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Identification of nonlinear dynamical systems has been popularized by sparse identification of the nonlinear dynamics (SINDy) via the sequentially thresholded least squares (STLS) algorithm. Many extensions SINDy have emerged in the literature to deal with experimental data which are finite in length and noisy. Recently, the computationally intensive method of ensembling bootstrapped SINDy models (E-SINDy) was proposed for model identification, handling finite, highly noisy data. While the extensions of SINDy are numerous, their sparsity-promoting estimators occasionally provide sparse approximations of the dynamics as opposed to exact recovery. Furthermore, these estimators suffer under multicollinearity, e.g. the irrepresentable condition for the Lasso. In this paper, we demonstrate that the Trimmed Lasso for robust identification of models (TRIM) can provide exact recovery under more severe noise, finite data, and multicollinearity as opposed to E-SINDy. Additionally, the computational cost of TRIM is asymptotically equal to STLS since the sparsity parameter of the TRIM can be solved efficiently by convex solvers. We compare these methodologies on challenging nonlinear systems, specifically the Lorenz 63 system, the Bouc Wen oscillator from the nonlinear dynamics benchmark of No\"el and Schoukens, 2016, and a time delay system describing tool cutting dynamics. This study emphasizes the comparisons between STLS, reweighted $\ell_1$ minimization, and Trimmed Lasso in identification with respect to problems faced by practitioners: the problem of finite and noisy data, the performance of the sparse regression of when the library grows in dimension (multicollinearity), and automatic methods for choice of regularization parameters.

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to "lift" these arguments to certain infinite posets over which Auslander-Reiten theory is not available. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature.

We prove lower bounds for the randomized approximation of the embedding $\ell_1^m \rightarrow \ell_\infty^m$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N \in \mathbb{R}^{n \times m}$. These lower bounds reflect the increasing difficulty of the problem for $m \to \infty$, namely, a term $\sqrt{\log m}$ in the complexity $n$. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity $n$ only exhibits a $(\log\log m)$-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order $n^{1/2} ( \log n)^{-1/2}$.

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n\geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant, $S_h^n$, converges to the optimal Hardy constant $S^n$ no slower than $O(1/\vert \log h \vert)$. We also show that the convergence is no faster than $O(1/\vert \log h \vert^2)$ if $n=1$ or if $n\geq 3$, the domain is the unit ball, and the finite element discretization exploits the rotational symmetry of the problem. Our estimates are compared to exact values for $S_h^n$ obtained computationally.

We propose a new discrete choice model, called the generalized stochastic preference (GSP) model, that incorporates non-rationality into the stochastic preference (SP) choice model, also known as the rank- based choice model. Our model can explain several choice phenomena that cannot be represented by any SP model such as the compromise and attraction effects, but still subsumes the SP model class. The GSP model is defined as a distribution over consumer types, where each type extends the choice behavior of rational types in the SP model. We build on existing methods for estimating the SP model and propose an iterative estimation algorithm for the GSP model that finds new types by solving a integer linear program in each iteration. We further show that our proposed notion of non-rationality can be incorporated into other choice models, like the random utility maximization (RUM) model class as well as any of its subclasses. As a concrete example, we introduce the non-rational extension of the classical MNL model, which we term the generalized MNL (GMNL) model and present an efficient expectation-maximization (EM) algorithm for estimating the GMNL model. Numerical evaluation on real choice data shows that the GMNL and GSP models can outperform their rational counterparts in out-of-sample prediction accuracy.

We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.

We consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modeled by incompressible surface Navier-Stokes-Cahn-Hilliard-like equations with bending forces. We derive this model using the Lagrange-D'Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor-Hood element for the surface Navier-Stokes part, a semi-implicit approach for the Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified sub-problems.

High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\text{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.

Developing an efficient computational scheme for high-dimensional Bayesian variable selection in generalised linear models and survival models has always been a challenging problem due to the absence of closed-form solutions for the marginal likelihood. The RJMCMC approach can be employed to samples model and coefficients jointly, but effective design of the transdimensional jumps of RJMCMC can be challenge, making it hard to implement. Alternatively, the marginal likelihood can be derived using data-augmentation scheme e.g. Polya-gamma data argumentation for logistic regression) or through other estimation methods. However, suitable data-augmentation schemes are not available for every generalised linear and survival models, and using estimations such as Laplace approximation or correlated pseudo-marginal to derive marginal likelihood within a locally informed proposal can be computationally expensive in the "large n, large p" settings. In this paper, three main contributions are presented. Firstly, we present an extended Point-wise implementation of Adaptive Random Neighbourhood Informed proposal (PARNI) to efficiently sample models directly from the marginal posterior distribution in both generalised linear models and survival models. Secondly, in the light of the approximate Laplace approximation, we also describe an efficient and accurate estimation method for the marginal likelihood which involves adaptive parameters. Additionally, we describe a new method to adapt the algorithmic tuning parameters of the PARNI proposal by replacing the Rao-Blackwellised estimates with the combination of a warm-start estimate and an ergodic average. We present numerous numerical results from simulated data and 8 high-dimensional gene fine mapping data-sets to showcase the efficiency of the novel PARNI proposal compared to the baseline add-delete-swap proposal.