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Albert Cohen

Publications and source records attributed to Albert Cohen.

At least 55 records · Page 3Linked to original sources
arXiv2016

Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $\mathbb{P}_Λ:={\rm span} \{y\mapsto y^ν\,: \, ν\in Λ\}$ with $Λ\subset \mathbb{N}_0^d$ over the domain $Γ:=[-1,1]^d$, based on the sampling of this function at points $y^1,\dots,y^m \in Γ$. The samples are independently drawn according to a given probability density $ρ$ belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. We restrict our attention to polynomial spaces associated with \emph{downward closed} sets $Λ$ of \emph{prescribed} cardinality $n$, and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in $L^2(Γ,dρ)$ with the best achievable polynomial approximation error when using downward closed sets of cardinality $n$. We establish conditions between the dimension $n$ and the size $m$ of the sample, under which these two errors are proven to be comparable. Our main finding is that the dimension $d$ enters only moderately in the resulting trade-off between $m$ and $n$, in terms of a logarithmic factor $\ln(d)$, and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named {\it anchored} sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon [2] which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs.

math.NA↗
arXiv2016

Optimal weighted least-squares methods

We consider the problem of reconstructing an unknown bounded function $u$ defined on a domain $X\subset \mathbb{R}^d$ from noiseless or noisy samples of $u$ at $n$ points $(x^i)_{i=1,\dots,n}$. We measure the reconstruction error in a norm $L^2(X,dρ)$ for some given probability measure $dρ$. Given a linear space $V_m$ with ${\rm dim}(V_m)=m\leq n$, we study in general terms the weighted least-squares approximations from the spaces $V_m$ based on independent random samples. The contribution of the present paper is twofold. From the theoretical perspective, we establish results in expectation and in probability for weighted least squares in general approximation spaces $V_m$. These results show that for an optimal choice of sampling measure $dμ$ and weight $w$, which depends on the space $V_m$ and on the measure $dρ$, stability and optimal accuracy are achieved under the mild condition that $n$ scales linearly with $m$ up to an additional logarithmic factor. The present analysis covers also cases where the function $u$ and its approximants from $V_m$ are unbounded, which might occur for instance in the relevant case where $X=\mathbb{R}^d$ and $dρ$ is the Gaussian measure. From the numerical perspective, we propose a sampling method which allows one to generate independent and identically distributed samples from the optimal measure $dμ$. This method becomes of interest in the multivariate setting where $dμ$ is generally not of tensor product type. We illustrate this for particular examples of approximation spaces $V_m$ of polynomial type, where the domain $X$ is allowed to be unbounded and high or even infinite dimensional, motivated by certain applications to parametric and stochastic PDEs.

math.NA↗
arXiv2016

Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to solutions of the PDE, considering both Taylor and Legendre series. Our results considerably improve on previously known estimates of this type, in particular taking into account structural features of the affine parametrization of the coefficient. Moreover, the results carry over to more general Jacobi polynomial expansions. We demonstrate that the new bounds are sharp in certain model cases and we illustrate them by numerical experiments.

math.NA↗
arXiv2016

Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence $y=(y_j)_{j\geq 1}$ of scalar random variables. One may then apply high-dimensional approximation methods to the solution map $y\mapsto u(y)$. Although Karhunen-Loève representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that they do not generally yield optimal approximation rates. Motivated by these results, we construct wavelet-type representations of stationary Gaussian random fields defined on bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using Karhunen-Loève representations. Our construction is based on a periodic extension of the random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matérn covariances.

math.NA↗
arXiv2015

Stochastic Optimal Control for Online Seller under Reputational Mechanisms

In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller's reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from (16) and used effectively in the Stochastic Optimal Control of Advertising (12). This model of reputation is combined with the effect of online reputation on sales price empirically verified in (9). We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller's reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct.

math.OC↗
arXiv2015

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

Elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are considered. We study the $\ell^p$ summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Loève representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion.

math.NA↗
arXiv2015

Data Assimilation in Reduced Modeling

We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from $m$ measurements obtained through known linear functionals on $\mathcal{H}$. Problems of this type are well studied \cite{MRW} under an assumption that $u$ belongs to a prescribed model class, e.g. a known compact subset of $\mathcal{H}$. Motivated by reduced modeling for parametric partial differential equations, this paper considers another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\mathcal{H}$ of dimension $n$, or more generally, how well $u$ can be approximated by each $k$-dimensional subspace $V_k$ of a sequence of nested subspaces $V_0\subset V_1\cdots\subset V_n$. A recovery algorithm for the one-space formulation, proposed in \cite{MPPY}, is proven here to be optimal and to have a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. The major contribution of the present paper is to analyze the multi-space case for which it is shown that the set of all $u$ satisfying the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multi-space problem is to identify any point in this intersection which can provide a much better accuracy than in the one-space problem. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem. A detailed analysis of one of them provides a posteriori performance estimates for the iterates, stopping criteria, and convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$.

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arXiv2015

Orthogonal Matching Pursuit under the Restricted Isometry Property

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary $\mathcal{D}$ in a Hilbert space $\mathcal{H}$. Given an element $f\in \mathcal{H}$, OMP generates a sequence of approximations $f_n$, $n=1,2,\dots$, each of which is a linear combination of $n$ dictionary elements chosen by a greedy criterion. It is studied whether the approximations $f_n$ are in some sense comparable to {\em best $n$ term approximation} from the dictionary. One important result related to this question is a theorem of Zhang \cite{TZ} in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers $n$-sparse signal, whenever the dictionary $\mathcal{D}$ satisfies a Restricted Isometry Property (RIP) of order $An$ for some constant $A$, and that the procedure is also stable in $\ell^2$ under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang's theorem, formulated in the general context of $n$ term approximation from a dictionary in arbitrary Hilbert spaces $\mathcal{H}$. Namely, it is shown that OMP generates near best $n$ term approximations under a similar RIP condition.

math.NA↗
arXiv2015

Approximation of high-dimensional parametric PDEs

Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analyzing effective numerical methodd which fully exploit these properties and, in turn, are immune to the growth in dimensionality. The first part of this article studies the smoothness and approximability of the solution map, that is, the map $a\mapsto u(a)$ where $a$ is the parameter value and $u(a)$ is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of $n$-term approximations to the solution map for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best $n$-term approximation, sparsity, and $n$-widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms. The second part of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.

math.AP↗
arXiv2015

Kolmogorov widths under holomorphic mappings

If $L$ is a bounded linear operator mapping the Banach space $X$ into the Banach space $Y$ and $K$ is a compact set in $X$, then the Kolmogorov widths of the image $L(K)$ do not exceed those of $K$ multiplied by the norm of $L$. We extend this result from linear maps to holomorphic mappings $u$ from $X$ to $Y$ in the following sense: when the $n$ widths of $K$ are $O(n^{-r})$ for some $r\textgreater{}1$, then those of $u(K)$ are $O(n^{-s})$ for any $s \textless{} r-1$, We then use these results to prove various theorems about Kolmogorov widths of manifolds consisting of solutions to certain parametrized PDEs. Results of this type are important in the numerical analysis of reduced bases and other reduced modeling methods, since the best possible performance of such methods is governed by the rate of decay of the Kolmogorov widths of the solution manifold.

math.AP↗
arXiv2015

Kolmogorov widths and low-rank approximations of parametric elliptic PDEs

Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the $n$-widths.

math.NA↗
arXiv2014

Classification algorithms using adaptive partitioning

Algorithms for binary classification based on adaptive tree partitioning are formulated and analyzed for both their risk performance and their friendliness to numerical implementation. The algorithms can be viewed as generating a set approximation to the Bayes set and thus fall into the general category of set estimators. In contrast with the most studied tree-based algorithms, which utilize piecewise constant approximation on the generated partition [IEEE Trans. Inform. Theory 52 (2006) 1335-1353; Mach. Learn. 66 (2007) 209-242], we consider decorated trees, which allow us to derive higher order methods. Convergence rates for these methods are derived in terms the parameter $α$ of margin conditions and a rate $s$ of best approximation of the Bayes set by decorated adaptive partitions. They can also be expressed in terms of the Besov smoothness $β$ of the regression function that governs its approximability by piecewise polynomials on adaptive partition. The execution of the algorithms does not require knowledge of the smoothness or margin conditions. Besov smoothness conditions are weaker than the commonly used Hölder conditions, which govern approximation by nonadaptive partitions, and therefore for a given regression function can result in a higher rate of convergence. This in turn mitigates the compatibility conflict between smoothness and margin parameters.

math.ST↗
arXiv2014

Automatic Detection of Performance Anomalies in Task-Parallel Programs

To efficiently exploit the resources of new many-core architectures, integrating dozens or even hundreds of cores per chip, parallel programming models have evolved to expose massive amounts of parallelism, often in the form of fine-grained tasks. Task-parallel languages, such as OpenStream, X10, Habanero Java and C or StarSs, simplify the development of applications for new architectures, but tuning task-parallel applications remains a major challenge. Performance bottlenecks can occur at any level of the implementation, from the algorithmic level (e.g., lack of parallelism or over-synchronization), to interactions with the operating and runtime systems (e.g., data placement on NUMA architectures), to inefficient use of the hardware (e.g., frequent cache misses or misaligned memory accesses); detecting such issues and determining the exact cause is a difficult task. In previous work, we developed Aftermath, an interactive tool for trace-based performance analysis and debugging of task-parallel programs and run-time systems. In contrast to other trace-based analysis tools, such as Paraver or Vampir, Aftermath offers native support for tasks, i.e., visualization, statistics and analysis tools adapted for performance debugging at task granularity. However, the tool currently does not provide support for the automatic detection of performance bottlenecks and it is up to the user to investigate the relevant aspects of program execution by focusing the inspection on specific slices of a trace file. In this paper, we present ongoing work on two extensions that guide the user through this process.

cs.DC↗
arXiv2014

Sampling and reconstruction of solutions to the Helmholtz equation

We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain $Ω$ from their values at scattered points $x_1,\dots,x_n\subset Ω$. This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose of reconstructing this field over a region of interest $Ω$ contained in a larger domain $D$ in which the acoustic field propagates. In many applied settings, the shape of $D$ and the boundary conditions on its border are unknown. Our reconstruction method is based on the approximation of a general solution $u$ by linear combinations of Fourier-Bessel functions or plane waves. We analyze the convergence of the least-squares estimates to $u$ using these families of functions based on the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes the amount of regularization needed to guarantee the convergence of the least squares estimate towards $u$, in terms of a condition that depends on the dimension of the approximation subspace, the sample size $n$ and the distribution of the samples. It reveals the advantage of using non-uniform distributions that have more points on the boundary of $Ω$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.

math.NA↗
arXiv2011

The Potential of Synergistic Static, Dynamic and Speculative Loop Nest Optimizations for Automatic Parallelization

Research in automatic parallelization of loop-centric programs started with static analysis, then broadened its arsenal to include dynamic inspection-execution and speculative execution, the best results involving hybrid static-dynamic schemes. Beyond the detection of parallelism in a sequential program, scalable parallelization on many-core processors involves hard and interesting parallelism adaptation and mapping challenges. These challenges include tailoring data locality to the memory hierarchy, structuring independent tasks hierarchically to exploit multiple levels of parallelism, tuning the synchronization grain, balancing the execution load, decoupling the execution into thread-level pipelines, and leveraging heterogeneous hardware with specialized accelerators. The polyhedral framework allows to model, construct and apply very complex loop nest transformations addressing most of the parallelism adaptation and mapping challenges. But apart from hardware-specific, back-end oriented transformations (if-conversion, trace scheduling, value prediction), loop nest optimization has essentially ignored dynamic and speculative techniques. Research in polyhedral compilation recently reached a significant milestone towards the support of dynamic, data-dependent control flow. This opens a large avenue for blending dynamic analyses and speculative techniques with advanced loop nest optimizations. Selecting real-world examples from SPEC benchmarks and numerical kernels, we make a case for the design of synergistic static, dynamic and speculative loop transformation techniques. We also sketch the embedding of dynamic information, including speculative assumptions, in the heart of affine transformation search spaces.

cs.DC↗
arXiv2011

A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(ϕ_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by Gérard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces.

math.FA↗
arXiv2011

Adaptive and anisotropic piecewise polynomial approximation

We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) the properties that describe an optimal partition for f, (ii) the smoothness properties of f that govern the rate of convergence of the approximation in the Lp-norms, and (iii) fast refinement algorithms that generate near optimal partitions. While these results constitute a fairly established theory in the univariate case and in the multivariate case when dealing with elements of isotropic shape, the approximation theory for adaptive and anisotropic elements is still building up. We put a particular emphasis on some recent results obtained in this direction.

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