Abstract. In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small-to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumann's alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware.

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The Metropolis algorithm for simulated annealing has been applied to earthquake location. The method combines linear and non-linear search techniques, and is able to locate events reliably and efficiently. A separation of the spatial and temporal components of the search improves performance. This arises from decoupling the strong correlation between depth and origin time and by taking advantage of the low computational cost of re-computing the misfit for multiple origin times. In addition, a method of generating new models is applied which progressively concentrates attention on more favourable regions while still allowing the algorithm to avoid local minima in the misfit function. In contrast to other non-linear algorithms there is no requirement to explicitly delineate bounds on the hypocentral coordinates.The simulated-annealing technique is an example of a global optimization routine. Consequently, it does not require the computation of derivatives and so can be used with arrival times of multiple phases, azimuth and slowness information and any type of velocity model, including laterally heterogeneous 3-D models, without modification to the basic algorithm. In addition, robust statistical functions describing the data misfit can be easily incorporated.The performance of the algorithm is illustrated on three events with different types of data, including one event with array information. Traveltimes are calculated relative to the l-D imp91 velocity model. In each case, starting from widely separated initial locations the algorithm converges to a region of a few cubic kilometres. This region represents the neighbourhood of the global minimum of the misfit function. By including the maximum amount of available information and using robust statistical functions, the final locations are more accurate than those obtained using linear methods. In addition, the algorithm is able to locate the immediate region of the global minimum with much less computational effort than standard non-linear algorithms.

S U M M A R Y This paper shows how the performance of a fully non-linear earthquake location scheme can be improved by taking advantage of problem-specific information in the location procedure. The genetic algorithm is best viewed as a method of parameter space sampling that can be used for optimization problems. It has been applied successfully in regional and teleseismic earthquake location when the network geometry is favourable. However, on a series of test events with unfavourable network geometries the performance of the genetic algorithm is found to be poor.We introduce a method to separate the spatial and temporal parameters in such a way that problems related to the strong trade-off between depth and origin time are avoided. Our modified algorithm has been applied to several test events. Performance over the unmodified algorithm is improved substantially and the computational cost is reduced. The algorithm is better suited to the determination of hypocentral location whether using arrival times, array information (slowness and azimuth) or a combination of both.A second type of modification is introduced which exploits the weak correlation between the epicentral parameters and depth. This algorithm also improves performance over the standard genetic algorithm search, except in circumstances where the depth and epicentre are not weakly correlated, which occurs when the azimuthal coverage is very poor, or when azimuth and slowness information are incorporated. On a shallow nuclear explosion with only teleseismic P arrivals available, the algorithm consistently converged to a depth very close to the true depth, indicating superior depth estimation for shallow earthquake locations over the unmodified algorithm.