Solving Multiple-Block Separable Convex Minimization Problems Using Two-Block Alternating Direction Method of Multipliers

In this paper, we consider solving multiple-block separable convex minimization problems using alternating direction method of multipliers (ADMM). Motivated by the fact that the existing convergence theory for ADMM is mostly limited to the two-block case, we analyze in this paper, both theoretically and numerically, a new strategy that first transforms a multi-block problem into an equivalent two-block problem (either in the primal domain or in the dual domain) and then solves it using the standard two-block ADMM. In particular, we derive convergence results for this two-block ADMM approach to solve multi-block separable convex minimization problems, including an improved O(1/\epsilon) iteration complexity result. Moreover, we compare the numerical efficiency of this approach with the standard multi-block ADMM on several separable convex minimization problems which include basis pursuit, robust principal component analysis and latent variable Gaussian graphical model selection. The numerical results show that the multiple-block ADMM, although lacks theoretical convergence guarantees, typically outperforms two-block ADMMs

New Methods in Gravitational and Seismic Reflection Exploration

For the purpose of regional gravity survey, a completely automatic terrain correction method has been developed. The advantages of previous methods developed by others have been taken over with some modifications, also a new idea for the inner zone correction has been presented. At first, the whole area under investigation is divided into a grid of equal squares of convenient size for the automatic computation. The terrain effect of the far distant zone (r>50 km) is neglected. The terrain effect of the distant zone (30<r<50 km) is evaluated by approximating the prism as a vertical line with all mass centred on it, so, the line mass formula is used for this compulation. The terrain correction of the intermediate zone (2<r≤30 km) is estimated by approximating prism as a segment of a hollow cylinder of different sizes. Specifically, the side of the prism is treated as 4 km in the zone (20<r≤30 km), 2 km in the zone (15<r≤20 km) and 1 km in the zone (2<r≤15 km). The terrain effect of near zone 2 (0.5<r<2 km) is calculated by approximating the terrain as a vertical prism with a horizontal lower face and an upper face constantly sloping towards the station. A simplified formula is used for this computation. The terrain effect of near zone 1 (r≤0.5 km), that is, the square with the gravity station inside, is obtained by triangulating that square with an additional four elevation values provided at the four corners of the square. Since these four heights are read directly from four points on the Ordnance Survey map, they are relatively accurate, so that the triangulated prisms will more approach the real terrain. The gravitational effect of individual prisms in near zone 1 is obtained by integrating gravity over the volume. As a result, the rather complicated formulae are derived. The software MATERRAIN has been developed on the VAX/UNIX operating system not only to make terrain corrections, but to make the free-air and Bouguer corrections. The output from the program is a Bouguer anomaly. The method is tested by the gravity data in the Southern Uplands of Scotland and the results are satisfactory. It is found that some of the original terrain corrections provided by the BGS are underestimated and need to be modified. The method is entirely automatic and easy to use. With respect to reflection seismology, a new experiment was conducted aimed at understanding the wave propagation in volcanic rocks, finding new means of obtaining conventional reflection seismic data, and extracting the weak signals in the presence of noise. To accommodate this, a new areal 'RAZOR' array was designed. Three-component geophones lie on one of two concentric circles of radii 75 and 130 m. The determination of the array dimension is based on several factors such as the wavelength of signal, the true dip of deep reflectors. Three-component seismic data were acquired over the basalt in the Midland Valley of Scotland using an MDS-10 Data System. The SEG-Y data were transformed into an ASCII-coded format and then rotated onto a new coordinate system. The study of characteristics of field data shows that 3-component seismograms are characterised by strong reverberations lasting as long as 500 ms. The reverberation patterns vary from station to station. The horizontal components exhibit larger amplitudes and lower frequency than the vertical component. Furthermore, the data from the inner stations are believed to be more affected by surface conditions than the data from the outer stations. The display of the vertical and radial components from the outer stations shows a line of reflection events at about 420 ms; there are no clear events on the transverse section. By applying a spatial directional filter to each component of seismic data, it is shown that there is more information in the horizontal component passing through the filter than the vertical component. This is attributed to the far larger amplitudes of the horizontal components, which may dominate the polarisation direction of particle motions. The energy variation diagram of each shot shows quantitatively that the radial component receives much more energy than the others. In order to extract weak signals in the presence of noise, a bandpass frequency filter with a low cut-off of 20 Hz and a slope of 30 dB/octave, and a high cut-off of 60 Hz and a slope of 70 dB/octave is applied. The filtered data reveal that the filter can reject part of the low frequency reverberations (<20 Hz) and high frequency noise. For most of high reverberations within the bandwidth, the filter does little to improve the data. Predictive deconvolution filtering shows that it is very good at compressing the wavelets and attenuating the amplitude of reverberations