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Register a new userNon-asymptotic $ell_1$ spaces with unique $ell_1$ asymptotic model
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A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we replace $c_0$ with $ell_1$ then this result is no longer true. Moreover, a stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting $ell_1$ as a unique asymptotic model whereas any subsequence of the basis generates a non-Asymptotic $ell_1$ subspace.
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We prove the support recovery for a general class of linear and nonlinear evolutionary partial differential equation (PDE) identification from a single noisy trajectory using $ell_1$ regularized Pseudo-Least Squares model~($ell_1$-PsLS). In any associative $mathbb{R}$-algebra generated by finitely many differentiation operators that contain the unknown PDE operator, applying $ell_1$-PsLS to a given data set yields a family of candidate models with coefficients $mathbf{c}(lambda)$ parameterized by the regularization weight $lambdageq 0$. The trace of ${mathbf{c}(lambda)}_{lambdageq 0}$ suffers from high variance due to data noises and finite difference approximation errors. We provide a set of sufficient conditions which guarantee that, from a single trajectory data denoised by a Local-Polynomial filter, the support of $mathbf{c}(lambda)$ asymptotically converges to the true signed-support associated with the underlying PDE for sufficiently many data and a certain range of $lambda$. We also show various numerical experiments to validate our theory.
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