Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. A possible reason for uncertainties can be inexact measurements (e.g. permeability or condactivity coefficents) or not full knowledge about the model.
The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo\`eve expansion (KLE) becomes important.
If input parameters are uncertain the computed solution will be also uncertain and it will be not one solution but a set of solutions. The final aim is to quantify uncertainties in the solution. It means to estimate propagation of input uncertainties to the solution. Another aim can be to compute the mean value or the variance of the solution.