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Nonlinear systems arising from time integrators like Backward Euler can sometimes be reformulated as optimization problems, known as incremental potentials. We show through a comprehensive experimental analysis that the widely used Projected Newton method, which relies on unconditional semidefinite projection of Hessian contributions, typically exhibits a reduced convergence rate compared to classical Newton's method. We demonstrate how factors like resolution, element order, projection method, material model and boundary handling impact convergence of Projected Newton and Newton. Drawing on these findings, we propose the hybrid method Project-on-Demand Newton, which projects only conditionally, and show that it enjoys both the robustness of Projected Newton and convergence rate of Newton. We additionally introduce Kinetic Newton, a regularization-based method that takes advantage of the structure of incremental potentials and avoids projection altogether. We compare the four solvers on hyperelasticity and contact problems. We also present a nuanced discussion of convergence criteria, and propose a new acceleration-based criterion that avoids problems associated with existing residual norm criteria and is easier to interpret. We finally address a fundamental limitation of the Armijo backtracking line search that occasionally blocks convergence, especially for stiff problems. We propose a novel parameter-free, robust line search technique to eliminate this issue.

We propose a new, computationally efficient, sparsity adaptive changepoint estimator for detecting changes in unknown subsets of a high-dimensional data sequence. Assuming the data sequence is Gaussian, we prove that the new method successfully estimates the number and locations of changepoints with a given error rate and under minimal conditions, for all sparsities of the changing subset. Moreover, our method has computational complexity linear up to logarithmic factors in both the length and number of time series, making it applicable to large data sets. Through extensive numerical studies we show that the new methodology is highly competitive in terms of both estimation accuracy and computational cost. The practical usefulness of the method is illustrated by analysing sensor data from a hydro power plant. An efficient R implementation is available.

For the iterative decoupling of elliptic-parabolic problems such as poroelasticity, we introduce time discretization schemes up to order $5$ based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As the main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer a significant advantage in dealing with complex geometries, eliminating the need for generating unstructured body-fitted meshes. However, current unfitted finite elements on nonlinear geometries are restricted to implicit (possibly high-order) level set geometries. In this work, we introduce a novel automatic computational pipeline to approximate solutions of partial differential equations on domains defined by explicit nonlinear boundary representations. For the geometrical discretization, we propose a novel algorithm to generate quadratures for the bulk and surface integration on nonlinear polytopes required to compute all the terms in unfitted finite element methods. The algorithm relies on a nonlinear triangulation of the boundary, a kd-tree refinement of the surface cells that simplify the nonlinear intersections of surface and background cells to simple cases that are diffeomorphically equivalent to linear intersections, robust polynomial root-finding algorithms and surface parameterization techniques. We prove the correctness of the proposed algorithm. We have successfully applied this algorithm to simulate partial differential equations with unfitted finite elements on nonlinear domains described by computer-aided design models, demonstrating the robustness of the geometric algorithm and showing high-order accuracy of the overall method.

The present study investigates to what degree the common variance of the factor score predictor with the original factor, i.e., the determinacy coefficient or the validity of the factor score predictor, depends on the mean-difference between groups. When mean-differences between groups in the factor score predictor are eliminated by means of covariance analysis, regression, or group specific norms, this may reduce the covariance of the factor score predictor with the common factor. It is shown that in a one-factor model with the same group mean-difference on all observed variables, the common factor cannot be distinguished from a common factor representing the group mean-difference. It is also shown that for common factor loadings equal or larger than .60, the elimination of a d = .50 mean-difference between two groups in the factor score predictor leads to only small decreases of the determinacy coefficient. A compensation-factor k is proposed allowing for the estimation of the number of additional observed variables necessary to recover the size of the determinacy coefficient before elimination of a group mean-difference. It turns out that for factor loadings equal or larger than .60 only a few additional items are needed in order to recover the initial determinacy coefficient after the elimination of moderate or large group mean-differences.

We propose a type-theoretic framework for describing and proving properties of quantum computations, in particular those presented as quantum circuits. Our proposal is based on an observation that, in the polymorphic type system of Coq, currying on quantum states allows us to apply quantum gates directly inside a complex circuit. By introducing a discrete notion of lens to control this currying, we are further able to separate the combinatorics of the circuit structure from the computational content of gates. We apply our development to define quantum circuits recursively from the bottom up, and prove their correctness compositionally.

We consider the problem of constructing multiple conditional randomization tests. They may test different causal hypotheses but always aim to be nearly independent, allowing the randomization p-values to be interpreted individually and combined using standard methods. We start with a simple, sequential construction of such tests, and then discuss its application to three problems: evidence factors for observational studies, lagged treatment effect in stepped-wedge trials, and spillover effect in randomized trials with interference. We compare the proposed approach with some existing methods using simulated and real datasets. Finally, we establish a general sufficient condition for constructing multiple nearly independent conditional randomization tests.

While ChatGPT is a well-known artificial intelligence chatbot being used to answer human's questions, one may want to discover its potential in advancing software testing. We examine the capability of ChatGPT in advancing the intelligence of software testing through a case study on metamorphic testing (MT), a state-of-the-art software testing technique. We ask ChatGPT to generate candidates of metamorphic relations (MRs), which are basically necessary properties of the object program and which traditionally require human intelligence to identify. These MR candidates are then evaluated in terms of correctness by domain experts. We show that ChatGPT can be used to generate new correct MRs to test several software systems. Having said that, the majority of MR candidates are either defined vaguely or incorrect, especially for systems that have never been tested with MT. ChatGPT can be used to advance software testing intelligence by proposing MR candidates that can be later adopted for implementing tests; but human intelligence should still inevitably be involved to justify and rectify their correctness.

Mediation analysis assesses the extent to which the exposure affects the outcome indirectly through a mediator and the extent to which it operates directly through other pathways. As the most popular method in empirical mediation analysis, the Baron-Kenny approach estimates the indirect and direct effects of the exposure on the outcome based on linear structural equation models. However, when the exposure and the mediator are not randomized, the estimates may be biased due to unmeasured confounding among the exposure, mediator, and outcome. Building on Cinelli and Hazlett (2020), we derive general omitted-variable bias formulas in linear regressions with vector responses and regressors. We then use the formulas to develop a sensitivity analysis method for the Baron-Kenny approach to mediation in the presence of unmeasured confounding. To ensure interpretability, we express the sensitivity parameters to correspond to the natural factorization of the joint distribution of the direct acyclic graph for mediation analysis. They measure the partial correlation between the unmeasured confounder and the exposure, mediator, outcome, respectively. With the sensitivity parameters, we propose a novel measure called the "robustness value for mediation" or simply the "robustness value", to assess the robustness of results based on the Baron-Kenny approach with respect to unmeasured confounding. Intuitively, the robustness value measures the minimum value of the maximum proportion of variability explained by the unmeasured confounding, for the exposure, mediator and outcome, to overturn the results of the point estimate or confidence interval for the direct and indirect effects. Importantly, we prove that all our sensitivity bounds are attainable and thus sharp.

In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of $k$ tokens on a graph and the question is whether we can transform this configuration into a feasible solution (for some problem) via a bounded number $b$ of small modification steps. In this work, we study solution discovery variants of polynomial-time solvable problems, namely Spanning Tree Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut Discovery in the unrestricted token addition/removal model, the token jumping model, and the token sliding model. In the unrestricted token addition/removal model, we show that all four discovery variants remain in P. For the toking jumping model we also prove containment in P, except for Vertex/Edge Cut Discovery, for which we prove NP-completeness. Finally, in the token sliding model, almost all considered problems become NP-complete, the exception being Spanning Tree Discovery, which remains polynomial-time solvable. We then study the parameterized complexity of the NP-complete problems and provide a full classification of tractability with respect to the parameters solution size (number of tokens) $k$ and transformation budget (number of steps) $b$. Along the way, we observe strong connections between the solution discovery variants of our base problems and their (weighted) rainbow variants as well as their red-blue variants with cardinality constraints.