This work presents a new robust black-box numerical solver that can reliably solve many MRI typical ordinary differential equations. Our adaptive Petrov-Galerkin (PG) method can solve challenging MRI problems with additional complexities such as B0- and B1- inhomogeneities, RF pulses, chemical exchange, and magnetization transfer (MT). We apply the proposed technique to solve an ODE-constrained optimization problem for pulse design via gradient descent. Our method reduces the time required to compute the gradients by three orders of magnitude.
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