Atmospheric temperature is one of the most important climate variables. This observational study presents detailed descriptions of the temperature variability imprinted in the 9-year brightness temperature data acquired by the Advanced Microwave Sounding Unit- Instrument A (AMSU-A) aboard Aqua since September 2002 over tropical oceans. A non-linear, adaptive method called the Ensemble Joint Multiple Extraction has been employed to extract the principal modes of variability in the AMSU-A/Aqua data. The semi-annual, annual, quasibiennial oscillation (QBO) modes and QBO–annual beat in the troposphere and the stratosphere have been successfully recovered. The modulation by the El Nin˜o/Southern oscillation (ENSO) in the troposphere was found and correlates well with the Multivariate ENSO Index. The longterm variations during 2002–2011 reveal a cooling trend (-0.5 K/decade at 10 hPa) in the tropical stratosphere; the trend below the tropical tropopause is not statistically significant due to the length of our data. A new tropospheric near-annual mode (period *1.6 years) was also revealed in the troposphere, whose existence was confirmed using National Centers for Environmental Prediction Reanalysis air temperature data. The near-annual mode in the troposphere is found to prevail in the eastern Pacific region and is coherent with a near-annual mode in the observed sea surface temperature over the Warm Pool region that has previously been reported. It remains a challenge for climate models to simulate the trends and principal modes of natural variability reported in this work.

In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a ∈ V (θ), V (θ) consists of the functions smoother than cos(θ(t)) and θ   0. This problem can be formulated as a nonlinear l0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is in H2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient.

Adaptive data analysis provides an important tool in extracting hidden physical information from multiscale data that arise from various applications. In this paper, we review two data-driven time-frequency analysis methods that we introduced recently to study trend and instantaneous frequency of nonlinear and nonstationary data. These methods are inspired by the empirical mode decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos((t))}, where a is assumed to be less oscillatory than cos((t)) and ′ > 0. This problem can be formulated as a nonlinear l0 optimization problem. We have proposed two methods to solve this nonlinear optimization problem. The first one is based on nonlinear basis pursuit and the second one is based on nonlinear matching pursuit. Convergence analysis has been carried out for the nonlinear matching pursuit method. Some numerical experiments are given to demonstrate the effectiveness of the proposed methods.

In a recent paper [11], Hou and Shi introduced a new adaptive data analysis method to analyze nonlinear and non-stationary data.The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form{a(t)cos(θ(t))}, where a∈V(θ), V(θ) consists of the functions that are less oscillatory than cos(θ(t))and θ0. This problem was formulated as a nonlinear L0 optimization problem and an iterative nonlinear matching pursuit method was proposed to solve this nonlinear optimization problem. In this paper, we prove the convergence of this nonlinear matching pursuit method under some scale separation assumptions on the signal. We consider both well- resolved and poorly sample dsignals, as well as signals with noise. In the case without noise, we prove that our method gives exact recovery of the original signal.