time trends when using NCC. We consider approaches where time trends are modeled as linear or as a step function, with steps at times where arms enter or leave the trial. For trials with continuous or binary outcomes, we investigate the type 1 error (t1e) rate and power of testing the efficacy of the newly added arm under a range of scenarios. In addition to scenarios where time trends are equal across arms, we investigate settings with trends that are different or not additive in the model scale. A step function model fitted on data from all arms gives increased power while controlling the t1e, as long as the time trends are equal for the different arms and additive on the model scale. This holds even if the trend's shape deviates from a step function if block randomisation is used. But if trends differ between arms or are not additive on the model scale, t1e control may be lost. The efficiency gained by using step function models to incorporate NCC can outweigh potential biases. However, the specifics of the trial, plausibility of different time trends, and robustness of results should be considered

We present a loosely coupled, non-iterative time-splitting scheme based on Robin-Robin coupling conditions. We apply a novel unified analysis for this scheme applied to both a Parabolic/Parabolic coupled system and a Parabolic/Hyperbolic coupled system. We show for both systems that the scheme is stable, and the error converges as $\mathcal{O}\big(\Delta t \sqrt{T +\log{\frac{1}{\Delta t}}}\big)$, where $\Delta t$ is the time step

We consider the problem of optimizing a portfolio of financial assets, where the number of assets can be much larger than the number of observations. The optimal portfolio weights require estimating the inverse covariance matrix of excess asset returns, classical solutions of which behave badly in high-dimensional scenarios. We propose to use a regression-based joint shrinkage method for estimating the partial correlation among the assets. Extensive simulation studies illustrate the superior performance of the proposed method with respect to variance, weight, and risk estimation errors compared with competing methods for both the global minimum variance portfolios and Markowitz mean-variance portfolios. We also demonstrate the excellent empirical performances of our method on daily and monthly returns of the components of the S&P 500 index.

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.

Ribbon decomposition is a way to obtain a surface with boundary (compact, not necessarily oriented) from a collection of disks by joining them with narrow ribbons attached to segments of the boundary. Counting ribbon decompositions gives rise to a "twisted" version of the classical Hurwitz numbers (studied earlier in \cite{CD} in a different context) and of the cut-and-join equation. We also provide an algebraic description of these numbers and an explicit formula for them in terms of zonal polynomials., Comment: 21 pages, 4 figures. v2 : The previous version of the paper was revised (including correction of some theorems) and completely rewritten ; v3 : minor corrections

Understanding dynamics of an outbreak like that of COVID-19 is important in designing effective control measures. This study aims to develop an agent based model that compares changes in infection progression by manipulating different parameters in a synthetic population. Model input includes population characteristics like age, sex, working status etc. of each individual and other factors influencing disease dynamics. Depending on number of epicentres of infection, location of primary cases, sensitivity, proportion of asymptomatic and frequency or duration of lockdown, our simulator tracks every individual and hence infection progression through community over time. In a closed community of 10000 people, it is seen that without any lockdown, number of cases peak around 6th week and wanes off around 15th week. If primary case is located inside dense population cluster like slums, cases peak early and wane off slowly. With introduction of lockdown, cases peak at slower rate. If sensitivity of identifying infection decreases, cases and deaths increase. Number of cases declines with increase in proportion of asymptomatic cases. The model is robust and provides reproducible estimates with realistic parameter values. It also guides in identifying measures to control outbreak in a community. It is flexible in accommodating different parameters like infectivity period, yield of testing, socio-economic strata, daily travel, awareness level, population density, social distancing, lockdown etc. and can be tailored to study other infections with similar transmission pattern., Comment: 6 pages, 4 figures, 1 table, 1 video

In this paper we discuss a hybridised method for FEM-BEM coupling. The coupling from both sides use a Nitsche type approach to couple to the trace variable. This leads to a formulation that is robust and flexible with respect to approximation spaces and can easily be combined as a building block with other hybridised methods. Energy error norm estimates and the convergence of Jacobi iterations are proved and the performance of the method is illustrated on some computational examples.

In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements. To obtain stability we add a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that in the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the $L^2$-norm error converges with the expected order $O(h^{k+\frac12})$. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if discontinuities have been present in the solution at some time during the evolution.

We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual $L^2$-norm. However, we are still able to recover the optimal a priori $L^2$-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.

We present a new approach for the mechanically consistent modelling and simulation of fluid-structure interactions with contact. The fundamental idea consists of combining a relaxed contact formulation with the modelling of seepage through a porous layer of co-dimension 1 during contact. For the latter, a Darcy model is considered in a thin porous layer attached to a solid boundary in the limit of infinitesimal thickness. In combination with a relaxation of the contact conditions the computational model is both mechanically consistent and simple to implement. We analyse the approach in detailed numerical studies with both thick- and thin-walled solids, within a fully Eulerian and an immersed approach for the fluid-structure interaction and using fitted and unfitted finite element discretisations.

We study distributed agreement in microbial distributed systems under stochastic population dynamics and competitive interactions. Motivated by recent applications in synthetic biology, we examine how the presence and absence of direct competition among microbial species influences their ability to reach majority consensus. In this problem, two species are designated as input species, and the goal is to guarantee that eventually only the input species which had the highest initial count prevails. We show that direct competition dynamics reach majority consensus with high probability even when the initial gap between the species is small, i.e., $\Omega(\sqrt{n\log n})$, where $n$ is the initial population size. In contrast, we show that absence of direct competition is not robust: solving majority consensus with constant probability requires a large initial gap of $\Omega(n)$. To corroborate our analytical results, we use simulations to show that these consensus dynamics occur within practical biological time scales.

In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $L^2$-norm for smooth flows in the pre-asymptotic high Reynolds number regime.

We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More precisely, the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator., Comment: 32 pages, 9 figures