English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Often, when solving forward, inverse or data assimilation problems, only apart of the solution is needed. As a model, we consider the stationarydiffusion problem. We demonstrate an algorithm that can compute only a part ora functional of the solution, without calculating the full inversion operatorand the complete solution. It is a well-known fact about partial differentialequations that the solution at each discretisation point depends on thesolutions at all other discretisation points. Therefore, it is impossible tocompute the solution only at one point, without calculating the solution at allother points. The standard numerical methods like a conjugate gradient or Gausselimination compute the whole solution and/or the complete inverse operator. Wesuggest a method which can compute the solution of the given partialdifferential equation 1) at a point; 2) at few points; 3) on an interface; or afunctional of the solution, without computing the solution at all points. Therequired storage cost and computational resources will be lower as in thestandard approach.  With this new method, we can speed up, for instance, computation of theinnovation in filtering or the likelihood distribution, which measures the datamisfit (mismatch). Further, we can speed up the solution of the regression,Bayesian inversion, data assimilation, and Kalman filter update problems.  Applying additionally the hierarchical matrix approximation, we reduce thecubic computational cost to almost linear $\mathcal{O}(k^2n \log^2 n)$, where$k\ll n$ and $n$ is the number of degrees of freedom.  Up to the hierarchical matrix approximation error, the computed solution isexact. One of the disadvantages of this method is the need to modify theexisting deterministic solver.

Partial inversion of the elliptic operator to speed up computation of likelihood in Bayesian inference