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We introduce a Bayesian framework for mixed-type multivariate regression using shrinkage priors. Our method enables joint analysis of mixed continuous and discrete outcomes and facilitates variable selection from the $p$ covariates. Our model can be implemented with a Gibbs sampling algorithm where all conditional distributions are tractable, leading to a simple one-step estimation procedure. We derive the posterior contraction rate for the one-step estimator when $p$ grows subexponentially with respect to sample size $n$. We further establish that subexponential growth is both necessary and sufficient for the one-step estimator to achieve posterior consistency. We then introduce a two-step variable selection approach that is suitable for large $p$. We prove that our two-step algorithm possesses the sure screening property. Moreover, our two-step estimator can provably achieve posterior contraction even when $p$ grows exponentially in $n$, thus overcoming a limitation of the one-step estimator. We demonstrate the utility of our method through simulation studies and applications to real datasets. R codes to implement our method are available at //github.com/raybai07/MtMBSP.

The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-$Q_1$ IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both $H^1$- and $L^2$- norms are proved and confirmed with numerical experiments.

This study proposes a hybrid deep-learning-metaheuristic framework with a bi-level architecture for road network design problems (NDPs). We train a graph neural network (GNN) to approximate the solution of the user equilibrium (UE) traffic assignment problem, and use inferences made by the trained model to calculate fitness function evaluations of a genetic algorithm (GA) to approximate solutions for NDPs. Using two NDP variants and an exact solver as benchmark, we show that our proposed framework can provide solutions within 5% gap of the global optimum results given less than 1% of the time required for finding the optimal results. Our framework can be utilized within an expert system for infrastructure planning to intelligently determine the best infrastructure management decisions. Given the flexibility of the framework, it can easily be adapted to many other decision problems that can be modeled as bi-level problems on graphs. Moreover, we observe many interesting future directions, thus we propose a brief research agenda for this topic. The key observation inspiring influential future research was that fitness function evaluation time using the inferences made by the GNN model for the genetic algorithm was in the order of milliseconds, which points to an opportunity and a need for novel heuristics that 1) can cope well with noisy fitness function values provided by neural networks, and 2) can use the significantly higher computation time provided to them to explore the search space effectively (rather than efficiently). This opens a new avenue for a modern class of metaheuristics that are crafted for use with AI-powered predictors.

This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the product function over a general compact domain. We also demonstrate this lower bound doesn't apply to normalized Lipschitz monomials over the unit cube. These findings suggest that shallow ReLU networks experience the curse of dimensionality when expressing functions with a Lipschitz parameter scaling with the dimension of the input, and that the expressive power of neural networks is more dependent on their depth rather than overall complexity.

This article proposes and analyzes the generalized weak Galerkin ({\rm g}WG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the {\rm g}WG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual $L^2$ norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed {\rm g}WG method.

We revisit classical Virtual Element approximations on polygonal and polyhedral decompositions. We also recall the treatment proposed for dealing with decompositions into polygons with curved edges. In the second part of the paper we introduce a couple of new ideas for the construction of VEM-approximations on domains with curved boundary, both in two and three dimensions. The new approach looks promising, although sound numerical tests should be made to validate the efficiency of the method.

We introduce new techniques for the parameterized verification of disjunctive timed networks (DTNs), i.e., networks of timed automata (TAs) that communicate via location guards that enable a transition only if at least one process is in a given location. This computational model has been considered in the literature before, and example applications are gossiping clock synchronization protocols or planning problems. We address the minimum-time reachability problem (minreach) in DTNs, and show how to efficiently solve it based on a novel zone-graph algorithm. We further show that solving minreach allows us to construct a summary TA capturing exactly the possible behaviors of a single TA within a DTN of arbitrary size. The combination of these two results enables the parameterized verification of DTNs, while avoiding the construction of an exponential-size cutoff-system required by existing results. Our techniques are also implemented, and experiments show their practicality.

Deadlocks are one of the most notorious concurrency bugs, and significant research has focused on detecting them efficiently. Dynamic predictive analyses work by observing concurrent executions, and reason about alternative interleavings that can witness concurrency bugs. Such techniques offer scalability and sound bug reports, and have emerged as an effective approach for concurrency bug detection, such as data races. Effective dynamic deadlock prediction, however, has proven a challenging task, as no deadlock predictor currently meets the requirements of soundness, high-precision, and efficiency. In this paper, we first formally establish that this tradeoff is unavoidable, by showing that (a) sound and complete deadlock prediction is intractable, in general, and (b) even the seemingly simpler task of determining the presence of potential deadlocks, which often serve as unsound witnesses for actual predictable deadlocks, is intractable. The main contribution of this work is a new class of predictable deadlocks, called sync(hronization)-preserving deadlocks. Informally, these are deadlocks that can be predicted by reordering the observed execution while preserving the relative order of conflicting critical sections. We present two algorithms for sound deadlock prediction based on this notion. Our first algorithm SPDOffline detects all sync-preserving deadlocks, with running time that is linear per abstract deadlock pattern, a novel notion also introduced in this work. Our second algorithm SPDOnline predicts all sync-preserving deadlocks that involve two threads in a strictly online fashion, runs in overall linear time, and is better suited for a runtime monitoring setting. We implemented both our algorithms and evaluated their ability to perform offline and online deadlock-prediction on a large dataset of standard benchmarks.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.