Recursos de colección
Caltech Authors (171.365 recursos)
Repository of works by Caltech published authors.
Group = Applied & Computational Mathematics
Repository of works by Caltech published authors.
Group = Applied & Computational Mathematics
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...
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...
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,...
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...
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...
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...