In this paper we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.
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This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $\Pi \left(\cdot |Y = y\right)$, we want to infer whether a feature $\langle\varphi, u^\dagger\rangle$ of the unknown quantity of interest $u^\dagger$ is positive. This can be done by the so-called maximum a posteriori test. We provide a frequentistic analysis of this test's properties such as level and power, and prove that it is a regularized test in the sense of Kretschmann et al. (2024). Furthermore we provide lower bounds for its power under classical spectral source conditions in case of Gaussian priors. Numerical simulations illustrate its superior performance both in moderately and severely ill-posed situations.
In this paper, we introduce a new fraud-proof algorithm that offers an unprecedented combination of decentralization, security, and liveness. The resources that must be mobilized by an honest participant to defeat an adversary grow only logarithmically with what the adversary ultimately loses. As a consequence, there is no need to introduce high bonds that prevent an adversary from creating too many Sybils. This makes the system very inclusive and frees participants from having to pool resources among themselves to engage the protocol. Finally, the maximum delay to finalization also grows only logarithmically with total adversarial expenditure, with the smallest multiplicative factor to date. In summary: the entire dispute completes in 2--5 challenge periods, the only way to break consensus is to censor the honest party for more than one challenge period, and the costs of engaging in the dispute are minimal.
In this paper, we propose high order numerical methods to solve a 2D advection diffusion equation, in the highly oscillatory regime. We use an integrator strategy that allows the construction of arbitrary high-order schemes {leading} to an accurate approximation of the solution without any time step-size restriction. This paper focuses on the multiscale challenges {in time} of the problem, that come from the velocity, an $\varepsilon-$periodic function, whose expression is explicitly known. $\varepsilon$-uniform third order in time numerical approximations are obtained. For the space discretization, this strategy is combined with high order finite difference schemes. Numerical experiments show that the proposed methods {achieve} the expected order of accuracy, and it is validated by several tests across diverse domains and boundary conditions. The novelty of the paper consists of introducing a numerical scheme that is high order accurate in space and time, with a particular attention to the dependency on a small parameter in the time scale. The high order in space is obtained enlarging the interpolation stencil already established in [44], and further refined in [46], with a special emphasis on the squared boundary, especially when a Dirichlet condition is assigned. In such case, we compute an \textit{ad hoc} Taylor expansion of the solution to ensure that there is no degradation of the accuracy order at the boundary. On the other hand, the high accuracy in time is obtained extending the work proposed in [19]. The combination of high-order accuracy in both space and time is particularly significant due to the presence of two small parameters-$\delta$ and $\varepsilon$-in space and time, respectively.
After nearly two decades of research, the question of a quantum PCP theorem for quantum Constraint Satisfaction Problems (CSPs) remains wide open. As a result, proving QMA-hardness of approximation for ground state energy estimation has remained elusive. Recently, it was shown [Bittel, Gharibian, Kliesch, CCC 2023] that a natural problem involving variational quantum circuits is QCMA-hard to approximate within ratio N^(1-eps) for any eps > 0 and N the input size. Unfortunately, this problem was not related to quantum CSPs, leaving the question of hardness of approximation for quantum CSPs open. In this work, we show that if instead of focusing on ground state energies, one considers computing properties of the ground space, QCMA-hardness of computing ground space properties can be shown. In particular, we show that it is (1) QCMA-complete within ratio N^(1-eps) to approximate the Ground State Connectivity problem (GSCON), and (2) QCMA-hard within the same ratio to estimate the amount of entanglement of a local Hamiltonian's ground state, denoted Ground State Entanglement (GSE). As a bonus, a simplification of our construction yields NP-completeness of approximation for a natural k-SAT reconfiguration problem, to be contrasted with the recent PCP-based PSPACE hardness of approximation results for a different definition of k-SAT reconfiguration [Karthik C.S. and Manurangsi, 2023, and Hirahara, Ohsaka, STOC 2024].
In this paper, we investigate the two-dimensional extension of a recently introduced set of shallow water models based on a regularized moment expansion of the incompressible Navier-Stokes equations \cite{kowalski2017moment,koellermeier2020analysis}. We show the rotational invariance of the proposed moment models with two different approaches. The first proof involves the split of the coefficient matrix into the conservative and non-conservative parts and proves the rotational invariance for each part, while the second one relies on the special block structure of the coefficient matrices. With the aid of rotational invariance, the analysis of the hyperbolicity for the moment model in 2D is reduced to the real diagonalizability of the coefficient matrix in 1D. Then we analyze the real diagonalizability by deriving the analytical form of the characteristic polynomial. We find that the moment model in 2D is hyperbolic in most cases and weakly hyperbolic in a degenerate edge case. With a simple modification to the coefficient matrices, we fix this weakly hyperbolicity and propose a new global hyperbolic model. Furthermore, we extend the model to include a more general class of closure relations than the original model and establish that this set of general closure relations retains both rotational invariance and hyperbolicity.
In this paper, we introduce a highly accurate and efficient numerical solver for the radial Kohn--Sham equation. The equation is discretized using a high-order finite element method, with its performance further improved by incorporating a parameter-free moving mesh technique. This approach greatly reduces the number of elements required to achieve the desired precision. In practice, the mesh redistribution involves no more than three steps, ensuring the algorithm remains computationally efficient. Remarkably, with a maximum of $13$ elements, we successfully reproduce the NIST database results for elements with atomic numbers ranging from $1$ to $92$.
In this paper, we propose a novel eigenpair-splitting method, inspired by the divide-and-conquer strategy, for solving the generalized eigenvalue problem arising from the Kohn-Sham equation. Unlike the commonly used domain decomposition approach in divide-and-conquer, which solves the problem on a series of subdomains, our eigenpair-splitting method focuses on solving a series of subequations defined on the entire domain. This method is realized through the integration of two key techniques: a multi-mesh technique for generating approximate spaces for the subequations, and a soft-locking technique that allows for the independent solution of eigenpairs. Numerical experiments show that the proposed eigenpair-splitting method can dramatically enhance simulation efficiency, and its potential towards practical applications is also demonstrated well through an example of the HOMO-LUMO gap calculation. Furthermore, the optimal strategy for grouping eigenpairs is discussed, and the possible improvements to the proposed method are also outlined.
Interpreting data with mathematical models is an important aspect of real-world applied mathematical modeling. Very often we are interested to understand the extent to which a particular data set informs and constrains model parameters. This question is closely related to the concept of parameter identifiability, and in this article we present a series of computational exercises to introduce tools that can be used to assess parameter identifiability, estimate parameters and generate model predictions. Taking a likelihood-based approach, we show that very similar ideas and algorithms can be used to deal with a range of different mathematical modelling frameworks. The exercises and results presented in this article are supported by a suite of open access codes that can be accessed on GitHub.
In this paper, we study the Hermitian hulls of generalized Reed-Solomon (GRS) codes over finite fields. For a given class of GRS codes, by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time, we obtain three classes of GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, based on some known $q^2$-ary Hermitian self-orthogonal GRS codes with dimension $q-1$, we obtain several classes of $q^2$-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. It is worth noting that the dimension of these MDS codes can be taken from $q$ to $\frac{n}{2}$, and the parameters of these MDS codes can be more flexible by propagation rules. As an application, we derive three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes. Then, from the presently known GRS codes with Hermitian hulls, we can directly obtain many MDS EAQECCs with more flexible parameters. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters, and the distance of these codes can be taken from $q+1$ to $\frac{n+2}{2}$.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
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