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We prove the Marchenko-Pastur law for the eigenvalues of $p \times p$ samplecovariance matrices in two new situations where the data does not haveindependent coordinates. In the first scenario - the block-independent model -the $p$ coordinates of the data are partitioned into blocks in such a way thatthe entries in different blocks are independent, but the entries from the sameblock may be dependent. In the second scenario - the random tensor model - thedata is the homogeneous random tensor of order $d$, i.e. the coordinates of thedata are all $\binom{n}{d}$ different products of $d$ variables chosen from aset of $n$ independent random variables. We show that Marchenko-Pastur lawholds for the block-independent model as long as the size of the largest blockis $o(p)$ and for the random tensor model as long as $d = o(n^{1/3})$. Our maintechnical tools are new concentration inequalities for quadratic forms inrandom variables with block-independent coordinates, and for random tensors.

Traditionally, the field of computational Bayesian statistics has been divided into two main subfields: variational methods and Markov chain Monte Carlo (MCMC). In recent years, however, several methods have been proposed based on combining variational Bayesian inference and MCMC simulation in order to improve their overall accuracy and computational efficiency. This marriage of fast evaluation and flexible approximation provides a promising means of designing scalable Bayesian inference methods. In this paper, we explore the possibility of incorporating variational approximation into a state-of-the-art MCMC method, Hamiltonian Monte Carlo (HMC), to reduce the required expensive computation involved in the sampling procedure, which is the bottleneck for many applications of HMC in big data problems. To this end, we exploit the regularity in parameter space to construct a free-form approximation of the target distribution by a fast and flexible surrogate function using an optimized additive model of proper random basis, which can also be viewed as a single-hidden layer feedforward neural network. The surrogate function provides sufficiently accurate approximation while allowing for fast computation in the sampling procedure, resulting in an efficient approximate Bayesian inference algorithm. We demonstrate the advantages of our proposed method using both synthetic and real data problems.

For big data analysis, high computational cost for Bayesian methods often limits their applications in practice. In recent years, there have been many attempts to improve computational efficiency of Bayesian inference. Here we propose an efficient and scalable computational technique for a state-of-the-art Markov chain Monte Carlo methods, namely, Hamiltonian Monte Carlo. The key idea is to explore and exploit the structure and regularity in parameter space for the underlying probabilistic model to construct an effective approximation of its geometric properties. To this end, we build a surrogate function to approximate the target distribution using properly chosen random bases and an efficient optimization process. The resulting method provides a flexible, scalable, and efficient sampling algorithm, which converges to the correct target distribution. We show that by choosing the basis functions and optimization process differently, our method can be related to other approaches for the construction of surrogate functions such as generalized additive models or Gaussian process models. Experiments based on simulated and real data show that our approach leads to substantially more efficient sampling algorithms compared to existing state-of-the-art methods.

Markov Chain Monte Carlo (MCMC) algorithms play an important role in statistical inference problems dealing with intractable probability distributions. Recently, many MCMC algorithms such as Hamiltonian Monte Carlo (HMC) and Riemannian Manifold HMC have been proposed to provide distant proposals with high acceptance rate. These algorithms, however, tend to be computationally intensive which could limit their usefulness, especially for big data problems due to repetitive evaluations of functions and statistical quantities that depend on the data. This issue occurs in many statistic computing problems. In this paper, we propose a novel strategy that exploits smoothness (regularity) in parameter space to improve computational efficiency of MCMC algorithms. When evaluation of functions or statistical quantities are needed at a point in parameter space, interpolation from precomputed values or previous computed values is used. More specifically, we focus on HMC algorithms that use geometric information for faster exploration of probability distributions. Our proposed method is based on precomputing the required geometric information on a set of grids before running sampling algorithm and approximating the geometric information for the current location of the sampler using the precomputed information at nearby grids at each iteration of HMC. Sparse grid interpolation method is used for high dimensional problems. Tests on computational examples are shown to illustrate the advantages of our method.

In this paper, a new Navier-Stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open, traction, and free boundary conditions, which can be applied to simulations of fluid structure interaction, implicit solvent model for biomolecular applications and other free boundary or interface problems. For some problems of this type, the projection method and the augmented immersed interface method (IIM) do not work well or does not work at all. The proposed new Navier-Stokes solver is based on the local pressure boundary method, and a semi-implicit augmented IIM. A fast Poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future. The time discretization is based on a second order multi-step method. Numerical tests with exact solutions are presented to validate the accuracy of the method. Application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented.

A multi-scale framework is proposed for more realistic molecular dynamics simulations in continuum solvent models by coupling a molecular mechanics treatment of solute with a fluid mechanics treatment of solvent. This article reports our initial efforts to formulate the physical concepts necessary for coupling the two mechanics and develop a 3D numerical algorithm to simulate the solvent fluid via the Navier-Stokes equation. The numerical algorithm was validated with multiple test cases. The validation shows that the algorithm is effective and stable, with observed accuracy consistent with our design.

Parameterizing the bioluminescent source globally in Gaussians provides several advantages over voxel representation in bioluminescence tomography. It is mathematically unique to recover Gaussians [Med. Phys. 31(8), 2289 (2004)] and practically sufficient to approximate various shapes by Gaussians in diffusive medium. The computational burden is significantly reduced since much fewer unknowns are required. Besides, there are physiological evidences that the source can be modeled by Gaussians. The simulations show that the proposed model and algorithm significantly improves accuracy and stability in the presence of Gaussian or non- Gaussian sources, noisy data or the optical background mismatch. It is also validated through in vivo experimental data.

In this paper we study the regularization with both l1 and total-variation norm for bioluminescence tomography based on radiative transfer equation, compare l1 data fidelity with l2 data fidelity for different type of noise, and propose novel interior-point methods for solving related optimization problems. Simulations are performed to show that our approach is not only capable of preserving shapes, details and intensities of bioluminescent sources in the presence of sparse or non-sparse sources with angular-resolved or angular-averaged data, but also robust to noise, and thus is potential for efficient high-resolution imaging with only boundary data. (C) 2010 Optical Society of America

In this paper we study an l1-regularized multilevel approach for bioluminescence tomography based on radiative transfer equation with the emphasis on improving imaging resolution and reducing computational time. Simulations are performed to validate that our algorithms are potential for efficient high-resolution imaging. Besides, we study and compare reconstructions with boundary angular-averaged data, boundary angular-resolved data and internal angular-averaged data respectively. (C) 2010 Optical Society of America

In this paper, we study how to image both the location and the shape of extended targets using the response matrix obtained from inter-element response of an active array of transducers. In particular, the time reversal technique is used for efficient initial localization of the target and the level set method is used for shape reconstruction. We then show how to couple the location estimation and shape reconstruction in a complementary way to improve accuracy for range estimation. Resolution analysis for active arrays in remote sensing regime is also presented. We illustrate with numerical experiments which show that the method is capable of imaging objects with complicated shapes and that the method is robust with respect to noisy data. (C) 2004 Elsevier Inc. All rights reserved.