13,077 research outputs found
Minimizing the Euclidean Condition Number
This paper considers the problem of determining the row and/or column scaling of a matrix A that minimizes the condition number of the scaled matrix. This problem has been studied by many authors. For the cases of the ∞-norm and the 1-norm, the scaling problem was completely solved in the 1960s. It is the Euclidean norm case that has widespread application in robust control analyses. For example, it is used for integral controllability tests based on steady-state information, for the selection of sensors and actuators based on dynamic information, and for studying the sensitivity of stability to uncertainty in control systems.
Minimizing the scaled Euclidean condition number has been an open question—researchers proposed approaches to solving the problem numerically, but none of the proposed numerical approaches guaranteed convergence to the true minimum. This paper provides a convex optimization procedure to determine the scalings that minimize the Euclidean condition number. This optimization can be solved in polynomial-time with off-the-shelf software
A mathematical theory of semantic development in deep neural networks
An extensive body of empirical research has revealed remarkable regularities
in the acquisition, organization, deployment, and neural representation of
human semantic knowledge, thereby raising a fundamental conceptual question:
what are the theoretical principles governing the ability of neural networks to
acquire, organize, and deploy abstract knowledge by integrating across many
individual experiences? We address this question by mathematically analyzing
the nonlinear dynamics of learning in deep linear networks. We find exact
solutions to this learning dynamics that yield a conceptual explanation for the
prevalence of many disparate phenomena in semantic cognition, including the
hierarchical differentiation of concepts through rapid developmental
transitions, the ubiquity of semantic illusions between such transitions, the
emergence of item typicality and category coherence as factors controlling the
speed of semantic processing, changing patterns of inductive projection over
development, and the conservation of semantic similarity in neural
representations across species. Thus, surprisingly, our simple neural model
qualitatively recapitulates many diverse regularities underlying semantic
development, while providing analytic insight into how the statistical
structure of an environment can interact with nonlinear deep learning dynamics
to give rise to these regularities
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
Low-rank matrix approximations, such as the truncated singular value
decomposition and the rank-revealing QR decomposition, play a central role in
data analysis and scientific computing. This work surveys and extends recent
research which demonstrates that randomization offers a powerful tool for
performing low-rank matrix approximation. These techniques exploit modern
computational architectures more fully than classical methods and open the
possibility of dealing with truly massive data sets.
This paper presents a modular framework for constructing randomized
algorithms that compute partial matrix decompositions. These methods use random
sampling to identify a subspace that captures most of the action of a matrix.
The input matrix is then compressed---either explicitly or implicitly---to this
subspace, and the reduced matrix is manipulated deterministically to obtain the
desired low-rank factorization. In many cases, this approach beats its
classical competitors in terms of accuracy, speed, and robustness. These claims
are supported by extensive numerical experiments and a detailed error analysis
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or
implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k))
floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data
The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments
The rotation arises as the unique orthogonal
factor of the right polar decomposition of a given
invertible matrix . In the context of nonlinear elasticity
Grioli (1940) discovered a geometric variational characterization of as a unique energy-minimizing rotation. In preceding works, we have
analyzed a generalization of Grioli's variational approach with weights
(material parameters) and (Grioli: ). The
energy subject to minimization coincides with the Cosserat shear-stretch
contribution arising in any geometrically nonlinear, isotropic and quadratic
Cosserat continuum model formulated in the deformation gradient field and the microrotation field . The corresponding set of non-classical energy-minimizing
rotations represents a new relaxed-polar mechanism.
Our goal is to motivate this mechanism by presenting it in a relevant setting.
To this end, we explicitly construct a deformation mapping
which models an idealized nanoindentation and compare the corresponding optimal
rotation patterns with experimentally
obtained 3D-EBSD measurements of the disorientation angle of lattice rotations
due to a nanoindentation in solid copper. We observe that the non-classical
relaxed-polar mechanism can produce interesting counter-rotations. A possible
link between Cosserat theory and finite multiplicative plasticity theory on
small scales is also explored.Comment: 28 pages, 11 figure
Robust Loopshaping for Process Control
Strong trends in chemical engineering and plant operation have made the control of processes increasingly difficult and have driven the process industry's demand for improved control techniques. Improved control leads to savings in resources, smaller downtimes, improved safety, and reduced pollution. Though the need for improved process control is clear, advanced control methodologies have had only limited acceptance and application in industrial practice. The reason for this gap between control theory and practice is that existing control methodologies do not adequately address all of the following control system requirements and problems associated with control design:
* The controller must be insensitive to plant/model mismatch, and perform well under unmeasured or poorly modeled disturbances.
* The controlled system must perform well under state or actuator constraints.
* The controlled system must be safe, reliable, and easy to maintain.
* Controllers are commonly required to be decentralized.
* Actuators and sensors must be selected before the controller can be designed.
* Inputs and outputs must be paired before the design of a decentralized controller.
A framework is presented to address these control requirements/problems in a general, unified manner. The approach will be demonstrated on adhesive coating processes and distillation columns
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
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