Recursos de colección

Caltech Authors (171.365 recursos)

Repository of works by Caltech published authors.

Group = Applied & Computational Mathematics

Mostrando recursos 1 - 6 de 6

  1. Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data

    Tropp, Joel A.; Yurtsever, Alp; Udell, Madeleine; Cevher, Volkan
    Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a sequence of linear updates. Because of storage limitations, it may only be possible to retain a sketch of the psd matrix. This paper develops a new algorithm for fixed-rank psd approximation from a sketch. The approach combines the Nyström approximation with a novel mechanism for rank truncation. Theoretical analysis establishes that the proposed method can achieve any prescribed relative error in the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that...

  2. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast

    Owhadi, Houman; Zhang, Lei
    We construct finite-dimensional approximations of solution spaces of divergence form operators with L^∞-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution of space of these operators is compactly embedded in H^1 if source terms are in the unit ball of L^2 instead of the unit ball of H^−1. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for H^2. The H^1-error estimates show that O(h^−d)-dimensional spaces with basis elements localized to sub-domains of diameter O(h^∞ ln 1/h) (with α ∈ [1/2 , 1)) result in an O(h^(2−2α) accuracy...

  3. Optimal Uncertainty Quantification

    Owhadi, H.; Scovel, C.; Sullivan, T. J.; McKerns, M.; Ortiz, M.
    We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call Optimal Uncertainty Quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as extreme values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large,...

  4. Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

    Tao, Molei; Owhadi, Houman; Marsden, Jerrold E.
    We introduce a new class of integrators for stiff ODEs as well as SDEs. An example of subclass of systems that we treat are ODEs and SDEs that are sums of two terms one of which has large coefficients. These integrators are (i) Multiscale: they are based on ow averaging and so do not resolve the fast variables but rather employ step-sizes determined by slow variables (ii) Basis: the method is based on averaging the ow of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for...

  5. Flux Norm Approach to Homogenization Problems with non-separated Scales

    Berlyand, Leonid; Owhadi, Houman
    We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L^∞(Ω­), ­Ω ⊂ ℝ^d ) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general case of arbitrary bounded coefficients. For such problems we introduce explicit finite dimensional approximations of solutions with controlled error estimates, which we refer to as homogenization approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical concept of an ∈ family of...

  6. Discrete Geometric Structures in Homogenization and Inverse Homogenization with Application to EIT

    Desbrun, Mathieu; Donaldson, Roger D.; Owhadi, Houman
    We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions s(x) over the domain Ω. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of Ω when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. We explicitly give the transformations which map conductivity coefficients into divergence-free matrices and convex functions, as well as their respective inverses. Using...

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