Elliptic interface boundary value problems play a major role in numerous applications involving heat, fluids, materials, and proteins, to name a few. As an example, in implicit variational solvation, for the construction of biomolecular shapes, the electrostatic contributions satisfy the Poisson-Boltzmann equation with discontinuous dielectric constants across the interface. When interface motions are involved, one often needs not only accurate solution values, but accurate derivatives as well, such as the normal derivatives at the interface. We introduce here the Compact Coupling Interface Method (CCIM), a finite difference method for the elliptic interface problem with interfacial jump conditions. The CCIM can calculate solution values and their derivatives up to second-order accuracy in arbitrary ambient space dimensions. It combines elements of Chern and Shu's Coupling Interface Method and Mayo's approach for elliptic interface boundary value problems, leading to more compact finite difference stencils that are applicable to more general situations. Numerical results on a variety of geometric interfacial shapes and on complex protein molecules in three dimensions support the efficacy of our approach and reveal advantages in accuracy and robustness.
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Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context; in particular to develop underpinning theory to establish the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory for the solution operator defined by the cell-problem arising in homogenization for elliptic PDEs.
We propose a novel framework for analyzing the dynamics of distribution shift in real-world systems that captures the feedback loop between learning algorithms and the distributions on which they are deployed. Prior work largely models feedback-induced distribution shift as adversarial or via an overly simplistic distribution-shift structure. In contrast, we propose a coupled partial differential equation model that captures fine-grained changes in the distribution over time by accounting for complex dynamics that arise due to strategic responses to algorithmic decision-making, non-local endogenous population interactions, and other exogenous sources of distribution shift. We consider two common settings in machine learning: cooperative settings with information asymmetries, and competitive settings where a learner faces strategic users. For both of these settings, when the algorithm retrains via gradient descent, we prove asymptotic convergence of the retraining procedure to a steady-state, both in finite and in infinite dimensions, obtaining explicit rates in terms of the model parameters. To do so we derive new results on the convergence of coupled PDEs that extends what is known on multi-species systems. Empirically, we show that our approach captures well-documented forms of distribution shifts like polarization and disparate impacts that simpler models cannot capture.
Exploring the origin and properties of magnetic fields is crucial to the development of many fields such as physics, astronomy and meteorology. We focus on the edge element approximation and theoretical analysis of celestial dynamo system with quasi-vacuum boundary conditions. The system not only ensures that the magnetic field on the spherical shell is generated from the dynamo model, but also provides convenience for the application of the edge element method. We demonstrate the existence, uniqueness and stability of the solution to the system by the fixed point theorem. Then, we approximate the system using the edge element method, which is more efficient in dealing with electromagnetic field problems. Moreover, we also discuss the stability of the corresponding discrete scheme. And the convergence is demonstrated by later numerical tests. Finally, we simulate the three-dimensional time evolution of the spherical interface dynamo model, and the characteristics of the simulated magnetic field are consistent with existing work.
Due to the diffusion of IoT, modern software systems are often thought to control and coordinate smart devices in order to manage assets and resources, and to guarantee efficient behaviours. For this class of systems, which interact extensively with humans and with their environment, it is thus crucial to guarantee their correct behaviour in order to avoid unexpected and possibly dangerous situations. In this paper we will present a framework that allows us to measure the robustness of systems. This is the ability of a program to tolerate changes in the environmental conditions and preserving the original behaviour. In the proposed framework, the interaction of a program with its environment is represented as a sequence of random variables describing how both evolve in time. For this reason, the considered measures will be defined among probability distributions of observed data. The proposed framework will be then used to define the notions of adaptability and reliability. The former indicates the ability of a program to absorb perturbation on environmental conditions after a given amount of time. The latter expresses the ability of a program to maintain its intended behaviour (up-to some reasonable tolerance) despite the presence of perturbations in the environment. Moreover, an algorithm, based on statistical inference, is proposed to evaluate the proposed metric and the aforementioned properties. We use two case studies to the describe and evaluate the proposed approach.
We construct a compact fourth-order scheme, in space and time, for the time-dependent Maxwell's equations given as a first-order system on a staggered (Yee) grid. At each time step, we update the fields by solving positive definite second-order elliptic equations. We face the challenge of finding compatible boundary conditions for these elliptic equations while maintaining a compact stencil. The proposed scheme is compared computationally with a non-compact scheme and data-driven approach.
The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years. While it seems to defy the conventional wisdom that overfitting leads to poor prediction, recent research reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. This renders the Ridgeless interpolator a theoretically tractable proxy that offers useful insights into the mechanisms of modern machine learning methods. This paper takes a different perspective that aims at understanding the precise stochastic behavior of the Ridgeless interpolator as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which plays the role of the prescribed implicit regularization in the context of prediction risk. Our distributional characterizations hold for general random designs and extend uniformly to positively regularized Ridge estimators. As a demonstration of the analytic power of these characterizations, we derive approximate formulae for a general class of weighted $\ell_q$ risks for Ridge(less) estimators that were previously available only for $\ell_2$. Our theory also provides certain further conceptual reconciliation with the conventional wisdom: given any data covariance, a certain amount of regularization in Ridge regression remains beneficial for `most' signals across various statistical tasks including prediction, estimation and inference, as long as the noise level is non-trivial. Surprisingly, optimal tuning can be achieved simultaneously for all the designated statistical tasks by a single generalized or $k$-fold cross-validation scheme, despite being designed specifically for tuning prediction risk.
Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics
We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is ``close'' to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.
Mutual coherence is a measure of similarity between two opinions. Although the notion comes from philosophy, it is essential for a wide range of technologies, e.g., the Wahl-O-Mat system. In Germany, this system helps voters to find candidates that are the closest to their political preferences. The exact computation of mutual coherence is highly time-consuming due to the iteration over all subsets of an opinion. Moreover, for every subset, an instance of the SAT model counting problem has to be solved which is known to be a hard problem in computer science. This work is the first study to accelerate this computation. We model the distribution of the so-called confirmation values as a mixture of three Gaussians and present efficient heuristics to estimate its model parameters. The mutual coherence is then approximated with the expected value of the distribution. Some of the presented algorithms are fully polynomial-time, others only require solving a small number of instances of the SAT model counting problem. The average squared error of our best algorithm lies below 0.0035 which is insignificant if the efficiency is taken into account. Furthermore, the accuracy is precise enough to be used in Wahl-O-Mat-like systems.
Experimental and observational studies often lack validity due to untestable assumptions. We propose a double machine learning approach to combine experimental and observational studies, allowing practitioners to test for assumption violations and estimate treatment effects consistently. Our framework tests for violations of external validity and ignorability under milder assumptions. When only one assumption is violated, we provide semi-parametrically efficient treatment effect estimators. However, our no-free-lunch theorem highlights the necessity of accurately identifying the violated assumption for consistent treatment effect estimation. We demonstrate the applicability of our approach in three real-world case studies, highlighting its relevance for practical settings.
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