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Parallel computing was applied to the solution of an inverse problem arising from a hyperbolic system of two coupled linear first-order partial differential equations. A known sequential algorithm based on the method of invariant imbedding was parallelized by a mapping assigning processors to characteristics. Two implementations of an algorithm based on this mapping and suitable for a message-passing architecture, but differing in their relative demands on communication and local memory, are described. Timing estimates for these are developed, and various consequences of these, notably the corresponding estimates for efficiency, are compared with computational results obtained from implementations on a 64-node hypercube system.