How to invert fast a big pseudo-inverse matrix

Hello everybody I have to invert a big (m,n) matrix using a pseudo inverse. The pinv() function doesnt work because the matrix is to big. I program a image reconstruction and have to solve the equation A*x = b, where x,b are vectors and A is my matrix. To get x is need to invert my matrix. For the pseudo inverse I use the Tikhonov regularization, because it gets the best images. Maybe you can tell me if there is a faster way to invert my matrix. Now it is very time consuming and I dont know how to make it faster. The Tikhonov regularization works as follows: (A'*A + T'*T)^(-1)*A'. T is defined as lambda*(identity matrix) My code is

if true
  lambda = 250;  
tikhonovMatrix = lambda*speye(numTrans*numWinkel, numPixZ*numPixX);
tikhonovMatrixTrans = tikhonovMatrix';
tikhonovMatrixProd = tikhonovMatrixTrans*tikhonovMatrix;
yT2Trans = yT2';
yT2Prod = yT2Trans*yT2;
pseudoInverse_yT2 = yT2Prod + tikhonovMatrixProd;
bildVektor = reshape(newsinogram,[],1);
recBildVektor = yT2Trans*bildVektor;
recBildVektor2 = pseudoInverse_yT2\recBildVektor;
end