Z ⌦ km(t, x)k2 2 2⇢(t, x) + F(x, ⇢(t, x))dxdt | {z } Y(⇢,m) , min ⇢,m Y(⇢, m) + XC(⇢0,⇢1) (⇢, m) where XC(⇢0,⇢1) (⇢, m) = ( +1, (⇢, m) 62 C(⇢0, ⇢1), 0, (⇢, m) 2 C(⇢0, ⇢1). Proximal Gradient Descent method for MFP: 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : ⇣ ⇢k+ 1 2 , mk+ 1 2 ⌘ = ⇢k, mk ⌘k ⇢,mY ⇢k, mk , | {z } gradient (forward) descent easy and fast to compute for di↵erentiable F ⇢k+1, mk+1 = ProjC(⇢0,⇢1) ⇣ ⇢k+ 1 2 , mk+ 1 2 ⌘ , | {z } proximal (backward) descent = ⇣ ⇢k+ 1 2 , mk+ 1 2 ⌘ rt,m 1 t,mrt,m · ⇣ ⇢k+ 1 2 , mk+ 1 2 ⌘ . | {z } independent to the form of F fast to compute with Fast Cosine Transformation 4 18