6,512 research outputs found
Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms
Tensor rank and low-rank tensor decompositions have many applications in
learning and complexity theory. Most known algorithms use unfoldings of tensors
and can only handle rank up to for a -th order
tensor in . Previously no efficient algorithm can decompose
3rd order tensors when the rank is super-linear in the dimension. Using ideas
from sum-of-squares hierarchy, we give the first quasi-polynomial time
algorithm that can decompose a random 3rd order tensor decomposition when the
rank is as large as .
We also give a polynomial time algorithm for certifying the injective norm of
random low rank tensors. Our tensor decomposition algorithm exploits the
relationship between injective norm and the tensor components. The proof relies
on interesting tools for decoupling random variables to prove better matrix
concentration bounds, which can be useful in other settings
Thermo-micro-mechanical simulation of bulk metal forming processes
The newly proposed microstructural constitutive model for polycrystal
viscoplasticity in cold and warm regimes (Motaman and Prahl, 2019), is
implemented as a microstructural solver via user-defined material subroutine in
a finite element (FE) software. Addition of the microstructural solver to the
default thermal and mechanical solvers of a standard FE package enabled coupled
thermo-micro-mechanical or thermal-microstructural-mechanical (TMM) simulation
of cold and warm bulk metal forming processes. The microstructural solver,
which incrementally calculates the evolution of microstructural state variables
(MSVs) and their correlation to the thermal and mechanical variables, is
implemented based on the constitutive theory of isotropic
hypoelasto-viscoplastic (HEVP) finite (large) strain/deformation. The numerical
integration and algorithmic procedure of the FE implementation are explained in
detail. Then, the viability of this approach is shown for (TMM-) FE simulation
of an industrial multistep warm forging
Non-linear modeling of active biohybrid materials
Recent advances in engineered muscle tissue attached to a synthetic substrate motivates the development of appropriate constitutive and numerical models. Applications of active materials can be expanded by using robust, non-mammalian muscle cells, such as those of Manduca sexta. In this study, we propose a model to assist in the analysis of biohybrid constructs by generalizing a recently proposed constitutive law for Manduca muscle tissue. The continuum model accounts (i) for the stimulation of muscle fibers by introducing multiple stress-free reference configurations for the active and passive states and (ii) for the hysteretic response by specifying a pseudo-elastic energy function. A simple example representing uniaxial loading-unloading is used to validate and verify the characteristics of the model. Then, based on experimental data of muscular thin films, a more complex case shows the qualitative potential of Manduca muscle tissue in active biohybrid constructs
Polynomial-time Tensor Decompositions with Sum-of-Squares
We give new algorithms based on the sum-of-squares method for tensor
decomposition. Our results improve the best known running times from
quasi-polynomial to polynomial for several problems, including decomposing
random overcomplete 3-tensors and learning overcomplete dictionaries with
constant relative sparsity. We also give the first robust analysis for
decomposing overcomplete 4-tensors in the smoothed analysis model. A key
ingredient of our analysis is to establish small spectral gaps in moment
matrices derived from solutions to sum-of-squares relaxations. To enable this
analysis we augment sum-of-squares relaxations with spectral analogs of maximum
entropy constraints.Comment: to appear in FOCS 201
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