Algorithm for intersecting symbolic and approximate linear differential varieties

Abstract

This article provides algorithms for systems of approximate linear partial differential equations that exploit exact subsystems. Such exact systems have rational function coefficients over $\mathbb{Q}$ and can be reduced to forms (e.g. differential Gröbner bases) by a finite number of differentiations and eliminations using available computer implementations. We will use the rifsimp algorithm in Maple for this purpose. Such algorithms use solvers based on orderings (rankings) of their derivatives, are coordinate dependent, and are prone to instability when applied to approximate input. In contrast, our Geometric Involutive Form algorithm, uses a sequence of geometric differentiations (prolongations) and projections to complete approximate linear systems to geometric involutive form. In particular, it uses numerical linear algebra (especially the SVD) to monitor dimension criteria for termination. However, this latter method can be expensive as the size of the matrices rapidly increases with the number of variables and order of derivatives involved.Approximate differential systems in applications often have exact subsystems and this motivated us to develop the hybrid method described in this article. The first step of the method is to partition the input into an exact subsystem and an approximate subsystem. The exact subsystem is reduced by using our rifsimp algorithm. The reduced exact subsystem is used to simplify the approximate subsystem. The previous partition, reduction and simplification steps are repeated until no new exact equations are found. Then the reduced exact subsystem is used to simplify prolongations of the approximate subsystem. Checking that the jointly prolonged system is geometrically involutive is done by computing dimension criteria of the simplified prolonged approximate system and using the differential Hilbert function of the reduced exact system.Our algorithm is illustrated by determination of approximate symmetry properties of a gravitational potential for a gaseous cloud. It enables a significant reduction of the size of the coefficient matrices of prolongations involved in numerical computations compared to our previous approach. For instance, the dimension of the jet space used for approximate calculations is reduced from dim J7 = 1320 to dim J1 = 20.