473 research outputs found
A two-dimensional backward heat problem with statistical discrete data
Abstract
We focus on the nonhomogeneous backward heat problem of finding the initial temperature
θ
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,
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=
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0
)
{\theta=\theta(x,y)=u(x,y,0)}
such that
{
u
t
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a
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=
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,
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∈
Ω
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,
u
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=
0
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∈
∂
Ω
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,
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=
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,
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∈
Ω
¯
,
\left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f%
(x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\
\displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in%
\partial\Omega\times(0,T),\\
\displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle%
\in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm}
where
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(
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×
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{\Omega=(0,\pi)\times(0,\pi)}
. In the problem, the source
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{f=f(x,y,t)}
and the final data
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{h=h(x,y)}
are determined through random noise data
g
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j
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)
{g_{ij}(t)}
and
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{d_{ij}}
satisfying the regression models
<jats:disp-formula id="j_jiip-2016-0038_eq_999
EFFECT OF BLASTING ON THE STABILITY OF LINING DURING EXCAVATION OF NEW TUNNEL NEAR THE EXISTING TUNNEL
In recent years, experimental and numerical researches on the effect of blasting pressure on the stability of existing tunnels was widely obtained. However, the effect of the blasting pressure during excavation a new tunnel or expansion old tunnels on an existing tunnel has disadvantages and still unclear. Some researches were carried out to study the relationship of the observed Peak Particle Velocity (PPV) on the lining areas along the existing tunnel direction, due to either the lack of in situ test data or the difficulty in conducting field tests, particularly for tunnels that are usually old and vulnerable after several decades of service. This paper introduces using numerical methods with the field data investigations on the effect of the blasting in a new tunnel on the surrounding rock mass and on the existing tunnel. The research results show that not only predicting the tunnel lining damage zone under the impact of blast loads but also determination peak maximum of explosion at the same time at the surface of tunnel working
A Rigorous Framework for the Mean Field Limit of Multilayer Neural Networks
We develop a mathematically rigorous framework for multilayer neural networks
in the mean field regime. As the network's width increases, the network's
learning trajectory is shown to be well captured by a meaningful and
dynamically nonlinear limit (the \textit{mean field} limit), which is
characterized by a system of ODEs. Our framework applies to a broad range of
network architectures, learning dynamics and network initializations. Central
to the framework is the new idea of a \textit{neuronal embedding}, which
comprises of a non-evolving probability space that allows to embed neural
networks of arbitrary widths.
We demonstrate two applications of our framework. Firstly the framework gives
a principled way to study the simplifying effects that independent and
identically distributed initializations have on the mean field limit. Secondly
we prove a global convergence guarantee for two-layer and three-layer networks.
Unlike previous works that rely on convexity, our result requires a certain
universal approximation property, which is a distinctive feature of
infinite-width neural networks. To the best of our knowledge, this is the first
time global convergence is established for neural networks of more than two
layers in the mean field regime
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