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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.