The finite element methods are important and powerful tools for solving partial differential equations on complex geometries, but their high-order generalizations pose significant challenges in terms of robustness and mesh generation. In this paper, we introduce a high-order finite element method, which is insensitive to element quality and hence is more robust, and which requires only linear elements even for curved geometries and hence significantly simplifies meshing. Based on our recent work on Adaptive Extended Stencil FEM, or AES-FEM (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002 nme.5246), the proposed method replaces the traditional interpolatory FEM basis functions with generalized Lagrange polynomial basis functions, constructed via local weighted least-squares approximations. In this work, we show that AES-FEM can achieve high-order accuracy using linear elements to discretize the geometry and using the standard FEM hat functions as test functions even for curved geometries. This is in contrast to most high-order generalizations of FEM, which require high-order curved elements for curved geometries. We describe the selection of stencils and the treatment of boundary conditions for high-order AES-FEM, and show that even-degree polynomial basis functions are preferred for even-order PDEs. We present numerical results in 2D and 3D for the Poisson equation and the convection-diffusion equation and demonstrate up to sixth order accuracy with AES-FEM. Overall, AES-FEM is much easier to use than FEM for curved boundaries, while being more accurate, more stable, and more efficient in terms of runtime versus error over relatively fine meshes.
Consider a partial differential equation on a bounded, simply-connected domain @math , subject to the Dirichlet and Neumann boundary conditions @math denotes a linear differential operator, @math and @math are disjoint sets of the boundary with @math , and @math denotes the outward normal to @math . A finite element method, or more generally a weighted-residual method @cite_38 , introduces a set of test functions (a.k.a. weight functions) @math , and require the residual @math to be orthogonal to @math , i.e., Eq. ) and ) are the and of the PDE, respectively. In finite element methods, @math is typically (weakly) differentiate and has a local support. In addition, @math typically forms a partition of unity, i.e., @math , so that the weighted-residual formulation is global conservative in that @math
@cite_38
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This is the second article of a series of papers on stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the @math -dimensional lattice @math with random coefficients @math of the simplest type: They are identically distributed and independent from edge to edge. On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ''homogenized'' matrix @math is characterized by @math for any direction @math , where the random field @math (the ''corrector'') is the unique solution of @math in @math such that @math , @math is stationary and @math , @math denoting the ensemble average (or expectation). In order to approximate the homogenized coefficients @math , the corrector problem is usually solved in a box @math of size @math with periodic boundary conditions, and the space averaged energy on @math defines an approximation @math of @math . Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation @math converges almost surely to @math as @math . In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size @math , but replace the elliptic operator by @math with (typically) @math , as standard in the homogenization literature. We then replace the ensemble average by a space average on @math , and estimate the overall error on the homogenized coefficients in terms of @math and @math . Let us consider the differential equation @math with an @math from @math to @math and suppose that there exists a transformation @math from @math to @math ( @math ) such that @math obeys a differential equation @math with some function @math ; then the first equation is said to be lumpable to the second by @math . Here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction. We provided a series of necessary and sufficient conditions for the existence of such functions @math and @math ; some of them are formulated in terms of @math and @math only. Beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point ... Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set @math of planar convex polygons such that @math with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of @math to a finite subset in a natural way. An @math is a differentiable convex simple closed curve @math in the plane having an inscribed circle of radius @math and a circumscribed circle of radius @math such that the ratio @math is at least @math . Motivated by Richter and Rogers' result, we construct a set @math such that (1) @math contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) @math with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to @math , @math is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive @math and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets @math of @math such that each @math consists of almost-circles of accuracy @math and the convex geometry in question is represented by restricting the convex hull operator to @math . The affine-disjointness of @math and @math means that, in addition to @math , even @math is disjoint from @math for every non-degenerate affine transformation @math . We consider in this paper the problem of sampling a high-dimensional probability distribution @math having a density with respect to the Lebesgue measure on @math , known up to a normalization constant @math . Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that @math is continuously differentiable, @math is globally Lipschitz and @math is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order @math and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims. A range counting problem is specified by a set @math of size @math of points in @math , an integer weight @math associated to each point @math , and a range space @math . Given a query range @math , the target output is @math . Range counting for different range spaces is a central problem in Computational Geometry. We study @math -differentially private algorithms for range counting. Our main results are for the range space given by hyperplanes, that is, the halfspace counting problem. We present an @math -differentially private algorithm for halfspace counting in @math dimensions which achieves @math average squared error. This contrasts with the @math lower bound established by the classical result of Dinur and Nissim [PODS 2003] for arbitrary subset counting queries. We also show a matching lower bound on average squared error for any @math -differentially private algorithm for halfspace counting. Both bounds are obtained using discrepancy theory. For the lower bound, we use a modified discrepancy measure and bound approximation of @math -differentially private algorithms for range counting queries in terms of this discrepancy. We also relate the modified discrepancy measure to classical combinatorial discrepancy, which allows us to exploit known discrepancy lower bounds. This approach also yields a lower bound of @math for @math -differentially private orthogonal range counting in @math dimensions, the first known superconstant lower bound for this problem. For the upper bound, we use an approach inspired by partial coloring methods for proving discrepancy upper bounds, and obtain @math -differentially private algorithms for range counting with polynomially bounded shatter function range spaces. We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification(UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let @math be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality @math . We will develop methods that approximate any function @math that is sparse in the BOPB, that is, @math of the form @math with @math of cardinality @math . Our method has a runtime of just @math , uses only @math function evaluations on a fixed and nonadaptive grid, and not more than @math bits of memory. For @math , the runtime @math will be less than what is required to simply enumerate the elements of the basis @math ; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as "sublinear-time". This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size @math . Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math Let @math denote a set of polynomials with complex coefficients. Let @math denote any set of sample points . For any @math , the discrete polynomial transform of @math (with respect to @math and @math ) is defined as the collection of sums, @math , where @math for some associated weight function @math . These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of @math orthogonal polynomials of degree at most @math , we give an @math algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the @math operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail. We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a so-called Fourier integral operator (FIO) of the form @math at points given on a Cartesian grid. Here, @math is a frequency variable, @math is the Fourier transform of the input @math , @math is an amplitude, and @math is a phase function, which is typically as large as @math ; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size @math by @math would require @math operations. This paper develops a new numerical algorithm which requires @math operations and as low as @math in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel @math into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology. Let @math be a partially ordered set. If the Boolean lattice @math can be partitioned into copies of @math for some positive integer @math , then @math must satisfy the following two trivial conditions: (1) the size of @math is a power of @math , (2) @math has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if @math only satisfies condition (2), we can still almost partition @math into copies of @math . We prove that if @math has a unique maximal and minimal element, then there exists a constant @math such that all but at most @math elements of @math can be covered by disjoint copies of @math . We consider the functions @math defined as the @math th partial derivative of Lebesgue's singular function @math with respect to @math at @math . This sequence includes a multiple of the Takagi function as the case @math . We show that @math is continuous but nowhere differentiable for each @math , and determine the Holder order of @math . From this, we derive that the Hausdorff dimension of the graph of @math is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for @math using the binary expansion of @math , and use this to find the sets of points where @math and @math take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of @math and @math . We study Byzantine fault-tolerant distributed optimization of a sum of convex (cost) functions with real-valued scalar input ouput. In particular, the goal is to optimize a global cost function @math , where @math is the set of non-faulty agents, and @math is agent @math 's local cost function, which is initially known only to agent @math . In general, when some of the agents may be Byzantine faulty, the above goal is unachievable, because the identity of the faulty agents is not necessarily known to the non-faulty agents, and the faulty agents may behave arbitrarily. Since the above global cost function cannot be optimized exactly in presence of Byzantine agents, we define a weaker version of the problem. The goal for the weaker problem is to generate an output that is an optimum of a function formed as a convex combination of local cost functions of the non-faulty agents. More precisely, for some choice of weights @math for @math such that @math and @math , the output must be an optimum of the cost function @math . Ideally, we would like @math for all @math -- however, this cannot be guaranteed due to the presence of faulty agents. In fact, we show that the maximum achievable number of nonzero weights ( @math 's) is @math , where @math is the upper bound on the number of Byzantine agents. In addition, we present algorithms that ensure that at least @math agents have weights that are bounded away from 0. We also propose a low-complexity suboptimal algorithm, which ensures that at least @math agents have weights that are bounded away from 0, where @math is the total number of agents, and @math ( @math ) is the actual number of Byzantine agents. The @math -fold direct sum encoding of a string @math is a function @math that takes as input sets @math of size @math and outputs @math . In this paper we prove a direct sum testing theorem. We describe a three query test that accepts with probability one any function of the form @math for some @math and rejects with probability @math functions @math that are @math -far from being a direct sum encoding, where the constant behind the @math notation is independent of @math . This theorem has a couple of additional guises: Linearity testing: By identifying the subsets of @math with vectors in @math in the natural way, our result can be thought of as a linearity testing theorem for functions whose domain is restricted to the @math th layer of the hypercube (i.e., the set of @math -bit strings with Hamming weight @math ). Tensor power testing: By moving to @math notation, the direct sum encoding is equivalent (up to a difference thatis negligi... We introduce a notion of quasi regularity for points with respect to the inclusion @math , where @math is a nonlinear Frechet differentiable function from @math to @math . When @math is the set of minimum points of a convex real-valued function @math on @math and @math satisfies the @math -average Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function @math . Results are new even when the initial point is regular and @math is Lipschitz. We study compositions of the integer @math in which the first part, successive differences, and the last part are constrained to lie in prescribed sets @math , respectively. A simple condition on @math guarantees that the generating function @math has only a simple pole on its circle of convergence and this at @math , a function independent of @math and @math . Thus the number of compositions is asymptotic to @math for a suitable constant @math . We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of @math , such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If @math has only finitely many positive elements @math , or finitely many negative elements @math , then the largest part and number of distinct part sizes are almost surely @math . On the other hand, when both @math and @math have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely @math , and we give sufficient conditions for the largest part to be almost surely asymptotic to @math . This article offers a simplified approach to the distribution theory of randomly weighted averages or @math -means @math , for a sequence of i.i.d.random variables @math , and independent random weights @math with @math and @math . The collection of distributions of @math , indexed by distributions of @math , is shown to encode Kingman's partition structure derived from @math . For instance, if @math has Bernoulli @math distribution on @math , the @math th moment of @math is a polynomial function of @math which equals the probability generating function of the number @math of distinct values in a sample of size @math from @math : @math . This elementary identity illustrates a general moment formula for @math -means in terms of the partition structure associated with random samples from @math , first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters @math , found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an @math mean. The analysis of these random means includes the characterization of @math -means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L 'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989). We seek a global minimum of @math . The solution to @math will find local minima. Using the idea of simulated annealing, we consider the diffusion process, @math , @math , where @math is the n-dimensional standard Brownian motion and @math is the annealing rate which decreases to zero as t goes to @math . Under suitable condition on @math , we prove that @math converges weakly to a probability measure @math if for large t, @math with @math , where @math has a simple expression involving the action function of the dynamical system @math , @math concentrates on the global minima of U and is the weak limit of the Gibbs densities @math .The above result can also be formulated as follows: consider the Fokker–Planck equation (forward equation) [ t V(t,y) = ... Several algorithms with an approximation guarantee of @math are known for the Set Cover problem, where @math is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into @math , and we are required to cover at least @math elements from each color class @math , using the minimum number of sets. We give a randomized LP-rounding algorithm that is an @math approximation for the Partition Set Cover problem. Here @math denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which @math is known to be sublogarithmic in @math . We also extend the LP rounding algorithm to obtain @math approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an @math -approximation for any of these problems. A graph @math is called @math -full if @math contains a subgraph @math of @math edges such that, for any non-empty @math , @math holds. Here, @math denotes the set of vertices incident to @math . It is known that the family of edge sets of @math -full graphs forms a family of matroid, known as the sparsity matroid of @math . In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for @math -fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property @math or far from @math ). Depending on the values of @math and @math , it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for @math -edge-connected-orientability in the bounded-degree model, where an undirected graph @math is called @math -edge-connected-orientable if there exists an orientation @math of @math with a vertex @math such that @math contains @math arc-disjoint dipaths from @math to each vertex @math and @math arc-disjoint dipaths from each vertex @math to @math . A tester is called a one-sided error tester for @math if it always accepts a graph satisfying @math . We show, for @math and (proper) @math , any one-sided error tester for @math -fullness and @math -edge-connected-orientability requires @math queries. This paper is concerned with the problem min @math where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. We propose the class of methods @math , where P denotes projection on X with respect to a Hilbert space norm @math , @math denotes the Frechet derivative of f at @math with respect to another Hilbert space norm @math on H, and @math is a positive scalar stepsize. We thus remove an important restriction in the original proposal of Goldstein [1] and Levitin and Poljak [2], where the norms @math and @math must be the same. It is therefore possible to match the norm @math with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose @math on the basis of approximations to the Hessian of f so as to attain a typically superlinear rate of convergence. The resulting methods are particularly attrac...