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We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions to the FSI problems in the standard $L^2$ norm. In this article, we propose a new kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the $L^2$ norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.

An important aspect of developing reliable deep learning systems is devising strategies that make these systems robust to adversarial attacks. There is a long line of work that focuses on developing defenses against these attacks, but recently, researchers have began to study ways to reverse engineer the attack process. This allows us to not only defend against several attack models, but also classify the threat model. However, there is still a lack of theoretical guarantees for the reverse engineering process. Current approaches that give any guarantees are based on the assumption that the data lies in a union of linear subspaces, which is not a valid assumption for more complex datasets. In this paper, we build on prior work and propose a novel framework for reverse engineering of deceptions which supposes that the clean data lies in the range of a GAN. To classify the signal and attack, we jointly solve a GAN inversion problem and a block-sparse recovery problem. For the first time in the literature, we provide deterministic linear convergence guarantees for this problem. We also empirically demonstrate the merits of the proposed approach on several nonlinear datasets as compared to state-of-the-art methods.

In this paper, we study two graph convexity parameters: iteration time and general position number. The iteration time was defined in 1981 in the geodesic convexity, but its computational complexity was still open. The general position number was defined in the geodesic convexity and proved NP-hard in 2018. We extend these parameters to any graph convexity and prove that the iteration number is NP-hard in the $P_3$ convexity and, with this result, we can prove that the iteration time is also NP-hard in the geodesic convexity even in graphs with diameter two, a very natural question which was unsolved since 1981. These results are also important, since they are the last two missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and $P_3$ convexities. Finally, we also prove that the general position number of the monophonic convexity is NP-hard, W[1]-hard (parameterized by the size of the solution) and $n^{1-\varepsilon}$-inapproximable in polynomial time for any $\varepsilon>0$ unless P=NP, even in graphs with diameter two.

Empirical neural tangent kernels (eNTKs) can provide a good understanding of a given network's representation: they are often far less expensive to compute and applicable more broadly than infinite width NTKs. For networks with O output units (e.g. an O-class classifier), however, the eNTK on N inputs is of size $NO \times NO$, taking $O((NO)^2)$ memory and up to $O((NO)^3)$ computation. Most existing applications have therefore used one of a handful of approximations yielding $N \times N$ kernel matrices, saving orders of magnitude of computation, but with limited to no justification. We prove that one such approximation, which we call "sum of logits", converges to the true eNTK at initialization for any network with a wide final "readout" layer. Our experiments demonstrate the quality of this approximation for various uses across a range of settings.

Robust Markov decision processes (RMDPs) provide a promising framework for computing reliable policies in the face of model errors. Many successful reinforcement learning algorithms build on variations of policy-gradient methods, but adapting these methods to RMDPs has been challenging. As a result, the applicability of RMDPs to large, practical domains remains limited. This paper proposes a new Double-Loop Robust Policy Gradient (DRPG), the first generic policy gradient method for RMDPs. In contrast with prior robust policy gradient algorithms, DRPG monotonically reduces approximation errors to guarantee convergence to a globally optimal policy in tabular RMDPs. We introduce a novel parametric transition kernel and solve the inner loop robust policy via a gradient-based method. Finally, our numerical results demonstrate the utility of our new algorithm and confirm its global convergence properties.

U-statistics play central roles in many statistical learning tools but face the haunting issue of scalability. Significant efforts have been devoted into accelerating computation by U-statistic reduction. However, existing results almost exclusively focus on power analysis, while little work addresses risk control accuracy -- comparatively, the latter requires distinct and much more challenging techniques. In this paper, we establish the first statistical inference procedure with provably higher-order accurate risk control for incomplete U-statistics. The sharpness of our new result enables us to reveal how risk control accuracy also trades off with speed for the first time in literature, which complements the well-known variance-speed trade-off. Our proposed general framework converts the long-standing challenge of formulating accurate statistical inference procedures for many different designs into a surprisingly routine task. This paper covers non-degenerate and degenerate U-statistics, and network moments. We conducted comprehensive numerical studies and observed results that validate our theory's sharpness. Our method also demonstrates effectiveness on real-world data applications.

We consider the damped time-harmonic Galbrun's equation, which is used to model stellar oscillations. We introduce a discontinuous Galerkin finite element method (DGFEM) with $H(\operatorname{div})$-elements, which is nonconform with respect to the convection operator. We report a convergence analysis, which is based on the frameworks of discrete approximation schemes and T-compatibility. A novelty is that we show how to interprete a DGFEM as a discrete approximation scheme and this approach enables us to apply compact perturbation arguments in a DG-setting, and to circumvent any extra regularity assumptions on the solution. The advantage of the proposed $H(\operatorname{div})$-DGFEM compared to $H^1$-conforming methods is that we do not require a minimal polynomial order or any special assumptions on the mesh structure. The considered DGFEM is constructed without a stabilization term, which considerably improves the assumption on the smallness of the Mach number compared to other DG methods and $H^1$-conforming methods, and the obtained bound is fairly explicit. In addition, the method is robust with respect to the drastic changes of magnitude of the density and sound speed, which occur in stars. The convergence of the method is obtained without additional regularity assumptions on the solution, and for smooth solutions and parameters convergence rates are derived.

In this paper, we aim at unifying, simplifying and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central role in the unification and in the simplification of the analysis of most Lagrangian-based methods. Equipped with a nice primal algorithmic map, we then introduce a versatile generic scheme, which allows for the design and analysis of Faster LAGrangian (FLAG) methods with new provably sublinear rate of convergence expressed in terms of function values and feasibility violation of the original (non-ergodic) generated sequence. To demonstrate the power and versatility of our approach and results, we show that most well-known iconic Lagrangian-based schemes admit a nice primal algorithmic map, and hence share the new faster rate of convergence results within their corresponding FLAG.

This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.