Does this oscillator have a name?
This is a more or less a variant of the traditional LC (inductor-capacitor) oscillator using discrete amplification and drawn in a unique way.

LC oscillators operate via a feedback network 'shifting' the phase of a signal at the oscillation frequency to yield positive feedback.

This little circuit is similar, with one big caveat. Namely, it draws about 1A of current RMS at 5V. Secondly, the transistors dissipate nearly a watt RMS each - so would quickly overheat. At 3.3v, things look somewhat better at 400mW RMS each. At 1.5v, it is a reasonable 80mW each, and quiescent current is 'only' 280mA RMS. So definitely not efficient in any sense of the word. On the up-side, the output voltage is much higher than the supply voltage:



As for a theory of operation:


At the initial power-on state, the capacitors and inductor (LC) are uncharged, so at 0 volts. C's look like shorts, L looks like open. D1 prevents the power source from charging the LC. The PNP transistor Q2 sees a "low" level on it's base 'A', so turns on, which brings 'state' high.
As C3 is a much larger value than C1 or C2, more current from 'state' begins to flow into 'out' than can be provided by either R1+C1 or R2+C2. So 'out' voltage begins to rise, and C1 begins to equalize charge. L1 is also storing charge, and is looking less and less like an open circuit.
Before C1 can reach equilibrium, the voltage at NPN Q1 'B' has risen to the threshold voltage, so it starts to turn on.
With both transistors on and in their "linear region", everything balances for a nanosecond; however L1's stored charge begins collapsing, reversing polarity and discharging through C3 mostly, bringing 'state' down ever-so-slightly. This imbalances the network, and starts the oscillation.
D1 and D2 tend to "clip" levels at A and B (from C1 and C2.)




Have a play with it in LTspice:

Version 4
SHEET 1 880 680
WIRE 160 -224 16 -224
WIRE 336 -224 160 -224
WIRE 336 -208 336 -224
WIRE 160 -176 160 -224
WIRE 272 -128 224 -128
WIRE 336 -128 336 -144
WIRE 336 -128 272 -128
WIRE 416 -128 336 -128
WIRE 512 -128 480 -128
WIRE 336 -112 336 -128
WIRE 272 -96 272 -128
WIRE 16 -48 16 -224
WIRE 160 -16 160 -80
WIRE 256 -16 160 -16
WIRE 336 -16 336 -32
WIRE 336 -16 256 -16
WIRE 416 -16 336 -16
WIRE 512 -16 512 -128
WIRE 512 -16 480 -16
WIRE 576 -16 512 -16
WIRE 608 -16 576 -16
WIRE 336 0 336 -16
WIRE 608 32 608 -16
WIRE 160 48 160 -16
WIRE 272 96 272 64
WIRE 272 96 224 96
WIRE 336 96 336 80
WIRE 336 96 272 96
WIRE 416 96 336 96
WIRE 512 96 512 -16
WIRE 512 96 480 96
WIRE 336 112 336 96
WIRE 16 192 16 32
WIRE 160 192 160 144
WIRE 160 192 16 192
WIRE 336 192 336 176
WIRE 336 192 160 192
WIRE 608 192 608 112
WIRE 608 192 336 192
WIRE 336 208 336 192
FLAG 336 208 0
FLAG 576 -16 OUT
FLAG 256 -16 STATE
FLAG 272 -96 A
FLAG 272 64 B
SYMBOL npn 224 48 M0
SYMATTR InstName Q1
SYMATTR Value 2N3904
SYMBOL pnp 224 -80 R180
SYMATTR InstName Q2
SYMATTR Value 2N3906
SYMBOL voltage 16 -64 R0
SYMATTR InstName V1
SYMATTR Value 1.5v
SYMBOL res 320 -128 R0
SYMATTR InstName R1
SYMATTR Value 2.2k
SYMBOL res 320 -16 R0
SYMATTR InstName R2
SYMATTR Value 2.2k
SYMBOL diode 352 -144 R180
WINDOW 0 24 64 Left 2
WINDOW 3 24 0 Left 2
SYMATTR InstName D1
SYMATTR Value 1N4148
SYMBOL diode 352 176 R180
WINDOW 0 24 64 Left 2
WINDOW 3 24 0 Left 2
SYMATTR InstName D2
SYMATTR Value 1N4148
SYMBOL cap 480 -144 R90
WINDOW 0 0 32 VBottom 2
WINDOW 3 32 32 VTop 2
SYMATTR InstName C1
SYMATTR Value 0.001µ
SYMATTR SpiceLine V=25 Irms=83.8m Rser=2.70485 Lser=0 mfg="KEMET" pn="C0805C102K3RAC" type="X7R"
SYMBOL cap 480 80 R90
WINDOW 0 0 32 VBottom 2
WINDOW 3 32 32 VTop 2
SYMATTR InstName C2
SYMATTR Value 0.001µ
SYMATTR SpiceLine V=25 Irms=83.8m Rser=2.70485 Lser=0 mfg="KEMET" pn="C0805C102K3RAC" type="X7R"
SYMBOL cap 480 -32 R90
WINDOW 0 0 32 VBottom 2
WINDOW 3 32 32 VTop 2
SYMATTR InstName C3
SYMATTR Value 0.01µ
SYMATTR SpiceLine V=25 Irms=291m Rser=0.34258 Lser=0 mfg="KEMET" pn="C0805F103K3RAC" type="X7R"
SYMBOL ind 592 16 R0
SYMATTR InstName L1
SYMATTR Value 100µ
SYMATTR SpiceLine Ipk=0.3 Rser=1.35 Rpar=46700 Cpar=0 mfg="Bourns, Inc." pn="SRR4018-101Y"
TEXT 390 176 Left 2 !.tran 0.5m startup