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Our objective is to stabilise and accelerate the time-domain boundary element method (TDBEM) for the three-dimensional wave equation. To overcome the potential time instability, we considered using the Burton--Miller-type boundary integral equation (BMBIE) instead of the ordinary boundary integral equation (OBIE), which consists of the single- and double-layer potentials. In addition, we introduced a smooth temporal basis, i.e. the B-spline temporal basis of order $d$, whereas $d=1$ was used together with the OBIE in a previous study [Takahashi 2014]. Corresponding to these new techniques, we generalised the interpolation-based fast multipole method that was developed in \cite{takahashi2014}. In particular, we constructed the multipole-to-local formula (M2L) so that even for $d\ge 2$ we can maintain the computational complexity of the entire algorithm, i.e. $O(N_{\rm s}^{1+\delta} N_{\rm t})$, where $N_{\rm s}$ and $N_{\rm t}$ denote the number of boundary elements and the number of time steps, respectively, and $\delta$ is theoretically estimated as $1/3$ or $1/2$. The numerical examples indicated that the BMBIE is indispensable for solving the homogeneous Dirichlet problem, but the order $d$ cannot exceed 1 owing to the doubtful cancellation of significant digits when calculating the corresponding layer potentials. In regard to the homogeneous Neumann problem, the previous TDBEM based on the OBIE with $d=1$ can be unstable, whereas it was found that the BMBIE with $d=2$ can be stable and accurate. The present study will enhance the usefulness of the TDBEM for 3D scalar wave problems.

In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly via a phase-field indicator function. In our reduced order model, we approximate the parameter-dependent phase-field function with a discrete empirical interpolation method (DEIM) that enables affine decomposition of the associated linear and bilinear forms. In addition, we introduce a modification of DEIM that leads to non-negativity preserving approximations, thus guaranteeing positive-semidefiniteness of the system matrix. We also present a strategy for determining the required number of DEIM modes for a given number of reduced basis functions. We couple reduced basis functions on neighboring patches to enable the efficient simulation of large-scale problems that consist of repetitive subdomains. We apply our reduced basis framework to efficiently solve the inverse problem of characterizing the subsurface damage state of a complete in-situ leach mining site.

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in $\mathbb{R}^3$, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.

We consider a potential outcomes model in which interference may be present between any two units but the extent of interference diminishes with spatial distance. The causal estimand is the global average treatment effect, which compares counterfactual outcomes when all units are treated to outcomes when none are. We study a class of designs in which space is partitioned into clusters that are randomized into treatment and control. For each design, we estimate the treatment effect using a Horovitz-Thompson estimator that compares the average outcomes of units with all neighbors treated to units with no neighbors treated, where the neighborhood radius is of the same order as the cluster size dictated by the design. We derive the estimator's rate of convergence as a function of the design and degree of interference and use this to obtain estimator-design pairs in this class that achieve near-optimal rates of convergence under relatively minimal assumptions on interference. We prove that the estimators are asymptotically normal and provide a variance estimator. Finally, we discuss practical implementation of the designs by partitioning space using clustering algorithms.

Visible light communication (VLC) has been recognized as a promising technology for handling the continuously increasing quality of service and connectivity requirements in modern wireless communications, particularly in indoor scenarios. In this context, the present work considers the integration of two distinct modulation schemes, namely spatial modulation (SM) with space time block codes (STBCs), aiming at improving the overall VLC system reliability. Based on this and in order to further enhance the achievable transmission data rate, we integrate quasi-orthogonal STBC (QOSTBC) with SM, since relaxing the orthogonality condition of OSTBC ultimately provides a higher coding rate. Then, we generalize the developed results to any number of active light-emitting diodes (LEDs) and any M-ary pulse amplitude modulation size. Furthermore, we derive a tight and tractable upper bound for the corresponding bit error rate (BER) by considering a simple two-step decoding procedure to detect the indices of the transmitting LEDs and then decode the signal domain symbols. Notably, the obtained results demonstrate that QOSTBC with SM enhances the achievable BER compared to SM with repetition coding (RC-SM). Finally, we compare STBC-SM with both multiple active SM (MASM) and RC-SM in terms of the achievable BER and overall data rate, which further justifies the usefulness of the proposed scheme.

Random fields are mathematical structures used to model the spatial interaction of random variables along time, with applications ranging from statistical physics and thermodynamics to system's biology and the simulation of complex systems. Despite being studied since the 19th century, little is known about how the dynamics of random fields are related to the geometric properties of their parametric spaces. For example, how can we quantify the similarity between two random fields operating in different regimes using an intrinsic measure? In this paper, we propose a numerical method for the computation of geodesic distances in Gaussian random field manifolds. First, we derive the metric tensor of the underlying parametric space (the 3 x 3 first-order Fisher information matrix), then we derive the 27 Christoffel symbols required in the definition of the system of non-linear differential equations whose solution is a geodesic curve starting at the initial conditions. The fourth-order Runge-Kutta method is applied to numerically solve the non-linear system through an iterative approach. The obtained results show that the proposed method can estimate the geodesic distances for several different initial conditions. Besides, the results reveal an interesting pattern: in several cases, the geodesic curve obtained by reversing the system of differential equations in time does not match the original curve, suggesting the existence of irreversible geometric deformations in the trajectory of a moving reference traveling along a geodesic curve.

The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.

The emerging Industrial Internet of Things (IIoT) is driving an ever increasing demand for providing low latency services to massive devices over wireless channels. As a result, how to assure the quality-of-service (QoS) for a large amount of mobile users is becoming a challenging issue in the envisioned sixth-generation (6G) network. In such networks, the delay-optimal wireless access will require a joint channel and queue aware scheduling, whose complexity increases exponentially with the number of users. In this paper, we adopt the mean field approximation to conceive a buffer-aware multi-user diversity or opportunistic access protocol, which serves all backlogged packets of a user if its channel gain is beyond a threshold. A theoretical analysis and numerical results will demonstrate that not only the cross-layer scheduling policy is of low complexity but is also asymptotically optimal for a huge number of devices.

We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of the this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented.

Biomembranes adopt varying morphologies that are vital to cellular functions. Many studies use computational modeling to understand how various mechanochemical factors contribute to membrane shape transformations. Compared to approximation-based methods (e.g., finite element method), the class of discrete mesh models offers greater flexibility to simulate complex physics and shapes in three dimensions; its formulation produces an efficient algorithm while maintaining coordinate-free geometric descriptions. However, ambiguities in geometric definitions in the discrete context have led to a lack of consensus on which discrete mesh model is theoretically and numerically optimal; a bijective relationship between the terms contributing to both the energy and forces from the discrete and smooth geometric theories remains to be established. We address this and present an extensible framework, $\texttt{Mem3DG}$, for modeling 3D mechanochemical dynamics of membranes based on Discrete Differential Geometry (DDG) on triangulated meshes. The formalism of DDG resolves the inconsistency and provides a unifying perspective on how to relate the smooth and discrete energy and forces. To demonstrate, $\texttt{Mem3DG}$ is used to model a sequence of examples with increasing mechanochemical complexity: recovering classical shape transformations such as 1) biconcave disk, dumbbell, and unduloid and 2) spherical bud on spherical, flat-patch membrane; investigating how the coupling of membrane mechanics with protein mobility jointly affects phase and shape transformation. As high-resolution 3D imaging of membrane ultrastructure becomes more readily available, we envision Mem3DG to be applied as an end-to-end tool to simulate realistic cell geometry under user-specified mechanochemical conditions.