3,768 research outputs found
Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification
In this paper, we are concerned with the problem of determining the existence
of multiple equilibria in economic models. We propose a general and complete
approach for identifying multiplicities of equilibria in semi-algebraic
economies, which may be expressed as semi-algebraic systems. The approach is
based on triangular decomposition and real solution classification, two
powerful tools of algebraic computation. Its effectiveness is illustrated by
two examples of application.Comment: 24 pages, 5 figure
Coupling the reduced-order model and the generative model for an importance sampling estimator
In this work, we develop an importance sampling estimator by coupling the
reduced-order model and the generative model in a problem setting of
uncertainty quantification. The target is to estimate the probability that the
quantity of interest (QoI) in a complex system is beyond a given threshold. To
avoid the prohibitive cost of sampling a large scale system, the reduced-order
model is usually considered for a trade-off between efficiency and accuracy.
However, the Monte Carlo estimator given by the reduced-order model is biased
due to the error from dimension reduction. To correct the bias, we still need
to sample the fine model. An effective technique to reduce the variance
reduction is importance sampling, where we employ the generative model to
estimate the distribution of the data from the reduced-order model and use it
for the change of measure in the importance sampling estimator. To compensate
the approximation errors of the reduced-order model, more data that induce a
slightly smaller QoI than the threshold need to be included into the training
set. Although the amount of these data can be controlled by a posterior error
estimate, redundant data, which may outnumber the effective data, will be kept
due to the epistemic uncertainty. To deal with this issue, we introduce a
weighted empirical distribution to process the data from the reduced-order
model. The generative model is then trained by minimizing the cross entropy
between it and the weighted empirical distribution. We also introduce a penalty
term into the objective function to deal with the overfitting for more
robustness. Numerical results are presented to demonstrate the effectiveness of
the proposed methodology
Quantitatively Analyzing Phonon Spectral Contribution of Thermal Conductivity Based on Non-Equilibrium Molecular Dynamics Simulation I: From Space Fourier Transform
Probing detailed spectral dependence of phonon transport properties in bulk
materials is critical to improve the function and performance of structures and
devices in a diverse spectrum of technologies. Currently, such information can
only be provided by the phonon spectral energy density (SED) or equivalently
time domain normal mode analysis (TDNMA) methods in the framework of
equilibrium molecular dynamics simulation (EMD), but has not been realized in
non-equilibrium molecular dynamics simulations (NEMD) so far. In this paper we
generate a new scheme directly based on NEMD and lattice dynamics theory,
called time domain direct decomposition method (TDDDM), to predict the phonon
mode specific thermal conductivity. Two benchmark cases of Lennard-Jones (LJ)
Argon and Stillinger-Weber (SW) Si are studied by TDDDM to characterize
contributions of individual phonon modes to overall thermal conductivity and
the results are compared with that predicted using SED and TDNMA. Excellent
agreements are found for both cases, which confirm the validity of our TDDDM
approach. The biggest advantage of TDDDM is that it can be used to investigate
the size effect of individual phonon modes in NEMD simulations, which cannot be
tackled by SED and TDNMA in EMD simulations currently. We found that the phonon
modes with mean free path larger than the system size are truncated in NEMD and
contribute little to the overall thermal conductivity. The TDDDM provides
direct physical origin for the well-known strong size effects in thermal
conductivity prediction by NEMD
Incremental Learning Using a Grow-and-Prune Paradigm with Efficient Neural Networks
Deep neural networks (DNNs) have become a widely deployed model for numerous
machine learning applications. However, their fixed architecture, substantial
training cost, and significant model redundancy make it difficult to
efficiently update them to accommodate previously unseen data. To solve these
problems, we propose an incremental learning framework based on a
grow-and-prune neural network synthesis paradigm. When new data arrive, the
neural network first grows new connections based on the gradients to increase
the network capacity to accommodate new data. Then, the framework iteratively
prunes away connections based on the magnitude of weights to enhance network
compactness, and hence recover efficiency. Finally, the model rests at a
lightweight DNN that is both ready for inference and suitable for future
grow-and-prune updates. The proposed framework improves accuracy, shrinks
network size, and significantly reduces the additional training cost for
incoming data compared to conventional approaches, such as training from
scratch and network fine-tuning. For the LeNet-300-100 and LeNet-5 neural
network architectures derived for the MNIST dataset, the framework reduces
training cost by up to 64% (63%) and 67% (63%) compared to training from
scratch (network fine-tuning), respectively. For the ResNet-18 architecture
derived for the ImageNet dataset and DeepSpeech2 for the AN4 dataset, the
corresponding training cost reductions against training from scratch (network
fine-tunning) are 64% (60%) and 67% (62%), respectively. Our derived models
contain fewer network parameters but achieve higher accuracy relative to
conventional baselines
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