Course on novel time integration methods

Since about a dec­ade ago the scien­tific com­puting com­muni­ty has been con­front­ed with physi­cal limita­tions of the CPU/GPU manu­factur­ing pro­cess. As a result, the main in­crease in per­for­mance is not driven any­more by the in­crease in clock fre­quency, but by an in­crease in par­allel­ism. This poses a par­ticular chal­lenge for simu­lations that have to be fin­ished within a cer­tain wall clock time with ongo­ing expec­tations on using higher resolu­tions for higher accu­racy. This re­quires dis­rup­tive ap­proaches to cope with such situa­tions, and this com­pact course gave an intro­duc­tion to time inte­gra­tion meth­ods which are one of the most prom­ising ways out of this situa­tion.

The first day of the course began with a crash course on ad­vanced time inte­gra­tion tech­niques for ordi­nary differ­ential equa­tions (ODE) and their exten­sion to partial differ­ential equa­tions (PDE). This was fol­lowed by an intro­duc­tion to the Parareal meth­od for ODEs. Parareal be­longs to the class of paral­lel-in-time meth­ods with the under­lying idea of first calcu­lating a specu­lative solu­tion which is suc­ces­sively cor­rected in par­allel over the specu­lative solu­tions, i.e. paral­lel in time, in or­der to exploit addi­tional de­grees of free­dom for paral­leliza­tion on large super­com­put­ers.

On the sec­ond day a very short intro­duc­tion to the meth­ods of space dis­creti­zation (finite differ­ences and differ­ent types of Ga­lerkin meth­ods such as glob­al/local spec­tral meth­ods) and their rela­tion to time inte­gra­tion meth­ods was dis­cuss­ed. This plays a crucial role in the devel­op­ment of PDE solv­ers, which are often ig­nored in such a devel­op­ment. This creat­ed the basis for the con­tinua­tion of the Parareal meth­od for PDEs.

The third and fourth day ex­tend­ed the previ­ously ac­quired knowledge and under­stand­ing by expo­nential inte­gra­tion and its ration­al ap­prox­imations. This was fol­lowed by spec­tral de­ferred correc­tion (SDC) meth­ods and their exten­sion to multi­level SDC (ML-SDC). Final­ly, the stu­dents gained insight into the ideas of the Paral­lel Full Ap­prox­imation Scheme in Space and Time (PFASST), which com­pleted this course.

Overall, the stu­dents gained insight into the chal­lenges of time inte­gra­tion and nextgene­ration time inte­gra­tion meth­ods, in partic­ular paral­lel in time, and under­stood the inter­play of time and space dis­creti­zation schemes.

Text:
Dr. Martin Schreiber, Elite Graduate Program "Bavarian Graduate School of Computational Engineering"