Showing papers in "Foundations of Computational Mathematics in 2015"

Adaptive Restart for Accelerated Gradient Schemes

Abstract: In this paper we introduce a simple heuristic adaptive restart technique that can dramatically improve the convergence rate of accelerated gradient schemes. The analysis of the technique relies on the observation that these schemes exhibit two modes of behavior depending on how much momentum is applied at each iteration. In what we refer to as the `high momentum' regime the iterates generated by an accelerated gradient scheme exhibit a periodic behavior, where the period is proportional to the square root of the local condition number of the objective function. Separately, it is known that the optimal restart interval is proportional to this same quantity. This suggests a restart technique whereby we reset the momentum whenever we observe periodic behavior. We provide a heuristic analysis that suggests that in many cases adaptively restarting allows us to recover the optimal rate of convergence with no prior knowledge of function parameters.

Exact Support Recovery for Sparse Spikes Deconvolution

Abstract: This paper studies sparse spikes deconvolution over the space of measures We focus on the recovery properties of the support of the measure (ie, the location of the Dirac masses) using total variation of measures (TV) regularization This regularization is the natural extension of the $$\ell ^1$$l1 norm of vectors to the setting of measures We show that support identification is governed by a specific solution of the dual problem (a so-called dual certificate) having minimum $$L^2$$L2 norm Our main result shows that if this certificate is non-degenerate (see the definition below), when the signal-to-noise ratio is large enough TV regularization recovers the exact same number of Diracs We show that both the locations and the amplitudes of these Diracs converge toward those of the input measure when the noise drops to zero Moreover, the non-degeneracy of this certificate can be checked by computing a so-called vanishing derivative pre-certificate This proxy can be computed in closed form by solving a linear system Lastly, we draw connections between the support of the recovered measure on a continuous domain and on a discretized grid We show that when the signal-to-noise level is large enough, and provided the aforementioned dual certificate is non-degenerate, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero

A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis

Abstract: The purpose of this work is to study solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions These operators can be realized as the Dirichlet-to-Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces Motivated by the rapid decay of the solution to this problem, we propose a truncation that is suitable for numerical approximation We discretize this truncation using first degree tensor product finite elements We derive a priori error estimates in weighted Sobolev spaces The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes For anisotropic meshes instead, they are quasi-optimal in both order and regularity We present numerical experiments to illustrate the method's performance

Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis

Abstract: We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring.

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Abstract: In 2009, Chazal et al. introduced $$\epsilon $$∈-interleavings of persistence modules. $$\epsilon $$∈-interleavings induce a pseudometric $$d_\mathrm{I}$$dI on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $$\epsilon $$∈-interleavings and $$d_\mathrm{I}$$dI generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $$d_\mathrm{I}$$dI is equal to the bottleneck distance $$d_\mathrm{B}$$dB. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $$\epsilon $$∈-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $$\epsilon $$∈-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $$d_\mathrm{I}$$dI satisfies a universality property. This universality result is the central result of the paper. It says that $$d_\mathrm{I}$$dI satisfies a stability property generalizing one which $$d_\mathrm{B}$$dB is known to satisfy, and that in addition, if $$d$$d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $$d\le d_\mathrm{I}$$d≤dI. We also show that a variant of this universality result holds for $$d_\mathrm{B}$$dB, over arbitrary fields. Finally, we show that $$d_\mathrm{I}$$dI restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.

Metrics for Generalized Persistence Modules

Abstract: We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between `soft' and `hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.

Improved Bounds on Sample Size for Implicit Matrix Trace Estimators

Abstract: This article is concerned with Monte Carlo methods for the estimation of the trace of an implicitly given matrix $$A$$A whose information is only available through matrix-vector products. Such a method approximates the trace by an average of $$N$$N expressions of the form $$ \mathbf{w} ^t (A \mathbf{w} )$$wt(Aw), with random vectors $$ \mathbf{w} $$w drawn from an appropriate distribution. We prove, discuss and experiment with bounds on the number of realizations $$N$$N required to guarantee a probabilistic bound on the relative error of the trace estimation upon employing Rademacher (Hutchinson), Gaussian and uniform unit vector (with and without replacement) probability distributions. In total, one necessary bound and six sufficient bounds are proved, improving upon and extending similar estimates obtained in the seminal work of Avron and Toledo (JACM 58(2). Article 8, 2011) in several dimensions. We first improve their bound on $$N$$N for the Hutchinson method, dropping a term that relates to $$\mathrm{rank}(A)$$rank(A) and making the bound comparable with that for the Gaussian estimator. We further prove new sufficient bounds for the Hutchinson, Gaussian and unit vector estimators, as well as a necessary bound for the Gaussian estimator, which depend more specifically on properties of matrix $$A$$A. As such, they may suggest the type of matrix for which one distribution or another provides a particularly effective or relatively ineffective stochastic estimation method.

Robust Computation of Linear Models by Convex Relaxation

Abstract: Consider a data set of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called reaper, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors and uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments that confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when reaper can approximate this subspace.

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

Abstract: This paper is a sequel to our previous work (Kuo et al. in SIAM J Numer Anal, 2012) where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to finite element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented by a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as an infinite-dimensional integral in the parameter space. Here, the (single-level) error analysis of our previous work is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error (i.e., the root-mean-square error averaged over all shifts) is of order $${\fancyscript{O}}(h^2)$$O(h2), where $$h$$h is the finest FE mesh width, or $${\fancyscript{O}}(N^{-1+\delta })$$O(N-1+?) for arbitrary $$\delta >0$$?>0, where $$N$$N denotes the maximal number of QMC sampling points in the parameter space. For these scenarios, the total work for all PDE solves in the multi-level QMC FE method is essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension $$d=2$$d=2 with linear elements. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). As in our previous work, families of QMC rules with "POD weights" ("product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.

Finding Hidden Cliques of Size $$\sqrt{N/e}$$N/e in Nearly Linear Time

Abstract: Consider an Erdos---Renyi random graph in which each edge is present independently with probability $$1/2$$1/2, except for a subset $$\mathsf{C}_N$$CN of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The algorithm of Dekel et al. (ANALCO. SIAM, pp 67---75, 2011) provably identifies the clique $$\mathsf{C}_N$$CN in linear time, provided $$|\mathsf{C}_N|\ge 1.261\sqrt{N}$$|CN|?1.261N. Spectral methods can be shown to fail on cliques smaller than $$\sqrt{N}$$N. In this paper we describe a nearly linear-time algorithm that succeeds with high probability for $$|\mathsf{C}_N|\ge (1+{\varepsilon })\sqrt{N/e}$$|CN|?(1+?)N/e for any $${\varepsilon }>0$$?>0. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on $$N$$N vertices). For large-girth regular graphs of degree $$(\varDelta +1)$$(Δ+1) we prove that so-called local algorithms succeed if $$|\mathsf{C}_N|\ge (1+{\varepsilon })N/\sqrt{e\varDelta }$$|CN|?(1+?)N/eΔ and fail if $$|\mathsf{C}_N|\le (1-{\varepsilon })N/\sqrt{e\varDelta }$$|CN|≤(1-?)N/eΔ.

Transversality and Alternating Projections for Nonconvex Sets

Abstract: We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

Abstract: We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we conduct a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.

Extensions of Gauss Quadrature Via Linear Programming

Abstract: Gauss quadrature is a well-known method for estimating the integral of a continuous function with respect to a given measure as a weighted sum of the function evaluated at a set of node points. Gauss quadrature is traditionally developed using orthogonal polynomials. We show that Gauss quadrature can also be obtained as the solution to an infinite-dimensional linear program (LP): minimize the $$n$$nth moment among all nonnegative measures that match the $$0$$0 through $$n-1$$n-1 moments of the given measure. While this infinite-dimensional LP provides no computational advantage in the traditional setting of integration on the real line, it can be used to construct Gauss-like quadratures in more general settings, including arbitrary domains in multiple dimensions.

Local Convergence of an Algorithm for Subspace Identification from Partial Data

Abstract: Grassmannian rank-one update subspace estimation (GROUSE) is an iterative algorithm for identifying a linear subspace of $$\mathbb {R}^n$$Rn from data consisting of partial observations of random vectors from that subspace. This paper examines local convergence properties of GROUSE, under assumptions on the randomness of the observed vectors, the randomness of the subset of elements observed at each iteration, and incoherence of the subspace with the coordinate directions. Convergence at an expected linear rate is demonstrated under certain assumptions. The case in which the full random vector is revealed at each iteration allows for much simpler analysis and is also described. GROUSE is related to incremental SVD methods and to gradient projection algorithms in optimization.

A Milstein Scheme for SPDEs

Abstract: This article studies an infinite-dimensional analog of Milstein's scheme for finite-dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite-dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite-dimensional commutativity condition. In particular, a suitable infinite-dimensional analog of Milstein's algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation, stochastic reaction diffusion equations and a stochastic Burgers equation, showing significant computational savings.

Typical Real Ranks of Binary Forms

Abstract: We prove a conjecture of Comon and Ottaviani that typical real Waring ranks of bivariate forms of degree d take all integer values between $\lfloor \frac{d+2}{2}\rfloor$ and d. That is, we show that for all d and all $\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$ there exists a bivariate form f such that f can be written as a linear combination of mdth powers of real linear forms and no fewer, and additionally all forms in an open neighborhood of f also possess this property. Equivalently we show that for all d and any $\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$ there exists a symmetric real bivariate tensor t of order d such that t can be written as a linear combination of m symmetric real tensors of rank 1 and no fewer, and additionally all tensors in an open neighborhood of t also possess this property.

Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

Abstract: Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced by Dick (Ann Stat 39:1372---1398, 2011) and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the $$s$$s-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is $$ds$$ds, where the integer $$d$$d is the so-called interlacing factor. In this paper we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matousek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to $$\alpha \ge 1$$??1, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base $$b$$b and dimension $$ds$$ds, to which we then apply the interlacing scheme of order $$d$$d, we obtain a construction cost of the algorithm of order $$\mathcal {O}(dsmb^m)$$O(dsmbm) operations using $$\mathcal {O}(b^m)$$O(bm) memory in case of product weights, where $$b^m$$bm is the number of points in the polynomial lattice point set.

The Persistent Homology of a Self-Map

Abstract: Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility.

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Abstract: We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Cea lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an $$H^1$$H1-type Finsler norm and with the $$H^1$$H1-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem.

Stroboscopic Averaging for the Nonlinear Schrödinger Equation

Abstract: In this paper, we are concerned with an averaging procedure, namely Stroboscopic averaging, for highly oscillatory evolution equations posed in a (possibly infinite dimensional) Banach space, typically partial differential equations in a high-frequency regime where only one frequency is present We construct a high-order averaged system whose solution remains exponentially close to the exact one over long time intervals, possesses the same geometric properties (structure, invariants, ) as compared to the original system and is non-oscillatory We then apply our results to the nonlinear Schrodinger equation on the $$d$$d-dimensional torus $$\mathbb {T}^d$$Td, or in $$\mathbb {R}^d$$Rd with a harmonic oscillator, for which we obtain a hierarchy of Hamiltonian averaged models Our results are then illustrated numerically on several examples borrowed from the recent literature

Higher-Order Averaging, Formal Series and Numerical Integration III: Error Bounds

Abstract: In earlier papers, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct high-order averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scalar coefficients that are computed via simple recursions. Here we show how the coefficients/basis functions approach may be used advantageously to derive exponentially small error bounds for averaged systems and approximate first integrals.

Probabilistic Analysis of the Grassmann Condition Number

Abstract: We analyze the probability that a random m-dimensional linear subspace of $\mathbb{R}^{n}$ both intersects a regular closed convex cone $C\subseteq\mathbb{R}^{n}$ and lies within distance ? of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number [InlineEquation not available: see fulltext.] for the homogeneous convex feasibility problem ?x?C?0:Ax=0. The Grassmann condition number is a geometric version of Renegar's condition number, which we have introduced recently in Amelunxen and Burgisser (SIAM J. Optim. 22(3):1029---1041 (2012)). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of $A\in\mathbb {R}^{m\times n}$ are chosen i.i.d. standard normal, then for any regular cone C, we have [InlineEquation not available: see fulltext.]. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.

Nonlinear Approximation Rates and Besov Regularity for Elliptic PDEs on Polyhedral Domains

Abstract: We investigate the Besov regularity for solutions of elliptic PDEs. This is based on regularity results in Babuska---Kondratiev spaces. Following the argument of Dahlke and DeVore, we first prove an embedding of these spaces into the scale $$B^r_{\tau ,\tau }(D)$$B?,?r(D) of Besov spaces with $$\frac{1}{\tau }=\frac{r}{d}+\frac{1}{p}$$1?=rd+1p. This scale is known to be closely related to $$n$$n-term approximation w.r.t. wavelet systems, and also adaptive finite element approximation. Ultimately, this yields the rate $$n^{-r/d}$$n-r/d for $$u\in {\mathcal {K}}^m_{p,a}(D)\cap H^s_p(D)$$u?Kp,am(D)?Hps(D) for $$r\frac{1}{\tau }\ge \frac{1}{p}$$md+1p>1??1p, which in turn indeed yields the desired $$n$$n-term rate. As an intermediate step, we also prove an extension theorem for Kondratiev spaces.

Zigzag Zoology: Rips Zigzags for Homology Inference

Abstract: For $$n$$n points sampled near a compact set $$X$$X, the persistence barcode of the Rips filtration built from the sample contains information about the homology of $$X$$X as long as $$X$$X satisfies some geometric assumptions. The Rips filtration is prohibitively large; however, zigzag persistence can be used to keep the size linear in $$n$$n, with a constant factor depending only (exponentially) on the intrinsic dimension of $$X$$X. We present several species of Rips-like zigzags and compare them with respect to the signal-to-noise ratio, a measure of how well the underlying homology is represented in the persistence barcode relative to the noise in the barcode at the relevant scales. Some of these Rips-like zigzags have been available as part of the Dionysus library for several years while others are new. Interestingly, we show that some species of Rips zigzags will exhibit less noise than the (nonzigzag) Rips filtration itself. Thus, Rips zigzags can offer improvements in both size complexity and signal-to-noise ratio. Along the way, we develop new techniques for manipulating and comparing persistence barcodes from zigzag modules. In particular, we give methods for reversing arrows and removing spaces from a zigzag while controlling the changes occurring in its barcode. These techniques were developed to provide our theoretical analysis of the signal-to-noise ratio of Rips-like zigzags, but they are of independent interest as they apply to zigzag modules generally.

Localization of Matrix Factorizations

Abstract: Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal-decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QR factorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question of when and under what conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU, QR, Cholesky, and polar factorizations.

A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points

Abstract: A breakthrough paper written in 1993 by Shub and Smale unveiled the relationship between stable polynomials and points which minimize the discrete logarithmic energy on the Riemann sphere (a.k.a. elliptic Fekete points). This relationship has inspired advances in the study of both concepts, many of whose main properties are not well known yet. In this paper I prove an equivalent formulation for the problem of elliptic Fekete points and some consequences, including a (nonsharp) reciprocal of Shub and Smale's result and some novel nontrivial claims about these classical problems.

Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

Abstract: In this paper we review, based on massive, long-term, numerical simulations, the effect of islands on the statistical properties of the standard map for large parameter values. Different sources of discrepancy with respect to typical diffusion are identified, and their individual roles are compared and explained in terms of available limit models.

Degeneracy Loci and Polynomial Equation Solving

Abstract: Let $$V$$V be a smooth, equidimensional, quasi-affine variety of dimension $$r$$r over $$\mathbb {C}$$C, and let $$F$$F be a $$(p\times s)$$(p×s) matrix of coordinate functions of $$\mathbb {C}[V]$$C[V], where $$s\ge p+r$$s?p+r. The pair $$(V,F)$$(V,F) determines a vector bundle $$E$$E of rank $$s-p$$s-p over $$W:=\{x\in V \mid \mathrm{rk }F(x)=p\}$$W:={x?V?rkF(x)=p}. We associate with $$(V,F)$$(V,F) a descending chain of degeneracy loci of $$E$$E (the generic polar varieties of $$V$$V represent a typical example of this situation). The maximal degree of these degeneracy loci constitutes the essential ingredient for the uniform, bounded-error probabilistic pseudo-polynomial-time algorithm that we will design and that solves a series of computational elimination problems that can be formulated in this framework. We describe applications to polynomial equation solving over the reals and to the computation of a generic fiber of a dominant endomorphism of an affine space.

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Abstract: In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semiclassical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for the class $$C^{0,1}(\mathbb {R})$$C0,1(R) of globally Lipschitz coefficients and non-uniform grids obtained by means of $$C^{1,1}(\mathbb {R})$$C1,1(R)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space---time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory.

Sparse Differential Resultant for Laurent Differential Polynomials

Abstract: In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for a Laurent differential polynomial system to be Laurent differentially essential in terms of its support matrix. Then, the sparse differential resultant for a Laurent differentially essential system is defined, and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the Jacobi number and the size of the system.