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Numerical solution to an inverse problem for quasilinear parabolic equation

In the work, we consider a process described by a quasilinear parabolic differential equation under some initial and boundary conditions. We set a problem of identifying the heat conductivity coefficient as a function of temperature value. Problems in such a statement arise when studying qualitative properties of materials and media which are not yielding to direct measurements, but studied by means of indirect measurements. To solve the problem formulated, we propose to seek the unknown function (heat conductivity coefficient) on the class of piecewise constant functions. With this end in view, we quantize the set of the phase state's values (temperature) by means of predetermined values. Thus the problem of finding the unknown function is reduced to a problem of determining the finite-dimensional vector. To solve the latter problem, we propose to use first-order numerical minimization methods. With this end in view, we derive formulas for the gradient of the target functional in the space of optimizable parameters.