- •Uncertainties in proton therapy beam delivery.
- •Uncertainty Quantification methods in beam dynamics.
- •Analysis of errors in transport lines and their effects on the beam properties.
- •Optimized beam optics solutions using UQ methods.
2. Uncertainty Quantification theory
where is the vector of input parameters (with j = 1, , d) that represent possible sources of uncertainty. Due to practical limitations, the PCE series in Eq. (1) is truncated up to a certain order p.
2.1 Training points
2.2 PCE basis
where is, in our case, the univariate Legendre polynomial of degree associated to the input .
where the i-th coefficient is obtained by the orthogonal projection of u onto the i-th basis polynomial. The integral at nominator in Eq. (6) is evaluated numerically by quadrature rules [
where denotes the Kronecker delta.
2.4 Statistical information
2.5 Sensitivity analysis
where is the set of multi-indices that include i-th order only. In principle a large value of the index means that the associated parameter is important in the variation of a certain QoI.
2.6 Setup of UQ analysis
|d = 3|
|p = 2||27||10|
|p = 3||64||20|
|p = 4||125||35|
|d = 5|
|p = 2||243||21|
|p = 3||1024||56|
|p = 4||3125||126|
3. UQ analysis for beam transport lines
3.1 Variation in the quadrupole gradients
|Order of surrogate model||p = 3|
|Number of inputs||d = 3|
|Number of coefficients||= 20|
|Number of training points||64|
3.2 Uncertainties in the beam kinetic energy
|COMET energy ()||250.0 MeV||0.5 MeV|
|COMET energy||25 keV||250 keV|
|Half-wedge thick. ()||2.5 mm||0.2 mm|
4. UQ optimization in beam dynamics
4.1 Superconducting gantry project at PSI
4.2 Beam envelope optimization with UQ
|Order of surrogate model||p = 3|
|Number of inputs||d = 4|
|Number of training points||256|
|Train.||Surrogate model||High-fidelity model|
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