Multiply a table with a sub-table of another table

Question

Consider two grids of coordinates:

grid1 = Flatten[Table[{i, j}, {i, 1, 15}, {j, 1, 20}], 1]//N;
grid2 = Flatten[Table[{0.5 i, 0.5 j}, {i, 1, 45}, {j, 1, 60}], 1];

grid2 includes grid1 as a subset. In my realistic case, the pattern of the sub-set is more complicated than in this toy example.

Next, consider two tables:

tab1 = Join[grid1, {Exp[#[[1]]] Exp[-#[[2]]]} & /@ grid1, 2];
tab2 = Join[grid2, {Sin[#[[1]]] Tan[#[[2]]]} & /@ grid2, 2];

In my realistic case, they are pre-computed, I only know that tab2 is computed for grid2 and tab1 for grid1.

I want to multiply the elements of the third column of tab1 by the elements of the third column of tab2 for matching values of the first and the second column. I.e., the resulting table should be just tab1 but where the third column or the ith row is replaced by the multiplication of tab1[[i]][[3]]*tab2[[j]][[3]], with j belonging to the row of tab2 with the first and second columns coinciding with those of tab1[[i]]:

tab3=Join[grid1, {Sin[#[[1]]] Tan[#[[2]]] Exp[#[[1]]] Exp[-#[[2]]]} & /@ grid1, 2];

Could you please tell me how to obtain tab3 if having tab1, tab2 precomputed (so we cannot just use the expression above)?

Answer 1

KeyValueMap[
  Append, 
  Merge[
    KeyIntersection[{GroupBy[tab1, Most -> Last], GroupBy[tab2, Most -> Last]}], 
    Apply[Times]@*Flatten]]

Answer 2

If you are sure that every element of grid1 is contained in grid2, then you could do it with Nearest as follows:

tab1[[All, 3]] *= tab2[[Flatten@ Nearest[tab2[[All, 1 ;; 2]] -> "Index", tab1[[All, 1 ;; 2]]], 3]];

Now tab1 should be equal to tab3.