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We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.

In this work, we develop a numerical method to study the error estimates of the $\alpha$-stable central limit theorem under sublinear expectation with $\alpha \in(0,2)$, whose limit distribution can be characterized by a fully nonlinear integro-differential equation (PIDE). Based on the sequence of independent random variables, we propose a discrete approximation scheme for the fully nonlinear PIDE. With the help of the nonlinear stochastic analysis techniques and numerical analysis tools, we establish the error bounds for the discrete approximation scheme, which in turn provides a general error bound for the robust $\alpha$-stable central limit theorem, including the integrable case $\alpha \in(1,2)$ as well as the non-integrable case $\alpha \in(0,1]$. Finally, we provide some concrete examples to illustrate our main results and derive the precise convergence rates.

We expound on some known lower bounds of the quadratic Wasserstein distance between random vectors in $\mathbb{R}^n$ with an emphasis on affine transformations that have been used in manifold learning of data in Wasserstein space. In particular, we give concrete lower bounds for rotated copies of random vectors in $\mathbb{R}^2$ with uncorrelated components by computing the Bures metric between the covariance matrices. We also derive upper bounds for compositions of affine maps which yield a fruitful variety of diffeomorphisms applied to an initial data measure. We apply these bounds to various distributions including those lying on a 1-dimensional manifold in $\mathbb{R}^2$ and illustrate the quality of the bounds. Finally, we give a framework for mimicking handwritten digit or alphabet datasets that can be applied in a manifold learning framework.

We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions.

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least integer $k$ for which $D$ has a coloring with $k$ colors such that there is no monochromatic directed cycle in $D$. The digraphs considered here are finite and may have antiparallel arcs, but no parallel arcs. A digraph $D$ is called $k$-critical if each proper subdigraph $D'$ of $D$ satisfies $\vec{\chi}(D')<\vec{\chi}(D)=k$. For integers $k$ and $n$, let $\overrightarrow{\mathrm{ext}}(k,n)$ denote the minimum number of arcs possible in a $k$-critical digraph of order $n$. It is easy to show that $\overrightarrow{\mathrm{ext}}(2,n)=n$ for all $n\geq 2$, and $\overrightarrow{\mathrm{ext}}(3,n)\geq 2n$ for all possible $n$, where equality holds if and only if $n$ is odd and $n\geq 3$. As a main result we prove that if $n, k$ and $p$ are integers with $n=k+p$ and $2\leq p \leq k-1$, then $\overrightarrow{\mathrm{ext}}(k,n)=2({\binom{n}{2}} - (p^2+1))$, and we give an exact characterisation of $k$-critical digraphs for which equality holds. This generalizes a result about critical graphs obtained in 1963 by Tibor Gallai.

A $(a,b)$-coloring of a graph $G$ associates to each vertex a $b$-subset of a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoint. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, using necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices, more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice for which McDiarmid-Reed's conjecture asserts that they are all $(9,4)$-colorable. Computations on millions of such graphs generated randomly show that our tools allow to find a $(9,4)$-coloring for each of them except for one specific regular shape of graphs (that can be $(9,4)$-colored by an easy ad-hoc process). We thus obtain computational evidence towards the conjecture of McDiarmid\&Reed.

The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(\Omega)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(\Omega)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest.

We consider a matching problem in a bipartite graph $G=(A\cup B,E)$ where nodes in $A$ are agents having preferences in partial order over their neighbors, while nodes in $B$ are objects without preferences. We propose a polynomial-time combinatorial algorithm based on LP duality that finds a maximum matching or assignment in $G$ that is popular among all maximum matchings, if there exists one. Our algorithm can also be used to achieve a trade-off between popularity and cardinality by imposing a penalty on unmatched nodes in $A$. We also provide an $O^*(|E|^k)$ algorithm that finds an assignment whose unpopularity margin is at most $k$; this algorithm is essentially optimal, since the problem is $\mathsf{NP}$-complete and $\mathsf{W}_l[1]$-hard with parameter $k$. We also prove that finding a popular assignment of minimum cost when each edge has an associated binary cost is $\mathsf{NP}$-hard, even if agents have strict preferences. By contrast, we propose a polynomial-time algorithm for the variant of the popular assignment problem with forced/forbidden edges. Finally, we present an application in the context of housing markets.

Given a sample of size $N$, it is often useful to select a subsample of smaller size $n<N$ to be used for statistical estimation or learning. Such a data selection step is useful to reduce the requirements of data labeling and the computational complexity of learning. We assume to be given $N$ unlabeled samples $\{{\boldsymbol x}_i\}_{i\le N}$, and to be given access to a `surrogate model' that can predict labels $y_i$ better than random guessing. Our goal is to select a subset of the samples, to be denoted by $\{{\boldsymbol x}_i\}_{i\in G}$, of size $|G|=n<N$. We then acquire labels for this set and we use them to train a model via regularized empirical risk minimization. By using a mixture of numerical experiments on real and synthetic data, and mathematical derivations under low- and high- dimensional asymptotics, we show that: $(i)$~Data selection can be very effective, in particular beating training on the full sample in some cases; $(ii)$~Certain popular choices in data selection methods (e.g. unbiased reweighted subsampling, or influence function-based subsampling) can be substantially suboptimal.

We present a new algorithm for amortized inference in sparse probabilistic graphical models (PGMs), which we call $\Delta$-amortized inference ($\Delta$-AI). Our approach is based on the observation that when the sampling of variables in a PGM is seen as a sequence of actions taken by an agent, sparsity of the PGM enables local credit assignment in the agent's policy learning objective. This yields a local constraint that can be turned into a local loss in the style of generative flow networks (GFlowNets) that enables off-policy training but avoids the need to instantiate all the random variables for each parameter update, thus speeding up training considerably. The $\Delta$-AI objective matches the conditional distribution of a variable given its Markov blanket in a tractable learned sampler, which has the structure of a Bayesian network, with the same conditional distribution under the target PGM. As such, the trained sampler recovers marginals and conditional distributions of interest and enables inference of partial subsets of variables. We illustrate $\Delta$-AI's effectiveness for sampling from synthetic PGMs and training latent variable models with sparse factor structure.