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Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions to the FSI problems in the standard $L^2$ norm. In this article, we propose a new kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the $L^2$ norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.

We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

We present a space--time ultra-weak discontinuous Galerkin discretization of the linear Schr\"odinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal~$h$-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials, or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

Parameter inference for ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODE solutions are typically approximated by deterministic algorithms, new research on probabilistic solvers indicates that they produce more reliable parameter estimates by better accounting for numerical errors. However, many ODE systems are highly sensitive to their parameter values. This produces deep local minima in the likelihood function -- a problem which existing probabilistic solvers have yet to resolve. Here, we show that a Bayesian filtering paradigm for probabilistic ODE solution can dramatically reduce sensitivity to parameters by learning from the noisy ODE observations in a data-adaptive manner. Our method is applicable to ODEs with partially unobserved components and with arbitrary non-Gaussian noise. Several examples demonstrate that it is more accurate than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood.

We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the Monge-Ampere equation. The first HDG method is devised to solve the nonlinear elliptic Monge-Ampere equation by using Newton's method. The second HDG method is devised to solve a sequence of the Poisson equation until convergence to a fixed-point solution of the Monge-Ampere equation is reached. Numerical examples are presented to demonstrate the convergence and accuracy of the HDG methods. Furthermore, the HDG methods are applied to r-adaptive mesh generation by redistributing a given scalar density function via the optimal transport theory. This r-adaptivity methodology leads to the Monge-Ampere equation with a nonlinear Neumann boundary condition arising from the optimal transport of the density function to conform the resulting high-order mesh to the boundary. Hence, we extend the HDG methods to treat the nonlinear Neumann boundary condition. Numerical experiments are presented to illustrate the generation of r-adaptive high-order meshes on planar and curved domains.

We present and analyse a hybridized discontinuous Galerkin method for incompressible flow problems using non-affine cells, proving that it preserves a key invariance property that illudes most methods, namely that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not influence the velocity field. This invariance property can be preserved in the discrete problem if the incompressibility constraint is satisfied in a sufficiently strong sense. We derive sufficient conditions to guarantee discretely divergence-free functions are exactly divergence-free, and give examples of divergence-free finite elements on meshes containing triangular, quadrilateral, tetrahedral, or hexahedral cells generated by a (possibly non-affine) map from their respective reference cells. In the case of quadrilateral cells, we prove an optimal error estimate for the velocity field that does not depend on the pressure approximation. Our theoretical analysis is supported by numerical results.

An important aspect of developing reliable deep learning systems is devising strategies that make these systems robust to adversarial attacks. There is a long line of work that focuses on developing defenses against these attacks, but recently, researchers have began to study ways to reverse engineer the attack process. This allows us to not only defend against several attack models, but also classify the threat model. However, there is still a lack of theoretical guarantees for the reverse engineering process. Current approaches that give any guarantees are based on the assumption that the data lies in a union of linear subspaces, which is not a valid assumption for more complex datasets. In this paper, we build on prior work and propose a novel framework for reverse engineering of deceptions which supposes that the clean data lies in the range of a GAN. To classify the signal and attack, we jointly solve a GAN inversion problem and a block-sparse recovery problem. For the first time in the literature, we provide deterministic linear convergence guarantees for this problem. We also empirically demonstrate the merits of the proposed approach on several nonlinear datasets as compared to state-of-the-art methods.

In this work, we prove rigorous error estimates for a hybrid method introduced in [15] for solving the time-dependent radiation transport equation (RTE). The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. To focus the analysis, we assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion ($\text{P}_N$ method). Using a non-standard set of semi-norms, we obtain estimates of the form $C(\varepsilon,\sigma,\Delta t)N^{-s}$ where $s\geq 1$ denotes the regularity of the solution in angle, $\varepsilon$ and $\sigma$ are scattering parameters, $\Delta t$ is the time-step before reinitialization, and $C$ is a complicated function of $\varepsilon$, $\sigma$, and $\Delta t$. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic $\text{P}_N$ method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes.

The theory of Koopman operators allows to deploy non-parametric machine learning algorithms to predict and analyze complex dynamical systems. Estimators such as principal component regression (PCR) or reduced rank regression (RRR) in kernel spaces can be shown to provably learn Koopman operators from finite empirical observations of the system's time evolution. Scaling these approaches to very long trajectories is a challenge and requires introducing suitable approximations to make computations feasible. In this paper, we boost the efficiency of different kernel-based Koopman operator estimators using random projections (sketching). We derive, implement and test the new "sketched" estimators with extensive experiments on synthetic and large-scale molecular dynamics datasets. Further, we establish non asymptotic error bounds giving a sharp characterization of the trade-offs between statistical learning rates and computational efficiency. Our empirical and theoretical analysis shows that the proposed estimators provide a sound and efficient way to learn large scale dynamical systems. In particular our experiments indicate that the proposed estimators retain the same accuracy of PCR or RRR, while being much faster.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.