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In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem $-\nabla \cdot( a\nabla u)=f$ in $\Omega\backslash \Gamma$, where $\Gamma$ is a smooth interface inside $\Omega$. The variable scalar coefficient $a>0$ and source $f$ are possibly discontinuous across $\Gamma$. The hybrid FDMs utilize a 9-point compact stencil at any interior regular point of the grid and a 13-point stencil at irregular points near $\Gamma$. For interior regular points away from $\Gamma$, we obtain a sixth-order 9-point compact FDM satisfying the M-matrix property. Consequently, for the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs having the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. For irregular points near $\Gamma$, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient $a$, the source term $f$, the interface curve $\Gamma$, the two jump functions along $\Gamma$, and the functions on $\partial \Omega$. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing $\Gamma$ and are also applicable if the jump conditions on $\Gamma$ only depend on the geometry (e.g., curvature) of the curve $\Gamma$. Our numerical experiments confirm the sixth-order convergence in the $l_{\infty}$ norm of the proposed hybrid FDMs for the elliptic interface problem.

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. However, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer, compared to MDS matrices. In this paper, we study NMDS matrices, exploring their construction in both recursive and nonrecursive settings. We provide several theoretical results and explore the hardware efficiency of the construction of NMDS matrices. Additionally, we make comparisons between the results of NMDS and MDS matrices whenever possible. For the recursive approach, we study the DLS matrices and provide some theoretical results on their use. Some of the results are used to restrict the search space of the DLS matrices. We also show that over a field of characteristic 2, any sparse matrix of order $n\geq 4$ with fixed XOR value of 1 cannot be an NMDS when raised to a power of $k\leq n$. Following that, we use the generalized DLS (GDLS) matrices to provide some lightweight recursive NMDS matrices of several orders that perform better than the existing matrices in terms of hardware cost or the number of iterations. For the nonrecursive construction of NMDS matrices, we study various structures, such as circulant and left-circulant matrices, and their generalizations: Toeplitz and Hankel matrices. In addition, we prove that Toeplitz matrices of order $n>4$ cannot be simultaneously NMDS and involutory over a field of characteristic 2. Finally, we use GDLS matrices to provide some lightweight NMDS matrices that can be computed in one clock cycle. The proposed nonrecursive NMDS matrices of orders 4, 5, 6, 7, and 8 can be implemented with 24, 50, 65, 96, and 108 XORs over $\mathbb{F}_{2^4}$, respectively.

Given a dataset of $n$ i.i.d. samples from an unknown distribution $P$, we consider the problem of generating a sample from a distribution that is close to $P$ in total variation distance, under the constraint of differential privacy (DP). We study the problem when $P$ is a multi-dimensional Gaussian distribution, under different assumptions on the information available to the DP mechanism: known covariance, unknown bounded covariance, and unknown unbounded covariance. We present new DP sampling algorithms, and show that they achieve near-optimal sample complexity in the first two settings. Moreover, when $P$ is a product distribution on the binary hypercube, we obtain a pure-DP algorithm whereas only an approximate-DP algorithm (with slightly worse sample complexity) was previously known.

We study extensions of Fr\'{e}chet means for random objects in the space ${\rm Sym}^+(p)$ of $p \times p$ symmetric positive-definite matrices using the scaling-rotation geometric framework introduced by Jung et al. [\textit{SIAM J. Matrix. Anal. Appl.} \textbf{36} (2015) 1180-1201]. The scaling-rotation framework is designed to enjoy a clearer interpretation of the changes in random ellipsoids in terms of scaling and rotation. In this work, we formally define the \emph{scaling-rotation (SR) mean set} to be the set of Fr\'{e}chet means in ${\rm Sym}^+(p)$ with respect to the scaling-rotation distance. Since computing such means requires a difficult optimization, we also define the \emph{partial scaling-rotation (PSR) mean set} lying on the space of eigen-decompositions as a proxy for the SR mean set. The PSR mean set is easier to compute and its projection to ${\rm Sym}^+(p)$ often coincides with SR mean set. Minimal conditions are required to ensure that the mean sets are non-empty. Because eigen-decompositions are never unique, neither are PSR means, but we give sufficient conditions for the sample PSR mean to be unique up to the action of a certain finite group. We also establish strong consistency of the sample PSR means as estimators of the population PSR mean set, and a central limit theorem. In an application to multivariate tensor-based morphometry, we demonstrate that a two-group test using the proposed PSR means can have greater power than the two-group test using the usual affine-invariant geometric framework for symmetric positive-definite matrices.

Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian~optimization~task.

Compatible finite element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging since the velocity mass matrix is non-diagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate- and weather prediction model LFRic.

Motivated by the virtual machine scheduling problem in today's computing systems, we propose a new setting of stochastic bin-packing in service systems that allows the item sizes (job resource requirements) to vary over time. In this setting, items (jobs) arrive to the system, vary their sizes, and depart from the system following certain Markovian assumptions. We focus on minimizing the expected number of non-empty bins (active servers) in steady state, where the expectation in steady state is equal to the long-run time-average with probability $1$ under the Markovian assumptions. Our main result is a policy that achieves an optimality gap of $O(\sqrt{r})$ in the objective, where the optimal objective value is $\Theta(r)$ and $r$ is a scaling factor such that the item arrival intensity scales linearly with it. When specialized to the setting where the item sizes do not vary over time, our result improves upon the state-of-the-art $o(r)$ optimality gap. Our technical approach highlights a novel policy conversion framework that reduces the policy design problem to that in a single-bin (single-server) system.

We consider here the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (\,for example $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for H\"older potentials $A:M \to \mathbb{R}$ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When $T=\sigma$ and $M=\{0,1\}^\mathbb{N}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}(\mu_1,\mu_2)$ for a pair of equilibrium probabilities. If $D_{KL}(\mu_1,\mu_2)=0$, then $\mu_1=\mu_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between $\mu_1$ and $\mu_2$. A natural problem is: for a fixed probability $\mu_1\in \mathcal{N}$ consider the probability $\mu_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}(\mu_1,\mu_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

Learning a nonparametric system of ordinary differential equations (ODEs) from $n$ trajectory snapshots in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach to learning using the implicit formulation provided by vector-valued Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral form, which we subsequently minimize, we derive our learning algorithm. The minimization problem's solution for the vector field relies on multivariate occupation kernel functions associated with the solution trajectories. We validate our approach through experiments on highly nonlinear simulated and real data, where $d$ may exceed 100. We further demonstrate the versatility of the proposed method by learning a nonparametric first order quasilinear partial differential equation.

Game theory has by now found numerous applications in various fields, including economics, industry, jurisprudence, and artificial intelligence, where each player only cares about its own interest in a noncooperative or cooperative manner, but without obvious malice to other players. However, in many practical applications, such as poker, chess, evader pursuing, drug interdiction, coast guard, cyber-security, and national defense, players often have apparently adversarial stances, that is, selfish actions of each player inevitably or intentionally inflict loss or wreak havoc on other players. Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal strategy seeking techniques, prevailing algorithms, and practical applications. Finally, promising future research directions are also discussed for relevant adversarial games.