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The article highlights two papers published within the issue titled "Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm," by Steven Brams, Michael Jones and Christian Klamler, and "Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples," by Rodrigo Platte, Lloyd Trefethen and Arno Kuijlaars. The first paper discussed the mathematical problem of proportional division. The second one talked about potential theory, matrix iterations in the form of Krylov space methods and quadrature formulae.

This paper revisits the problem of finding the best rank-1 approximation to a symmetric tensor and makes three contributions. First, in contrast to the many long and lingering arguments in the literature, it offers a straightforward justification that generically the best rank-1 approximation to a symmetric tensor is symmetric. Second, in contrast to the typical workhorse in the practice for the low-rank tensor approximation, namely, the alternating least squares (ALS) technique which improves one factor a time, this paper proposes three alternative algorithms based on the singular value decomposition (SVD) that modifies two factors a time. One step of SVD-based iteration is superior to two steps of ALS iterations. Third, it is not only proved that the generalized Rayleigh quotients generated from the three SVD-based algorithms enjoy monotone convergence, but also that the iterates themselves converge. [ABSTRACT FROM AUTHOR]

This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514{540]. In this sequel, we quantitatively improve several of the main results of the rst paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set A ⊂ ℝ: Ǝa ∈ A such that ∣A(A + a)∣ ≳ ∣A∣3/2+1/186, ∣A(A - A)∣ ≳ ∣A∣3/2+1/34, ∣A(A + A)∣ ≳ ∣A∣ 3/2+5/242, ∣{(a1 + a2 + a3 + a4)² + log a5 : ai ∈ A}∣ ⪢ ∣A∣²/log ∣A∣. [ABSTRACT FROM AUTHOR]

Let F be a compact set of a Banach space X. This paper analyzes the "generalized empirical interpolation method," which, given a function f ∊ F, builds an interpolant Jn[f] in an n dimensional subspace Xn ⊂ X with the knowledge of n outputs (σi(f))ni=1, where σi ∊ X' and X' is the dual space of X. The space Xn is built with a greedy algorithm that is adapted to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ ⊂ X'. In this paper, we study the interpolation error maxf∊F ‖f - Jn[f]‖X by comparing it with the best possible performance on an n dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space. [ABSTRACT FROM AUTHOR]

This paper discusses the multiscale analysis of the initial-boundary value problem for three-dimensional (3D) time-dependent Maxwell's equations in composite materials. The new contributions in this paper are the determination of higher-order correctors and an explicit convergence rate for the approximate solution. Consequently, a multiscale hybrid finite element finite-difference time-domain (FE-FDTD) method is presented. The numerical results demonstrate that this multiscale method has potential applications in engineering electromagnetics. [ABSTRACT FROM AUTHOR]

We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in Rd, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 = p < 8) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time nO(log s/ε²}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time nO(log d/ε²). [ABSTRACT FROM AUTHOR]

In this paper we consider discrete inverse problems for which noise becomes negligible compared to data with increasing model norm. We introduce two novel definitions of regularization for characterizing inversion methods which provide approximations of ill-conditioned inverse oper- ators consistent with such noisy data. In particular, these definitions, respectively, require that the reconstruction error computed from normalized data (p-asymptotic regularization) and the rel- ative reconstruction error (p-relative regularization) go to zero as the model norm tends to infinity, 0 ≥ p < 1 being a parameter controlling the increase rate of the noise level. We investigate the relationship between these two definitions and we prove that they are all equivalent for positively homogeneous iterative algorithms with suitable stopping rules. This result has as a crucial conse- quence that such iterative algorithms realize regularization independently of the noise model. Then we give suficient conditions for such methods to be p-asymptotic and p-relative regularizations in a discrete setting and we prove that the classical expectation maximization algorithm for Poisson data and the Landweber algorithm, if suitably stopped, are regularization methods in this sense. We perform numerical simulations in the case of image deconvolution and computerized tomography to show that, in the presence of model-dependent noise, the reconstructions provided by the above men- tioned methods improve with increasing model norm as required by the p-asymptotic and p-relative regularization properties. More extensive studies on the p-asymptotic and p-relative regularizations for Tikhonov-type methods will be the object of future work. [ABSTRACT FROM AUTHOR]

Many dynamic decision problems involving uncertainty can be appropriately modeled as multistage stochastic programs. However, most practical instances are so large and/or complex that it is impossible to solve them on a single computer, especially due to memory limitations. Extending the work of [B. Sandikci, N. Kong, and A. J. Schaefer, Math. Program., 138 (2013), pp. 253{272] on two-stage stochastic mixed-integer programs, this paper considers general multistage stochastic programs and develops a bounding method based on scenario decomposition. This method is broadly applicable, as it does not assume any problem structure including convexity. Moreover, it naturally fits into a distributed computing environment. Computational experiments with large-scale instances (with up to 100 million scenarios, about 1.5 billion decision variables—85% binary—and 800 million constraints) demonstrate that the proposed method scales nicely with problem size and has immense potential to obtain high-quality solutions to practical instances within a reasonable time frame. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem governed by a regularized fracture model using a phase-field technique. To avoid the nondifferentiability due to the irreversibility constraint on the fracture growth, the phase-field fracture model is relaxed using a penalization approach. The existence of a solution to the penalized fracture model is shown, and the existence of at least one solution for the regularized optimal control problem is established. Moreover, the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss QP-approximations to the nonlinear optimization problem. A numerical example suggests that these can be used to obtain a fast convergent algorithm. [ABSTRACT FROM AUTHOR]

We seek discrete approximations to solutions u : Ω → ℝ of semilinear elliptic PDE of the form Δu + fs (u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in ℝd. The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0,π)³ ⊆ ℝ³. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531-2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443-464]. [ABSTRACT FROM AUTHOR]

In this paper we derive a framework for multiscale approximation of elliptic problems on standard and mixed form. The method presented is based on a splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part, based on the solution of decoupled localized subgrid problems. The fine scale approximation is then used to modify the coarse scale equations. A key feature of the method is that symmetry in the bilinear form is preserved in the discrete system. Other key features are a posteriori error bounds and adaptive algorithms based on these bounds. The adaptive algorithms are used for automatic tuning of the method parameters. In the last part of the paper we present numerical examples where we apply the framework to a problem in oil reservoir simulation. [ABSTRACT FROM AUTHOR]

Let A ∈ ℂN x N be a nonsingular complex matrix and b and c be complex vectors of length N. The goal of this paper is to investigate approaches for efficient approximations of the bilinear form c*A-1b. Equivalently, we wish to approximate the scalar value c*x, where x solves the linear system Ax = b. Here the matrix A can be very large or its elements can be too costly to compute so that A is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For A Hermitian positive definite, b*A-1b can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss-Christoffel quadrature. In this paper we propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach will be compared with existing ones using analytic arguments and numerical experiments. [ABSTRACT FROM AUTHOR]

This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable to large problems of this kind with over a million unknown entries. This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative, produces a sequence of matrices {Xk,Yk}, and at each step mainly performs a soft-thresholding operation on the singular values of the matrix Yk. There are two remarkable features making this attractive for low-rank matrix completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix; the second is that the rank of the iterates {Xk} is empirically nondecreasing. Both these facts allow the algorithm to make use of very minimal storage space and keep the computational cost of each iteration low. On the theoretical side, we provide a convergence analysis showing that the sequence of iterates converges. On the practical side, we provide numerical examples in which 1, 000 x 1, 000 matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate that our approach is amenable to very large scale problems by recovering matrices of rank about 10 with nearly a billion unknowns from just about 0.4% of their sampled entries. Our methods are connected with the recent literature on linearized Bregman iterations for ℓ1 minimization, and we develop a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms. [ABSTRACT FROM AUTHOR]

The computational singular perturbation (CSP) method of Lam and Goussis is an iterative method to reduce the dimensionality of systems of ordinary differential equations with multiple time scales. In [J. Nonlinear Sci., 14 (2004), pp. 59-91], the authors of this paper showed that each iteration of the CSP algorithm improves the approximation of the slow manifold by one order. In this paper, it is shown that the CSP method simultaneously approximates the tangent spaces to the fast fibers along which solutions relax to the slow manifold. Again, each iteration adds one order of accuracy. In some studies, the output of the CSP algorithm is postprocessed by linearly projecting initial data onto the slow manifold along these approximate tangent spaces. These projections, in turn, also become successively more accurate. [ABSTRACT FROM AUTHOR]

Let M be a 2 × 2 matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or finite-impulse-response filters) u1, u2, v1, v2 with real coefficients and symmetry such that [This symbol cannot be presented in ASCII format] and [Su1](z)[Sv2](z) = [Su2](z)[Sv1](z), where [Sp](z) = p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u1, u2, v1, v2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric multiresolution analysis tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two highpass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters. [ABSTRACT FROM AUTHOR]

In (SIAM J. Matrix Anal. Appl., 23 (2002), pp. 706–727, we developed numerical algorithms for computing sparse low-rank approximations of matrices, and we also provided a detailed error analysis of the proposed algorithms together with some numerical experiments. The low-rank approximations are constructed in a certain factored form with the degree of sparsity of the factors controlled by some user-specified parameters. In this paper, we cast the sparse low-rank approximation problem in the framework of penalized optimization problems. We discuss various approximation schemes for the penalized optimization problem which are more amenable to numerical computations. We also include some analysis to show the relations between the original optimization problem and the reduced one. We then develop a globally convergent discrete Newton-like iterative method for solving the approximate penalized optimization problems. We also compare the reconstruction errors of the sparse low-rank approximations computed by our new methods with those obtained using the methods in the earlier paper and several other existing methods for computing sparse low-rank approximations. Numerical examples show that the penalized methods are more robust and produce approximations with factors which have fewer columns and are sparser. [ABSTRACT FROM AUTHOR]

Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using a Krylov subspace method, such as the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is difficult because the error variances are the diagonal elements of a matrix expression involving the inverse of a given matrix. This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a convergent approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a new estimation-theoretic interpretation of the CG algorithm. This paper discusses this interpretation and convergence issues and presents numerical examples. The examples include a 105-dimensional estimation problem from oceanography. [ABSTRACT FROM AUTHOR]

Max-plus methods have previously been used to solve deterministic control problems. The methods are based on max-plus (or min-plus) expansions and can yield curse-of-dimensionalityfree numerical methods. In this paper, we explore min-plus methods for continuous-time stochastic control on a finite-time horizon. We first approximate the original value function via timediscretization. By generalizing the min-plus distributive property to continuum spaces, we obtain an algorithm for recursive computation of the time-discretized values, which we refer to as the idempotent distributed dynamic programming principle (IDDPP). Under the IDDPP, the value function at each step can be represented as an infimum of functions in a certain class. This is a min-plus expansion for the value function. For the specific class of problems considered here, we see that the class can be taken as that consisting of the quadratic functions. A means for reducing the numbers of constituent quadratic functions is discussed. [ABSTRACT FROM AUTHOR]

Biochemical processes in living cells are comprised of reactions with vastly varying speeds and molecular counts of the reactant species. Classical deterministic or stochastic approaches to modeling often fail to exploit this multiscale nature of the reaction systems. In this paper, we propose a jump-diffusion approximation to these types of multiscale systems that couples the two traditional modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into two sets, where one set is modeled by continuous stochastic differential equations and the other by discrete state processes. The methodology leads to a very efficient dynamic partitioning algorithm which has been implemented for several multiscale reaction systems. The gain in computational efficiency is evident in all these examples, which include a realistically sized model of a signal transduction cascade coupled to a gene expression dynamics. [ABSTRACT FROM AUTHOR]

Given a matrix M (not necessarily nonnegative) and a factorization rank r, seminonnegative matrix factorization (semi-NMF) looks for a matrix U with r columns and a nonnegative matrix V with r rows such that UV is the best possible approximation of M according to some metric. In this paper, we study the properties of semi-NMF from which we develop exact and heuristic algorithms. Our contribution is threefold. First, we prove that the error of a semi-NMF of rank r has to be smaller than the best unconstrained approximation of rank r - 1. This leads us to a new initialization procedure based on the singular value decomposition (SVD) with a guarantee on the quality of the approximation. Second, we propose an exact algorithm (that is, an algorithm that finds an optimal solution), also based on the SVD, for a certain class of matrices (including nonnegative irreducible matrices) from which we derive an initialization for matrices not belonging to that class. Numerical experiments illustrate that this second approach performs extremely well, and allows us to compute optimal semi-NMF decompositions in many situations. Finally, we analyze the computational complexity of semi-NMF proving its NP-hardness, already in the rank-one case (that is, for r = 1), and we show that semi-NMF is sometimes ill-posed (that is, an optimal solution does not exist). [ABSTRACT FROM AUTHOR]

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N x N matrix, the resulting factorization is a product of O(log N) sparse matrices, each with O(N) nonzero entries. Hence, it can be applied rapidly in O(N log N) operations. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms. [ABSTRACT FROM AUTHOR]

This paper discusses a few algorithms for updating the approximate singular value decomposition (SVD) in the context of information retrieval by latent semantic indexing (LSI) methods. A unifying framework is considered which is based on Rayleigh-Ritz projection methods. First, a Rayleigh-Ritz approach for the SVD is discussed and it is then used to interpret the Zha and Simon algorithms [SIAM J. Sci. Comput., 21 (1999), pp. 782-791]. This viewpoint leads to a few alternatives whose goal is to reduce computational cost and storage requirement by projection techniques that utilize subspaces of much smaller dimension. Numerical experiments show that the proposed algorithms yield accuracies comparable to those obtained from standard ones at a much lower computational cost. [ABSTRACT FROM AUTHOR]

Given nf sites, each equipped with one facility, and nc cities, fault tolerant facility location (FTFL) [K. Jain and V. V. Vazirani, APPROX '00: Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization, Spinger, New York, 2000, pp. 177-183] requires computing a minimum-cost connection scheme such that each city connects to a specified number of facilities. When each city connects to exactly one facility, FTFL becomes the classical uncapacitated facility location problem (UFL) that is well-known NP hard. The current best solution to FTFL admits an approximation ratio 1.7245 due to Byrka, Srinivasan, and Swamy applying the dependent rounding technique announced recently [Proceedings of IPCO, 2010, pp. 244-257], which improves the ratio 2.076 obtained by Swamy and Shmoys based on LP rounding [ACM Trans. Algorithms, 4 (2008), pp. 1-27]. In this paper, we study a variant of the FTFL problem, namely, fault tolerant facility allocation (FTFA), as another generalization of UFL by allowing each site to hold multiple facilities and show that we can obtain better solutions for this problem. We first give two algorithms with 1.81 and 1.61 approximation ratios in time complexity O(m R log m) and O(Rn³), respectively, where R is the maximum number of facilities required by any city, m = nfnc, and n = max{nf, nc}. Instead of applying the dual-fitting technique that reduces the dual problem's solution to fit the original problem as used in the literature [K. Jain et al., Journal of the ACM, 50 (2003), pp. 795-824; K. Jain, M. Mahdian, and A. Saberi, STOC'02: Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, New York, 2002, pp. 731-740; A. Saberi et al., Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques, Springer, New York, 2001, pp. 127-137], we propose a method called inverse dual-fitting that alters the original problem to fit the dual solution and show that this method is more effective for obtaining solutions of multifactor approximation. We show that applying inverse dual-fitting and factor-revealing techniques our second algorithm is also (1.11,1.78)- and (1,2)-approximation simultaneously. These results can be further used to achieve solutions of 1.52-approximation to FTFA and 4-approximation to the fault tolerant k-facility allocation problem in which the total number of facilities is bounded by k. These are currently the best bifactor and single-factor approximation ratios for the problems concerned. [ABSTRACT FROM AUTHOR]

In Part I of this paper we presented minimization principles and related theoretical results for the linear response eigenvalue problem. Here we develop best approximations for the few smallest eigenvalues with the positive sign via a structure-preserving subspace projection. Then we present four-dimensional subspace search conjugate gradient-like algorithms for simultaneously computing these eigenvalues and their associated eigenvectors. Finally, we present numerical examples to illustrate convergence behaviors of the proposed methods with and without preconditioning. [ABSTRACT FROM AUTHOR]

While the minimax (or maximin) strategy has become the standard and most agreed-upon solution for decision making in adversarial settings, as discussed in game theory, computer science, and other disciplines, its power arises from the use of mixed strategies, also known as probabilistic algorithms. Nevertheless, in adversarial settings we face the risk of information leakage about the actual strategy instantiation. Hence, real robust algorithms should take information leakage into account. In this paper we introduce the notion of adversarial leakage in games, namely, the ability of a player to learn the value of b binary predicates about the strategy instantiation of her opponent. Different leakage models are suggested and tight bounds on the effect of adversarial leakage as a function of the level of leakage (captured by b) are established. The complexity of computing optimal strategies under these adversarial leakage models is also addressed. Together, our study introduces a new framework for robust decision making and provides rigorous fundamental understanding of its properties. [ABSTRACT FROM AUTHOR]

We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows at the end of the process whether H has the specified property or not). We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and s-t cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree (MST), shortest paths, and minimum cut. Many of these results are the first nontrivial lower bounds for both exact and approximate distributed computation, and they resolve previous open questions. Moreover, our unconditional lower bound of approximating MST subsumes and improves upon the previous hardness of approximation bound of Elkin [M. Elkin, SIAM J. Comput., 36 (2006), pp. 433-456] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [D. Peleg and V. Rubinovich, SIAM J. Comput., 30 (2000), pp. 1427-1442]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm for any approximation factor. Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems. [ABSTRACT FROM AUTHOR]

In this work we study the problem of bipartite correlation clustering (BCC), a natural bipartite counterpart of the well-studied correlation clustering (CC) problem [N. Bansal, A. Blum, and S. Chawla, Machine Learning, 56 (2004), pp. 89-113], also referred to as graph editing [R. Shamir, R. Sharan, and D. Tsur, Discrete Appl. Math., 144 (2004), pp. 173-182]. Given a bipartite graph, the objective of BCC is to generate a set of vertex disjoint bicliques (clusters) that minimizes the symmetric difference to the original graph. The best-known approximation algorithm for BCC due to Amit [N. Amit, The Bicluster Graph Editing Problem, Master's Thesis, Tel Aviv University, Tel Aviv, Israel, 2004] guarantees an 11-approximation ratio. In this paper we present two algorithms. The first is a linear program based 4-approximation algorithm. Like the previous approximation algorithm, it requires solving a large convex problem, which becomes prohibitive even for modestly sized tasks. The second algorithm, and our main contribution, is a simple randomized combinatorial algorithm. It also achieves an expected 4-approximation factor, and it is trivial to implement and highly scalable. The analysis extends a method developed by Ailon, Charikar, and Newman in 2008, where a randomized pivoting algorithm was analyzed for obtaining a 3-approximation algorithm for CC. For analyzing our algorithm for BCC, considerably more sophisticated arguments are required in order to take advantage of the bipartite structure. Whether it is possible to achieve (or beat) the 4-approximation factor using a scalable and deterministic algorithm remains an open problem. [ABSTRACT FROM AUTHOR]

The semi-infinite program (SIP) is normally represented with infinitely many inequality constraints and has been studied extensively so far. However, there have been very few studies on the SIP involving conic constraints, even though it has important applications such as Chebyshev-like approximation, filter design, and so on. In this paper, we focus on the SIP with infinitely many conic constraints, called an SICP for short. We show that under the Robinson constraint qualification a local optimum of the SICP satisfies the KKT conditions that can be represented only with a finite subset of the conic constraints. We also introduce two exchange type algorithms for solving the convex SICP. We first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function. We then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict convexity assumption. We report some numerical results to examine the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We consider approximation algorithms for nonnegative polynomial optimization over unit spheres. Such optimization models have wide applications, e.g., in signal and image processing, high order statistics, and computer vision. Since polynomial functions are nonconvex, the problems under consideration are all NP-hard. In this paper, based on convex polynomial optimization relaxations, we propose polynomial-time approximation algorithms with new approximation bounds. Numerical results are reported to show the effectiveness of the proposed approximation algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a stability analysis of switched homogeneous systems on cones under arbitrary and optimal switching rules with extensions to conewise homogeneous or linear inclusions. Several interrelated approaches, such as the joint spectral radius approach and the generating function approach, are exploited to derive necessary and sufficient stability conditions and to develop suitable algorithms for stability tests. Specifically, the generalized joint spectral radius and the generalized joint lower spectral radius are introduced to characterize the radii of domains of strong and weak attraction. Furthermore, strong and weak generating functions and their radii of convergence are employed to derive stability conditions; their analytic properties, numerical approximations, and convergence analysis are established. Extensions to conewise homogeneous or linear inclusions are made to address state-dependent switching dynamics. Relations between different stability notions in the strong or weak sense are studied; Lyapunov techniques are used for stability analysis of the conewise linear inclusions. [ABSTRACT FROM AUTHOR]

In this paper, we show that for a symmetric tensor, its best symmetric rank-1 approximation is its best rank-1 approximation. Based on this result, a positive lower bound for the best rank-1 approximation ratio of a symmetric tensor is given. Furthermore, a higher order polynomial spherical optimization problem can be reformulated as a multilinear spherical optimization problem. Then, we present a modified power algorithm for solving the homogeneous polynomial spherical optimization problem. Numerical results are presented, illustrating the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

A Euclidean approximate sparse recovery system consists of parameters k, N, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, the system approximates x by x = D(Φx), which must satisfy ¦x̂ -- x¦2 “ C¦x -- xk¦2, where xk denotes the optimal k-term approximation to x. (The output x̂ may have more than k terms.) For each vector x, the system must succeed with probability at least ¾. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, D. In this paper, we give a system with m = O(k log(N/k)) measurements--matching a lower bound, up to a constant factor--and decoding time k logO(1) N, matching a lower bound up to a polylog(N) factor. We also consider the encode time (i.e., the time to multiply Φ by x), the time to update measurements (i.e., the time to multiply Φ by a 1-sparse x), and the robustness and stability of the algorithm (resilience to noise before and after the measurements). Our encode and update times are optimal up to log(k) factors. The columns of Φ have at most O(log²(k) log(N/k)) nonzeros, each of which can be found in constant time. Our full result, a fully polynomial randomized approximation scheme, is as follows. If x = xk + v1, where v1 and v2 (below) are arbitrary vectors (regarded as noise), then setting x̂ = D(Φx + v2), and for properly normalized Φ, we get ¦x -- x̂¦²2 ≤ (1 + ε) ¦ v1¦²2 + ε ¦ v2¦²2 using O((k/ε) log(N/k)) measurements and (k/ε) logO(1)(N) time for decoding. [ABSTRACT FROM AUTHOR]

This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems. We develop an abstract framework for such constructions, akin to an abstract Bramble--Hilbert-type lemma, and then apply it in the design of coarse spaces for discretizations of PDEs with highly varying coefficients. The stability and approximation bounds of the constructed interpolant are in the weighted L2 norm and are independent of the variations in the coefficients. Such spaces can be used, for example, in two-level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid or for numerical upscaling purposes. Some numerical illustration is provided. [ABSTRACT FROM AUTHOR]

This paper investigates backscatter data for the inverse obstacle problem in impedance tomography, when the obstacles are small. It is shown that under this circumstance the backscatter data are close approximations of a rational function that has second or fourth order poles at the locations of the obstacles. Furthermore, a numerical method is presented to locate the obstacles via the poles of certain Laurent-Padé approximations. Numerical experiments explore the potential of this algorithm also for extended obstacles, taking into consideration that the problem is severely ill-posed. [ABSTRACT FROM AUTHOR]

In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the best rank- one approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm. For finite dimensional general tensor spaces, third order finite dimensional symmetric tensor spaces, and finite biquadratic tensor spaces, we give positive lower bounds for the best rank-one approximation ratio. For finite symmetric tensor spaces and finite dimensional biquadratic tensor spaces, we give upper bounds for this ratio. [ABSTRACT FROM AUTHOR]

Radial basis function (RBF) approximation is all extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-Padé method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided. [ABSTRACT FROM AUTHOR]

A new algorithm is developed for computing etAB, where A is an n x n matrix and B is n x n0 with n0 « n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n x n0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etA B or a sequence Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree, using the recent analysis of Al-Mohy and Higham [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970 989], which provides sharp truncation error bounds expressed in terms of the quantities ||Ak

We study a block triangular preconditioner for finite element approximations of the linearized Navier-Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113]. In this paper we prove field-of-values type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRES algorithm applied to solve the system. Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical variant based on block triangular approximations of the velocity submatrix. [ABSTRACT FROM AUTHOR]

In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined ell Grassmannians or a product of Grassmannians. Specifically we defined BFGS and limited memory BFGS updates in local and global coordinates oil Grassmannians or a product, of these. We proved that, when local coordinates are used, our BFGS updates on Grassmaunians share the same optimality property as the usual BFGS updates on Euclidean spaces. When applied to the best multilinear rank approximation problem for general and symmetric tensors, our approach yields fast, robust, and accurate algorithms that exploit the special Grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order. Extensive numerical experiments are included to substantiate our claims. [ABSTRACT FROM AUTHOR]

Szemerédi's regularity lemma is a cornerstone result in extremal combinatorics. It (roughly) asserts that any dense graph is composed of a finite number of pseudorandom graphs. The regularity lemma has found many applications in theoretical computer science, and thus a lot of attention was given to designing algorithmic versions of this lemma. Our main results in this paper are the following: (i) We introduce a new approach to the problem of constructing regular partitions of graphs, which results in a surprisingly simple O(n) time algorithmic version of the regularity lemma, thus improving over the previous O(n2) time algorithms. Furthermore, unlike all the previous approaches for this problem (see [N. Alon and A. Naor, SIAM J. Comput., 35 (2006), pp. 787-803], [R. A. Duke, H. Lefmann, and V. Rödl, SIAM J. Comput., 24 (1995), pp. 598-620], [A. Frieze and R. Kannan, Electron. J. Combin., 6 (1999), article 17], [A. Frieze and R. Kannan, "The regularity lemma and approximation schemes for dense problems," in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12-20], and [Y. Kohayakawa, V. Rödl, and L. Thoma, SIAM J. Comput., 32 (2003), pp. 1210-1235]), which only guaranteed to find tower-size partitions, our algorithm will find a small regular partition, if one exists in the graph. (ii) For any constant r ≥ 3 we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n2r-1) time (deterministic) algorithms [A. Czygrinow and V. Rödl, SIAM J. Comput., 30 (2000), pp. 1041-1066], [A. Frieze and R. Kannan, "The regularity lemma and approximation schemes for dense problems," in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12-20]. These two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here: Given a (directed) hypergraph with bounded edge arities, a set of constraints on the set sizes and densities of a possible partition of its vertex set, and an approximation parameter, we provide in O(n) time a partition approximating the constraints if a partition satisfying them exists. We can also test in O(1) time for the existence of such a partition given the approximation parameter. This algorithm extends the result of Goldreich, Goldwasser, and Ron for graph partition problems [O. Goldreich, S. Goldwasser, and D. Ron, J. ACM, 45 (1998), pp. 653-750] and encompasses more recent hypergraph-related results such as the maximal constraint satisfaction approximation of [G. Andersson and L. Engebretsen, Random Structures Algorithms, 21 (2002), pp. 14-32]. [ABSTRACT FROM AUTHOR]

This paper shows how to compute O(√log n)-approximations to the Sparsest Cut and Balanced Separator problems in Õ(n²) time, thus improving upon the recent algorithm of Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. Their algorithm uses semidefinite programming and requires Õ (n9.5) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new general approximation method for set covering problems, based on the combination of randomized rounding of the (near-) optimal solution of the linear programming (LP) relaxation, leading to a partial integer solution and the application of a well-behaved approximation algorithm to complete this solution. If the value of the solution returned by the latter can be bounded in a suitable way, as is the case for the most relevant generalizations of bin packing, the method leads to improved approximation guarantees, along with a proof of tighter integrality gaps for the LP relaxation. For d-dimensional vector packing, we obtain a polynomial-time randomized algorithm with asymptotic approximation guarantee arbitrarily close to ln d + 1. For d = 2, this value is 1.693 … ; i.e., we break the natural 2 "barrier" for this case. Moreover, for small values of d this is a notable improvement over the previously known O(ln d) guarantee by Chekuri and Khanna [SIAM J. Comput., 33 (2004), pp. 837-851]. For two-dimensional bin packing with and without rotations, we obtain polynomial-time randomized algorithms with asymptotic approximation guarantee 1.525 . . . , improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by Jansen and van Stee [On strip packing with rotations, in Proceedings of the 37th Annual ACM Symposium on the Theory of Computing, 2005, pp. 755-761] for the problem with rotations and 1.691 … by Caprara [Math. Oper. Res., 33 (2008), pp. 203-215] for the problem without rotations. The previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by Fernandez de la Vega and Lueker [Combinatorica, 1 (1981), pp. 349-355]. We prove that their approximation scheme is "subset oblivious," which leads to numerous applications. [ABSTRACT FROM AUTHOR]

This paper presents a three-dimensional GPU-accelerated algebraic reconstruction method in a few-projection cone-beam setting with arbitrary acquisition geometry. To achieve artifact-reduced reconstructions in the challenging case of unconstrained geometry and extremely limited input data, we use linear methods and an artifact-avoiding projection algorithm to provide high reconstruction quality. We apply the conjugate gradient method in the linear case of Tikhonov regularization and the two-point-step-size gradient method in the nonlinear case of total variation regularization to solve the system of equations. By taking advantage of modern graphics hardware we achieve acceleration of up to two orders of magnitude over classical CPU implementations. [ABSTRACT FROM AUTHOR]

In this paper we develop new methods for computing k-dimensional invariant manifolds of delayed systems for k ≥ 2. Our current implementation is built for k = 2 only, but the numerical and algorithmic challenges encountered in this case will be also present for any k > 1. For small delays, we consider methods for approximating delay differential equations (DDEs) with ordinary differential equations (ODEs). Once these approximations are made, any existing method for computing invariant manifolds of ODEs can then be used directly. We derive bounds on errors incurred by the most natural of these approximations. For large delays, we extend to DDEs the method originally introduced by Krauskopf and Osinga [Chaos, 9 (1999), pp. 768-774] for invariant manifolds of ODEs. We test the convergence of the resulting algorithms numerically and further illustrate our approach by computing two-dimensional unstable manifolds of equilibria in the context of phase-conjugate feedback lasers. [ABSTRACT FROM AUTHOR]

In this paper we present a sublinear-time (1+ϵ)-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an n-point metric space. The running time of the algorithm is Õ (n/ϵO(1)). Since the full description of an n-point metric space is of size ϴ(n2), the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. We also show that no deterministic algorithm can achieve a B-approximation in o(n2/B3) time. Furthermore, it has been previously shown that no o(n2) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum. [ABSTRACT FROM AUTHOR]

In this paper we present algorithms for approximating real band-limited signals by multiple Gaussian chirps. These algorithms do not rely on matching pursuit ideas. They are hierarchical, and, at each stage, the number of terms in a given approximation depends only on the number of positive-valued maxima and negative-valued minima of a signed amplitude function characterizing part of the signal. Like the algorithms used in [J. M. Greenberg, Z. Wang, and J. Li, IEEE Trans. Signal Process., 55 (2007), pp. 734-741] and unlike previous methods, our chirplet approximations require neither a complete dictionary of chirps nor complicated multidimensional searches to obtain suitable choices of chirp parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with the evaluation of the partial Fourier transform ux = Σ|k|

This paper shows how to compute O(√log n)-approximations to the Sparsest Cut and Balanced Separator problems in Õ(n²) time, thus improving upon the recent algorithm of Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. Their algorithm uses semidefinite programming and requires Õ (n9.5) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani. [ABSTRACT FROM AUTHOR]

In Part II of this paper we first refine the analysis of error-free vector transformations presented in Part I. Based on that we present an algorithm for calculating the rounded-to-nearest result of s := Σ pi for a given vector of floating-point numbers pi, as well as algorithms for directed rounding. A special algorithm for computing the sign of s is given, also working for huge dimensions. Assume a floating-point working precision with relative rounding error unit eps. We define and investigate a K-fold faithful rounding of a real number r. Basically the result is stored in a vector Resν of K nonoverlapping floating-point numbers such that Σ Resν approximates r with relative accuracy epsK, and replacing ResK by its floating-point neighbors in Σ Resν forms a lower and upper bound for r. For a given vector of floating-point numbers with exact sum s, we present an algorithm for calculating a K-fold faithful rounding of s using solely the working precision. Furthermore, an algorithm for calculating a faithfully rounded result of the sum of a vector of huge dimension is presented. Our algorithms are fast in terms of measured computing time because they allow good instruction-level parallelism, they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require some extra precision. The only operations used are standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision. Certain constants used in the algorithms are proved to be optimal. [ABSTRACT FROM AUTHOR]

The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as a linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear semidefinite low-rank approximation of the semidefinite dual. We introduce an algorithm based on this approach and analyze its convergence properties. The article is concluded by numerical experiments proving the effectiveness of the new approach. [ABSTRACT FROM AUTHOR]