Offering throughput guarantees for cellular wireless networks, carrying real-time traffic, is of interest to both the network operators and the customers. In this article, we formulate an optimization problem which aims at maximizing the throughput that can be guaranteed to the mobile users. By building on results obtained by Borst and Whiting and by assuming that the distributions of the users' carrier-to-noise ratios are known, we find the solution to this problem for users with different channel quality distributions, for both the scenario where all the users have the same throughput guarantees, and the scenario where all the users have different throughput guarantees. Based on these solutions, we also propose two simple and low complexity adaptive scheduling algorithms that perform significantly better than other well-known scheduling algorithms. We further develop an expression for the approximate throughput guarantee violation probability for users in time-slotted networks with the given cumulants of the distribution of bit-rate in a time-slot, and a given distribution for the number of timeslots allocated within a time-window.

S U M M A R YWe put forward the idea of using a Block Low-Rank (BLR) multifrontal direct solver to efficiently solve the linear systems of equations arising from a finite-difference discretization of the frequency-domain Maxwell equations for 3-D electromagnetic (EM) problems. The solver uses a low-rank representation for the off-diagonal blocks of the intermediate dense matrices arising in the multifrontal method to reduce the computational load. A numerical threshold, the so-called BLR threshold, controlling the accuracy of low-rank representations was optimized by balancing errors in the computed EM fields against savings in floating point operations (flops). Simulations were carried out over large-scale 3-D resistivity models representing typical scenarios for marine controlled-source EM surveys, and in particular the SEG SEAM model which contains an irregular salt body. The flop count, size of factor matrices and elapsed run time for matrix factorization are reduced dramatically by using BLR representations and can go down to, respectively, 10, 30 and 40 per cent of their full-rank values for our largest system with N = 20.6 million unknowns. The reductions are almost independent of the number of MPI tasks and threads at least up to 90 × 10 = 900 cores. The BLR savings increase for larger systems, which reduces the factorization flop complexity from O(N 2 ) for the full-rank solver to O(N m ) with m = 1.4-1.6. The BLR savings are significantly larger for deep-water environments that exclude the highly resistive air layer from the computational domain. A study in a scenario where simulations are required at multiple source locations shows that the BLR solver can become competitive in comparison to iterative solvers as an engine for 3-D controlled-source electromagnetic Gauss-Newton inversion that requires forward modelling for a few thousand right-hand sides.

We have developed an efficient numerical scheme for fast multimodel 3D electromagnetic simulations by applying a Schur complement approach to a frequency-domain finite-difference method. The scheme is based on direct solvers and developed with constrained inversion algorithms in view. Such algorithms normally need many forward modeling jobs with different resistivities for the target zone and/or background formation. We geometrically divide the computational domain into two subdomains: an anomalous subdomain, the resistivities of which were permitted to change, and a background subdomain, having fixed resistivities. The system matrix is partially factorized by precomputing a Schur complement to eliminate unknowns associated with the background subdomain. The Schur complement system is then solved to compute fields inside the anomalous subdomain.Finally, the background subdomain fields are computed using inexpensive local substitutions. For each successive simulation, only the relatively small Schur complement system has to be solved, which results in significant computational savings. We applied this approach to two moderately sized 3D problems in marine controlled-source electromagnetic modeling: (1) a deepwater model in which the resistivities of the seawater and the air layer were kept fixed and (2) a model in which focused inversion was performed in a scenario in which the resistivities of the background formation, the air layer, and the seawater were known. We found a significant reduction of the modeling time in inversion that depended on the relative sizes of the constrained and unconstrained volumes: the smaller the unconstrained volume, the larger the savings. Specifically, for a focused inversion of the Troll oil field in the North Sea, the gain amounted up to 80% of the total modeling time.

Frequency-domain methods, which are typically applied to 3D magnetotelluric (MT) modeling, require solving a system of linear equations for every frequency of interest. This is memory and computationally intensive. We developed a finite-difference time-domain algorithm to perform 3D MT modeling in a marine environment in which Maxwell’s equations are solved in a so-called fictitious-wave domain. Boundary conditions are efficiently treated via convolutional perfectly matched layers, for which we evaluated optimized parameter values obtained by testing over a large number of models. In comparison to the typically applied frequency-domain methods, two advantages of the finite-difference time-domain method are (1) that it is an explicit, low-memory method that entirely avoids the solution of systems of linear equations and (2) that it allows the computation of the electromagnetic field unknowns at all frequencies of interest in a single simulation. We derive a design criterion for vertical node spacing in a nonuniform grid using dispersion analysis as a starting point. Modeling results obtained using our finite-difference time-domain algorithm are compared with results obtained using an integral equation method. The agreement was found to be very good. We also discuss a real data inversion example in which MT modeling was done with our algorithm.

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