Shape restriction, like monotonicity or convexity, imposed on a function of interest, such as a regression or density function, allows for its estimation without smoothness assumptions. The concept of $k$-monotonicity encompasses a family of shape restrictions, including decreasing and convex decreasing as special cases corresponding to $k=1$ and $k=2$. We consider Bayesian approaches to estimate a $k$-monotone density. By utilizing a kernel mixture representation and putting a Dirichlet process or a finite mixture prior on the mixing distribution, we show that the posterior contraction rate in the Hellinger distance is $(n/\log n)^{- k/(2k + 1)}$ for a $k$-monotone density, which is minimax optimal up to a polylogarithmic factor. When the true $k$-monotone density is a finite $J_0$-component mixture of the kernel, the contraction rate improves to the nearly parametric rate $\sqrt{(J_0 \log n)/n}$. Moreover, by putting a prior on $k$, we show that the same rates hold even when the best value of $k$ is unknown. A specific application in modeling the density of $p$-values in a large-scale multiple testing problem is considered. Simulation studies are conducted to evaluate the performance of the proposed method.
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The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.
In its additive version, Bohr-Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution $f(x)$ to the equation $\Delta f(x)=\ln x$ on the open half-line $(0,\infty)$ is the log-gamma function $f(x)=\ln\Gamma(x)$, where $\Delta$ denotes the classical difference operator and $\Gamma(x)$ denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr-Mollerup's theorem by considering the functional equation $\Delta f(x)=g(x)$, where $g$ can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions $f(x)$ arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr-Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application.
In this paper the authors study a non-linear elliptic-parabolic system, which is motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation. The goal is to determine several scalar parameters in the coupled model in an optimal manner by utilizing a reliable reduced-order approach based on the reduced basis (RB) method. However, the states are coupled through a strongly non-linear function, and this makes the evaluation of online-efficient error estimates difficult. First the well-posedness of the system is proved. Then a Galerkin finite element and RB discretization are described for the coupled system. To certify the RB scheme hierarchical a-posteriori error estimators are utilized in an adaptive trust-region optimization method. Numerical experiments illustrate good approximation properties and efficiencies by using only a relatively small number of reduced basis functions.
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in analytic language. We extend the theory of graph limits to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which we call an orderon. As a special case, this yields limit objects for matrices whose rows and columns are ordered, and for dynamic graphs that expand (via vertex insertions) over time. Along the way, we devise an ordered locality-preserving variant of the cut distance between ordered graphs, showing that two graphs are close with respect to this distance if and only if they are similar in terms of their ordered subgraph frequencies. We show that the space of orderons is compact with respect to this distance notion, which is key to a successful analysis of combinatorial objects through their limits. We derive several applications of the ordered limit theory in extremal combinatorics, sampling, and property testing in ordered graphs. In particular, we prove a new ordered analogue of the well-known result by Alon and Stav [RS\&A'08] on the furthest graph from a hereditary property; this is the first known result of this type in the ordered setting. Unlike the unordered regime, here the random graph model $G(n, p)$ with an ordering over the vertices is not always asymptotically the furthest from the property for some $p$. However, using our ordered limit theory, we show that random graphs generated by a stochastic block model, where the blocks are consecutive in the vertex ordering, are (approximately) the furthest. Additionally, we describe an alternative analytic proof of the ordered graph removal lemma [Alon et al., FOCS'17].
This paper proposes a non-centered parameterization based infinite-dimensional mean-field variational inference (NCP-iMFVI) approach for solving the hierarchical Bayesian inverse problems. This method can generate available estimates from the approximated posterior distribution efficiently. To avoid the mutually singular obstacle that occurred in the infinite-dimensional hierarchical approach, we propose a rigorous theory of the non-centered variational Bayesian approach. Since the non-centered parameterization weakens the connection between the parameter and the hyper-parameter, we can introduce the hyper-parameter to all terms of the eigendecomposition of the prior covariance operator. We also show the relationships between the NCP-iMFVI and infinite-dimensional hierarchical approaches with centered parameterization. The proposed algorithm is applied to three inverse problems governed by the simple smooth equation, the Helmholtz equation, and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of solving the iMFVI problem formulated by large-scale linear and nonlinear statistical inverse problems, and verify the mesh-independent property.
A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection-diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.
In this paper, we propose a method for estimating model parameters using Small-Angle Scattering (SAS) data based on the Bayesian inference. Conventional SAS data analyses involve processes of manual parameter adjustment by analysts or optimization using gradient methods. These analysis processes tend to involve heuristic approaches and may lead to local solutions.Furthermore, it is difficult to evaluate the reliability of the results obtained by conventional analysis methods. Our method solves these problems by estimating model parameters as probability distributions from SAS data using the framework of the Bayesian inference. We evaluate the performance of our method through numerical experiments using artificial data of representative measurement target models.From the results of the numerical experiments, we show that our method provides not only high accuracy and reliability of estimation, but also perspectives on the transition point of estimability with respect to the measurement time and the lower bound of the angular domain of the measured data.
We investigate the fixed-budget best-arm identification (BAI) problem for linear bandits in a potentially non-stationary environment. Given a finite arm set $\mathcal{X}\subset\mathbb{R}^d$, a fixed budget $T$, and an unpredictable sequence of parameters $\left\lbrace\theta_t\right\rbrace_{t=1}^{T}$, an algorithm will aim to correctly identify the best arm $x^* := \arg\max_{x\in\mathcal{X}}x^\top\sum_{t=1}^{T}\theta_t$ with probability as high as possible. Prior work has addressed the stationary setting where $\theta_t = \theta_1$ for all $t$ and demonstrated that the error probability decreases as $\exp(-T /\rho^*)$ for a problem-dependent constant $\rho^*$. But in many real-world $A/B/n$ multivariate testing scenarios that motivate our work, the environment is non-stationary and an algorithm expecting a stationary setting can easily fail. For robust identification, it is well-known that if arms are chosen randomly and non-adaptively from a G-optimal design over $\mathcal{X}$ at each time then the error probability decreases as $\exp(-T\Delta^2_{(1)}/d)$, where $\Delta_{(1)} = \min_{x \neq x^*} (x^* - x)^\top \frac{1}{T}\sum_{t=1}^T \theta_t$. As there exist environments where $\Delta_{(1)}^2/ d \ll 1/ \rho^*$, we are motivated to propose a novel algorithm $\mathsf{P1}$-$\mathsf{RAGE}$ that aims to obtain the best of both worlds: robustness to non-stationarity and fast rates of identification in benign settings. We characterize the error probability of $\mathsf{P1}$-$\mathsf{RAGE}$ and demonstrate empirically that the algorithm indeed never performs worse than G-optimal design but compares favorably to the best algorithms in the stationary setting.
A geometric graph is an abstract graph along with an embedding of the graph into the Euclidean plane which can be used to model a wide range of data sets. The ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances or metrics on these objects. In this work, we study the interleaving distance on geometric graphs, where functor representations of data can be compared by finding pairs of natural transformations between them. However, in many cases, particularly those of the set-valued functor variety, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from the work of Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation. Specifically, we call collections $\phi = \{\phi_U\mid U\}$ and $\psi = \{\psi_U\mid U\}$ which do not necessarily form a true interleaving an \textit{assignment}. In the case of embedded graphs, we impose a grid structure on the plane, treat this as a poset endowed with the Alexandroff topology $K$, and encode the embedded graph data as functors $F: \mathbf{Open}(K) \to \mathbf{Set}$ where $F(U)$ is the set of connected components of the graph inside of the geometric realization of the set $U$. We then endow the image with the extra structure of a metric space and define a loss function $L(\phi,\psi)$ which measures how far the required diagrams of an interleaving are from commuting. Then for a pair of assignments, we use this loss function to bound the interleaving distance, with an eye toward computation and approximation of the distance. We expect these ideas are not only useful in our particular use case of embedded graphs, but can be extended to a larger class of interleaving distance problems where computational complexity creates a barrier to use in practice.
For a long time, the Von Neumann has been a successful model of computation for sequential computing .Many models including the dataflow model have been unsuccessfully developed to emulate the same results in parallel computing. It is widely accepted that high performance computation is better-achieved using parallel architectures and is seen as the basis for future computational architectures with the ever-increasing need for high performance computation. We describe a new model of parallel computation known as the Arithmetic Deduction Model (AriDem) which has some similarities with the Von Neumann. A theoretical evaluation conducted on this model in comparison with the predominant von Neumann model indicated AriDeM to be more efficient in resources utilization. In this paper, we conduct an empirical evaluation of the model and the results reflect the output of the theoretical evaluation.
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