443 research outputs found
Formula of Entropy along Unstable Foliations for Diffeomorphisms with Dominated Splitting
Metric entropies along a hierarchy of unstable foliations are investigated
for diffeomorphisms with dominated splitting. The analogues of Ruelle's
inequality and Pesin's formula, which relate the metric entropy and Lyapunov
exponents in each hierarchy, are given
Low rank approximation method for perturbed linear systems with applications to elliptic type stochastic PDEs
In this paper, we propose a low rank approximation method for efficiently
solving stochastic partial differential equations. Specifically, our method
utilizes a novel low rank approximation of the stiffness matrices, which can
significantly reduce the computational load and storage requirements associated
with matrix inversion without losing accuracy. To demonstrate the versatility
and applicability of our method, we apply it to address two crucial uncertainty
quantification problems: stochastic elliptic equations and optimal control
problems governed by stochastic elliptic PDE constraints. Based on varying
dimension reduction ratios, our algorithm exhibits the capability to yield a
high precision numerical solution for stochastic partial differential
equations, or provides a rough representation of the exact solutions as a
pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal
control problems allows a diverse range of gradient-based unconstrained
optimization methods, rendering it particularly appealing for computationally
intensive large-scale problems. Numerical experiments are conducted and the
results provide strong validation of the feasibility and effectiveness of our
algorithm
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