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Recently emerging large-scale biomedical data pose exciting opportunities for scientific discoveries. However, the ultrahigh dimensionality and non-negligible measurement errors in the data may create difficulties in estimation. There are…
This paper presents a probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques. Such uncertainties can arise in the selection of an…
Let $A$ be a compact set in $\Rp$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^{s,A}$ is the unique Borel probability measure with support in $A$ that minimizes $$…
The Loschmidt echo measures the sensitivity to perturbations of quantum evolutions. We study its short time decay in classically chaotic systems. Using perturbation theory and throwing out all correlation imposed by the initial state and…
Weak measurement is a novel technique for parameter estimation with higher precision. In this paper we develop a general theory for the parameter estimation based on weak measurement technique with arbitrary postselection. The previous weak…
The robustness of dynamical systems against external perturbations is crucial in engineering; however, it is often overlooked for the lack of methods for rapidly computing it. This paper proposes a novel algorithm for estimating the…
Delay embedding---a method for reconstructing dynamical systems by delay coordinates---is widely used to forecast nonlinear time series as a model-free approach. When multivariate time series are observed, several existing frameworks can be…
The predictability problem for systems with different characteristic time scales is investigated. It is shown that even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability…
This paper presents new results concerning the observer design for wide classes of nonlinear systems with both sampled and delayed measurements. By using a small gain approach we provide sufficient conditions, which involve both the delay…
While the existing stochastic control theory is well equipped to handle dynamical systems with stochastic uncertainties, a paradigm shift using distance measure based decision making is required for the effective further exploration of the…
A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff…
In this paper, we first provide a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients, which can be applied to stochastic systems with coefficients of…
Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are…
Assuming that a reflected Ornstein-Uhlenbeck state process is observed at discrete time instants, we propose generalized moment estimators to estimate all drift and diffusion parameters via the celebrated ergodic theorem. With the sampling…
Time-invariant finite-dimensional systems, under reasonable continuity assumptions, exhibit the property that if solutions exist for all future times, the set of vectors reachable from a bounded set of initial conditions over bounded time…
We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our…
We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders.…
In this paper it is showed that if a time-varying uncertain system is robustly completely detectable then there exists an estimator for this system, i.e. we can estimate asymptotically the state vector of the system. Moreover, if a…
Understanding, predicting and controlling laminar-turbulent boundary-layer transition is crucial for the next generation aircraft design. However, in real flight experiments, or wind tunnel tests, often only sparse sensor measurements can…
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The…