Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.     

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

Interpolation of Sparse Multivariate Polynomials over Large Finite Fields with Applications

We develop a Las Vegas-randomized algorithm which performs interpolation of sparse multivariate polynomials over finite fields. Our algorithm can be viewed as the first successful adaptation of the sparse polynomial interpolation algorithm for the complex field developed by M. Ben-Or and P. Tiwari (1988, in “Proceedings of the 20th ACM Symposium on the Theory of Computing,” pp. 301–309, Assoc. Comput. Mach., New York) to the case of finite fields. It improves upon a previous result by D. Y. Grigoriev, M. Karpinski, and M. F. Singer (1990, SIAM J. Comput.19, 1059–1063) and is by far the most time efficient algorithm (time and processor efficient parallel algorithm) for the problem when the finite field is large. As applications, we obtain Monte Carlo-randomized parallel algorithms for sparse multivariate polynomial factorization and GCD over finite fields. The efficiency of these algorithms improves upon that of the previously known algorithms for the respective problems.     

Computing Jacobi Symbols modulo Sparse Integers and Polynomials and Some Applications

We describe a polynomial time algorithm to compute Jacobi symbols of exponentially large integers of special form, including so-called sparse integers which are exponentially large integers with only polynomially many nonzero binary digits. In a number of papers sequences of Jacobi symbols have been proposed as generators of cryptographically secure pseudorandom bits. Our algorithm allows us to use much larger moduli in such constructions. We also use our algorithm to design a probabilistic polynomial time test which decides if a given integer of the aforementioned type is a perfect square (assuming the Extended Riemann Hypothesis). We also obtain analogues of these results for polynomials over finite fields. Moreover, in this case the perfect square testing algorithm is unconditional. These results can be compared with many known NP-hardness results for some natural problems on sparse integers and polynomials.     

Generalized Normal Derivatives and Their Applications in DDMs with Nonmatching Grids and DG Methods

A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.     

Recent Developments In Harmonic Approximation,With Applications

A theorem of J.L. Walsh (1929) says that if E is a compact subset of Rⁿ with connected complement and if u is harmonic on a neighbourhood of E, then u can be uniformly approximated on E by functions harmonic on the whole of Rⁿ. In Part I of this article we survey some generalizations of Walsh’s theorem from the period 1980–94. In Part II we discuss applications of Walsh’s theorem and its generalizations to four diverse topics: universal harmonic functions, the Radon transform, the maximum principle, and the Dirichlet problem.     

正交设计的最新发展和应用-回归分析在正交设计的应用

正交设计有许多新发展,本系列讲座介绍其中的一些便于应用的结果,共有四部份：回归分析在正交设计中的应用,均匀正交设计,正交设计的Ｄ－最优性,以及正交设计的投影性质。本讲强调回归分析用于正交设计的建模、估计、减少参数数目方面的应用     

Some Recent Progress and Applications in Graph Minor Theory

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists Supported in part by the Research Grant P1–0297 and by the CRC program On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia     

Sparse Regression by Projection and Sparse Discriminant Analysis

Recent years have seen active developments of various penalized regression methods, such as LASSO and elastic net, to analyze high-dimensional data. In these approaches, the direction and length of the regression coefficients are determined simultaneously. Due to the introduction of penalties, the length of the estimates can be far from being optimal for accurate predictions. We introduce a new framework, regression by projection, and its sparse version to analyze high-dimensional data. The unique nature of this framework is that the directions of the regression coefficients are inferred first, and the lengths and the tuning parameters are determined by a cross-validation procedure to achieve the largest prediction accuracy. We provide a theoretical result for simultaneous model selection consistency and parameter estimation consistency of our method in high dimension. This new framework is then generalized such that it can be applied to principal components analysis, partial least squares, and canonical correlation analysis. We also adapt this framework for discriminant analysis. Compared with the existing methods, where there is relatively little control of the dependency among the sparse components, our method can control the relationships among the components. We present efficient algorithms and related theory for solving the sparse regression by projection problem. Based on extensive simulations and real data analysis, we demonstrate that our method achieves good predictive performance and variable selection in the regression setting, and the ability to control relationships between the sparse components leads to more accurate classification. In supplementary materials available online, the details of the algorithms and theoretical proofs, and R codes for all simulation studies are provided.     

Sparse matrices

One gives a survey of methods and programs for solving large sparse spectral problems based on the Lanczos algorithm. Practically all the important works on this topic are reflected in this survey. One also considers applications of the variants of the Lanczos method to the solution of symmetric indefinite systems of linear equations and to a series of other problems of linear algebra.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 20, pp. 179–260, 1982.     

Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

A dimension reduction method based on the “Nonlinear Level set Learning” (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.     

Grids in random graphs

Threshold probabilities for the existence in a random graph on n vertices of a graph isomorphic to a given graph of order Cn and average degree at least three are investigated. In particular it is proved that the random graph G(n, p) on n vertices with edge probability contains a square grid on En/2 vertices. © 1994 John Wiley & Sons, Inc.     

Sparse Partition Regularity

Our aim in this paper is to prove Deuber's conjecture on sparsepartition regularity, that for every m, p and c there existsa subset of the natural numbers whose (m,p,c)-sets have highgirth and chromatic number. More precisely, we show that forany mp, c, k and g there is a subset S of the natural numbersthat is sufficiently rich in (m,p,c)-sets that whenever S isk-coloured there is a monochromatic (m,p,c)-set, yet is so sparsethat its (m,p,c)-sets do not form any cycles of length lessthan g. Our main tools are some extensions of Neetil–Rödlamalgamation and a Ramsey theorem of Bergelson, Hindman andLeader. As a sideline, we obtain a Ramsey theorem for productsof trees that may be of independent interest. 2000 MathematicsSubject Classification 05D10.     

Sparse ramsey graphs

IfH is a Ramsey graph for a graphG thenH is rich in copies of the graphG. Here we prove theorems in the opposite direction. We find examples ofH such that copies ofG do not form short cycles inH. This provides a strenghtening also, of the following well-known result of Erdős: there exist graphs with high chromatic number and no short cycles. In particular, we solve a problem of J. Spencer. Dedicated to Paul Erdős on his seventieth birthday     

Sparse color-critical hypergraphs

In this paper we obtain estimates for the least number of edges ann-uniformr-color-critical hypergraph of orderm may have.     

Sparse pseudorandom distributions

The existence of sparse pseudorandom distributions is proved. These are probability distributions concentrated in a very small set of strings, yet it is infeasible for any polynomial-time algorithm to distinguish between truly random coins and coins selected according to these distributions. It is shown that such distributions can be generated by (nonpolynomial) probabilistic algorithms, while probabilistic polynomial-time algorithms cannot even approximate all the pseudorandom distributions. Moreover, we show the existence of evasive pseudorandom distributions which are not only sparse, but also have the property that no polynomial-time algorithm may find an element in their support, except for a negligible probability. All these results are proved independently of any intractability assumption.