arXiv Math @arxiv_math@qoto.org

Data assimilation plays a crucial role in numerical modeling, enabling the integration of real-world observations into mathematical models to enhance the accuracy and predictive capabilities of simulations. This approach is widely applied in fields such as meteorology, oceanography, and environmental science, where the dynamic nature of systems demands continuous updates to model states. However, the calibration of models in these high-dimensional, nonlinear systems poses significant challenges. In this paper, we explore a novel calibration methodology using diffusion generative models. We generate synthetic data that statistically aligns with a given set of observations (in this case the increments of the numerical approximation of a solution of a partial differential equation). This allows us to efficiently implement a model reduction and assimilate data from a reference system state modeled by a highly resolved numerical solution of the rotating shallow water equation of order 104 degrees of freedom into a stochastic system having two orders of magnitude less degrees of freedom. To do so, the new samples are incorporated into a particle filtering methodology augmented with tempering and jittering for dynamic state estimation, a method particularly suited for handling complex and multimodal distributions. This work demonstrates how generative models can be used to improve the predictive accuracy for particle filters, providing a more computationally efficient solution for data assimilation and model calibration.

In this article, we study the problem of recovering symmetric $m$-tensor fields (including vector fields) supported in a unit disk $\mathbb{D}$ from a set of generalized V-line transforms, namely longitudinal, transverse, and mixed V-line transforms, and their integral moments. We work in a circular geometric setup, where the V-lines have vertices on a circle, and the axis of symmetry is orthogonal to the circle. We present two approaches to recover a symmetric $m$-tensor field from the combination of longitudinal, transverse, and mixed V-line transforms. With the help of these inversion results, we are able to give an explicit kernel description for these transforms. We also derive inversion algorithms to reconstruct a symmetric $m$-tensor field from its first $(m+1)$ moment longitudinal/transverse V-line transforms.

The best uniform rational approximation of the Sign function on two intervals separated by zero was explicitly found by E.I. Zolotarëv in 1877. The natural extension of this problem to three bands was solved by E.Stiefel in 1961. We indicate the solutions overlooked by the prominent geometer and study their properties.

The enumeration of polymer coverings on two-dimensional rectangular lattices is considered as "intractable". We prove that the number of coverings of $s$ polymer satisfies a simple recurrence relation $ \sum_{i=0}^{2s} (-1)^i \binom{2s}{i} a_{n-i, m-i} = 2^s {(2s)!} / {s!} $ on a $n \times m$ rectangular lattice with open boundary conditions in both directions.

The aim of this paper is to study $K$-frames for quaternionic Hilbert spaces. First, we present the quaternionic version of Douglas's theorem and then investigate $K$-frames for a quaternionic Hilbert space $\mathcal{H}$, where $K \in \mathbb{B}(\mathcal{H})$. Given two quaternionic Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, along with two right $\mathbb{H}$-linear bounded operators $K_1 \in \mathbb{B}(\mathcal{H}_1)$ and $K_2 \in \mathbb{B}(\mathcal{H}_2)$, we study the $K_1 \oplus K_2$-frames for the super space $\mathcal{H}_1 \oplus \mathcal{H}_2$ and their relationship with $K_1$-frames and $K_2$-frames for $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively. We also explore the $K_1 \oplus K_2$-duality in relation to $K_1$-duality and $K_2$-duality.

The aim of this paper is to examine the large-scale behavior of dynamical optimal transport on stationary random graphs embedded in $\R^n$. Our primary contribution is a stochastic homogenization result that characterizes the effective behavior of the discrete problems in terms of a continuous optimal transport problem, where the homogenized energy density results from the geometry of the discrete graph.

In this work we derive a functional equation in terms of the Hurwitz-Lerch zeta function along with definite integrals in terms of the incomplete gamma and Hurwitz-Lerch zeta functions. The method used in these derivations is contour integration. Special cases in terms of fundamental constants are produced.

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody algebra. In this paper, we show that this Lie algebra can be made into a sheaf of Lie algebras over $T^*[V/G]=[μ^{-1}(0)/G]$, where $μ: T^*V\to \mathfrak{g}^*$ is the moment map. We identify this sheaf of Lie algebras with the tangent Lie algebra of the stack $T^*[V/G]$. Moreover, we show that there is an equivalence of braided tensor categories between the bounded derived category of graded modules of $\mathfrak{d}_{\mathfrak{g}, V}$ and graded perfect complexes of $[μ^{-1}(0)/G]$.

We consider a strategic decision-making problem where a logistics provider (LP) seeks to locate collection and delivery points (CDPs) with the objective to reduce total logistics costs. The customers maximize utility that depends on their perception of home delivery service as well as the characteristics of the CDPs, including their location. At the strategic planning level, the LP does not have complete information about customers' preferences and their exact location. We introduce a mixed integer non-linear formulation of the problem and propose two linear reformulations. The latter involve sample average approximations and closest assignment constraints, and in one of the formulations we use scenario aggregation to reduce its size. We solve the formulations with a general-purpose solver using a standard Benders decomposition method. Based on extensive computational results and a realistic case study, we find that the problem can be solved efficiently. However, the level of uncertainty in the instances determines which approach is the most efficient. We use an entropy measure to capture the level of uncertainty that can be computed prior to solving. Furthermore, the results highlight the value of accurate demand modeling, as customer preferences have an important impact on the solutions and associated costs.

Let $ μ_1 $ and $ μ_2 $ be two, in general complex-valued, Borel measures on the real line such that $ \mathrm{supp} \,μ_1 =[α_1,β_1] < \mathrm{supp}\,μ_2 =[α_2,β_2] $ and $ dμ_i(x) = -ρ_i(x)dx/2π\mathrm{i} $, where $ ρ_i(x) $ is the restriction to $ [α_i,β_i] $ of a function non-vanishing and holomorphic in some neighborhood of $ [α_i,β_i] $. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $ (n_1,n_2) $ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $ \min\{n_1,n_2\} $.

The aim of the current paper is to determine the necessary and sufficient conditions for the weights $\mathbf{q}=\{q_k\}$, ensuring that the sequence of operators $\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $ associated with Walsh system, is convergent almost everywhere for all integrable function $f$. The article also examines the convergence of a sequence of tensor product operators denoted as $\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$ involving functions of two variables. We point out that recent research by Gát and Karagulyan (2016) demonstrated that this sequence of tensor product operators cannot converge almost everywhere for every integrable function. In this paper, the necessary and sufficient conditions for the weight are provided which ensure that the sequence of the mentioned operators converges in measure on $L_{1}$.

How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is parametric but extremely rich: it contains all exponential polynomials over $\mathbb{C}$ with total degree $s$, including harmonic oscillations with $s$ arbitrary frequencies. Geometrically, this class corresponds to the projection onto $\mathbb{C}^{n}$ of the union of all shift-invariant subspaces of $\mathbb{C}^\mathbb{Z}$ of dimension $s$. We show that the statistical complexity of this class, as measured by the squared minimax radius of the $(1-δ)$-confidence $\ell_2$-ball, is nearly the same as for the class of $s$-sparse signals, namely $O\left(s\log(en) + \log(δ^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).$ Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rest upon an approximation-theoretic one: we show that finite-dimensional shift-invariant subspaces admit compactly supported reproducing kernels whose Fourier spectra have nearly the smallest possible $\ell_p$-norms, for all $p \in [1,+\infty]$ at once.

It is well-known that to establish the almost everywhere convergence of a sequence of operators on $L_1$-space, it is sufficient to obtain a weak $(1,1)$-type inequality for the maximal operator corresponding to the sequence of operators. However, in practical applications, the establishment of the mentioned inequality for the maximal operators is very tricky and difficult job. In the present paper, the main aim is a novel outlook at above mentioned inequality for the tensor product of two weighted one-dimensional Walsh-Fourier series. Namely, our main idea is naturally to consider uniformly boundedness conditions for the sequences which imply the weak type estimation for the maximal operator of the tensor product. More precisely, we are going to establish weak type of inequality for the tensor product of two dimensional maximal operators while having the uniform boundedness of the corresponding one dimensional operators. Moreover, the convergence of the tensor product of operators is established at the two dimensional Walsh-Lebesgue points.