Bayesian Calibration and Uncertainty Analysis for Computationally Expensive Models Using Optimization and Radial Basis Function Approximation

We presenta Bayesian approach to model calibration when evaluation of the model is computationally expensive. Here, calibration is a nonlinear regression problem: given a data vector Y corresponding to the regression model f(β), find plausible values of β. As an intermediate step, Y and f are embedded into a statistical model allowing transformation and dependence. Typically, this problem is solved by sampling from the posterior distribution of β given Y using MCMC. To reduce computational cost, we limit evaluation of f to a small number of points chosen on a high posterior density region found by optimization.Then,we approximate the logarithm of the posterior density using radial basis functions and use the resulting cheap-to-evaluate surface in MCMC.We illustrate our approach on simulated data for a pollutant diffusion problem and study the frequentist coverage properties of credible intervals. Our experiments indicate that our method can produce results similar to those when the true “expensive” posterior density is sampled by MCMC while reducing computational costs by well over an order of magnitude.     

Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f^**_x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.     

On an inverse problem for inhomogeneous thermoelastic rod

In recent years, different fields of engineering have been increasingly incorporating functionally graded materials with variable physical properties that significantly improve a quality of elements of designs. The efficiency of practical application of thermoelastic inhomogeneous materials depends on knowledge of exact laws of heterogeneity, and to define them it is necessary to solve coefficient inverse problems of thermoelasticity.In the present research a scheme of solving the inverse problem for an inhomogeneous thermoelastic rod is presented. Two statements of the inverse problem are considered: in the Laplace transform space and in the actual space. The direct problem solving is reduced to a system of the Fredholm integral equations of the 2nd kind in the Laplace transform space and an inversion of the solutions obtained on the basis of the theory of residues. The inverse problem solving is reduced to an iterative procedure, at its each step it is necessary to solve the Fredholm integral equation of the 1st kind; to solve it the Tikhonov method is used. Specific examples of a reconstruction of variable characteristics required are given.     

Identification of the plastic behavior of aluminum plates under free air explosions using inverse methods and full-field measurements

This article describes an inverse method for the identification of the plastic behavior of aluminum plates subjected to sudden blast loads. The method uses full-field optical measurements taken during the first milliseconds of a free air explosion and the finite element method for the numerical prediction of the blast response. The identification is based on a damped least-squares solution according to the Levenberg–Marquardt formulation. Three different rate-dependent plasticity models are examined. First, a combined model based on linear strain hardening and the strain rate term of the Cowper–Symonds model, secondly, the Johnson–Cook model and finally, a combined model based on a bi-exponential relation for the strain hardening term and the strain rate term of the Cowper–Symonds model. A validation of the method and its sensitivity to measurement uncertainties is first provided according to virtual measurements generated with the finite element method. Next, the plastic behavior of aluminum is identified using measurements from real free air explosions obtained from a controlled detonation of C4. The results show that inverse methods can be successfully applied for the identification of the plastic behavior of metals subjected to blast waves. In addition, the material parameters identified with inverse methods enable the numerical prediction of the material’s response with increased accuracy.     

基于空时级联滤波的拖船自噪声抵消方法

拖船自噪声属于近场多途强干扰信号,具有显著角度扩展现象,是影响拖曳阵探测性能的重要因素之一。在逆波束形成自噪声抵消方法基础上,结合空域和时域滤波,提出了空时级联滤波的拖船自噪声抵消方法。首先,由方位估计器估计拖船自噪声方位,经过波束形成空域滤波估计得到自噪声信号;然后,利用自噪声空域滤波输出设计最优维纳滤波器,将自噪声抵消在阵元级。基于简正波声场模型的仿真和实际海试数据的处理表明,该方法能够有效抵消拖船自噪声,性能优于逆波束形成方法。     

A single hop architecture exploiting cooperative beamforming for wireless sensor networks

In this paper, we propose a single hop architecture for a cooperative wireless sensor network and analyze the attained distributed beamforming gain performance using the theory of random arrays. All nodes in the system transmit a single carrier such that the signals add up constructively towards the direction of the fusion center. The potential directive beamforming gains are investigated for different sensor network densities which are expressed as the number of nodes per carrier wavelength squared. The multiple access capability of the sensor network is achieved by employing an on-off keying orthogonal signaling technique, which is usually employed in atmospheric optical systems. Finally, we investigate the average loss in directivity gain when the received signal from each sensor node follows a Ricean distribution. The results show that high directive gains can be achieved in practical wireless sensor networks using simple sensor nodes.     

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain. In many practice applications, we cannot know the initial temperature during heat process, therefore we consider a non-characteristic Cauchy problem for the heat equation. The method of fundamental solutions is applied to solve this problem. Due to ill-posedness of this problem, we first discretize the problem and then regularize it in the form of discrete equation. Numerical tests are conducted for showing the effectiveness of the proposed method.     

基于逆向空间偏移拉曼光谱技术的多层混浊样品深层分析研究

空间偏移拉曼光谱（SORS）能够准确、快速、无损检测多层混浊介质样品深层生化构成信息。该研究通过搭建集成化逆向SORS光谱分析装置,在实现逆向SORS和背散射式拉曼光谱两种不同的光谱检测模式的基础上,检测与分析了不同空间偏移量（Δs）条件下双/三层组织模型内的深层拉曼光谱信息,并根据几何光学理论和投影测量原理,量化标定了Δs与锥透镜空间位置之间的关系,这为精确控制光谱检测条件提供了保障。为了验证该装置的检测能力,采用由羊肩胛骨/对乙酰氨基酚组成的双层模型和猪皮/硅橡胶/对乙酰氨基酚组成的三层模型,获得不同Δs条件下包含样品表层和深层信息的混合光谱。并进一步对该混合光谱进行面积归一化处理,观察到随着Δs的增大样品表层的拉曼贡献逐渐减小,而第二层以及第三层的拉曼贡献逐渐增大的现象。在此基础上,通过选择模型中每层物质的拉曼特征峰计算其相对拉曼强度,分析研究了相对拉曼强度、空间偏移量与样品厚度三者之间关系,即当Δs增大时相对拉曼强度比值随之增加,这清晰地表明深层物质的拉曼强度增加。然而,在同一Δs条件下,相对拉曼强度随着表层物质厚度的增大而减小。以上实验结果表明,我们搭建的集成化逆向SORS光谱分析装置可从深度达8 mm的生物模型下获取光谱信息,并证明了该装置在经皮无损探测方面的应用价值。