395 research outputs found
Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator
We consider the problems of extreming the first eigenvalue and the
fundamental gap of a sub-elliptic operator with Dirichlet boundary condition,
when the potential is subjected to a -norm constraint. The existence
results for weak solutions, compact embedding theorem and spectral theory for
sub-elliptic equation are given. Moreover, we provide the specific
characteristics of the corresponding optimal potential function
Optimal Actuator Location of the Norm Optimal Controls for Degenerate Parabolic Equations
This paper focuses on investigating the optimal actuator location for
achieving minimum norm controls in the context of approximate controllability
for degenerate parabolic equations. We propose a formulation of the
optimization problem that encompasses both the actuator location and its
associated minimum norm control. Specifically, we transform the problem into a
two-person zero-sum game problem, resulting in the development of four
equivalent formulations. Finally, we establish the crucial result that the
solution to the relaxed optimization problem serves as an optimal actuator
location for the classical problem
Null controllability of two kinds of coupled parabolic systems with switching control
The focus of this paper is on the null controllability of two kinds of
coupled systems including both degenerate and non-degenerate equations with
switching control. We first establish the observability inequality for
measurable subsets in time for such coupled system, and then by the HUM method
to obtain the null controllability. Next, we investigate the null
controllability of such coupled system for segmented time intervals. Notably,
these results are obtained through spectral inequalities rather than using the
method of Carleman estimates. Such coupled systems with switching control, to
the best of our knowledge, are among the first to discuss
Observability inequalities for the backward stochastic evolution equations and their applications
The present article delves into the investigation of observability
inequalities pertaining to backward stochastic evolution equations. We employ a
combination of spectral inequalities, interpolation inequalities, and the
telegraph series method as our primary tools to directly establish
observability inequalities. Furthermore, we explore three specific equations as
application examples: a stochastic degenerate equation, a stochastic fourth
order parabolic equation and a stochastic heat equation. It is noteworthy that
these equations can be rendered null controllability with only one control in
the drift term to each system
Data-Free Quantization with Accurate Activation Clipping and Adaptive Batch Normalization
Data-free quantization is a task that compresses the neural network to low
bit-width without access to original training data. Most existing data-free
quantization methods cause severe performance degradation due to inaccurate
activation clipping range and quantization error, especially for low bit-width.
In this paper, we present a simple yet effective data-free quantization method
with accurate activation clipping and adaptive batch normalization. Accurate
activation clipping (AAC) improves the model accuracy by exploiting accurate
activation information from the full-precision model. Adaptive batch
normalization firstly proposes to address the quantization error from
distribution changes by updating the batch normalization layer adaptively.
Extensive experiments demonstrate that the proposed data-free quantization
method can yield surprisingly performance, achieving 64.33% top-1 accuracy of
ResNet18 on ImageNet dataset, with 3.7% absolute improvement outperforming the
existing state-of-the-art methods.Comment: submitted to ICML202
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