Rationale 1
• The demand for eruption scenario forecast is pres-sing.
• Volcanic systems are largely out of direct observa-tion.
• Some of the quantities which determine volcanic pro-cesses are uncertain.
• Taking into account these uncertainties in predictivemodels can be exceedingly demanding.
• We are therefore developing the application of sto-chastic methods to largely reduce the computationalcosts.
What is stochastic quantization? 2
A practical situation:
• the random vector X = (X1, . . . , Xd) is part of theinput data of a numerical code ! and the randomvariable Y is one relevant model output;
• the probability density function f(x) of X is assumedto be known;
• there is a maximum number N of a!ordable simula-tions.
Strategy ! stochastic quantization method:
• find N values of X,!x(1)1 , . . . , x(1)
d
", . . . ,
!x(N)1 , . . . , x(N)
d
",
and N corresponding weights,#w(1), . . . , w(N)
$, with
%Ni=1 w(i) = 1, so that the resulting discrete distribu-
tion is the “best” approximation of f(x);
known
input
distribution
unknown
output
distributionModel
!
quantizationStochastic
Model
!
inputdiscretization
outputdiscretization
prob
abilit
yde
nsi
typr
obab
ilit
yde
nsi
ty
prob
abilit
yde
nsi
typr
obab
ilit
yde
nsi
ty
variable X1
variable X1
variable Y
variable Y
N = 10
N = 10
x(1) x(10) y(1) y(10)
w1
w2
w2
w1w10 w10
. . . . . . . . .. . .
• for i = 1, . . . , N compute
y(i) = !!x(i)1 , . . . , x(i)
d
"
and give it the weight w(i); the resulting discre-te distribution is an approximation of the unknowndistribution of Y .
Figure 1: approximation of input and output distribu-tions. The orange arrows represent performed computa-tions, while the blue one represents a computation oftenout of reach in real situations.
The single parameter input case, d=1 3
• Introduce a distance between two probability distri-butions, in the case in which X is a scalar quantity:if F (x) is the cumulative distribution function as-sociated with the density f(x) and F (x) is the oneassociated with its discretization f(x), we define thedistance between f and f as follows:
d(f, f) =! xmax
xmin
"""F (x)! F (x)"""dx, (1)
where xmin and xmax are the minimum and maximumpossible values of X.
countinuous
cumulative
prob
abilit
y
F (x)
variable X0
1
discrete
cumulative
prob
abilit
y
F (x)
variable Xx(1) x(10)
w1
w2
w10
0
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
. . . . . .. . .
• The procedure consists in searching for N points!x(1), . . . , x(N)
"
and N corresponding weights!w(1), . . . , w(N)
"
that minimize the quantity d(f, f).
distance
between
distributions
prob
abilit
y
F (x)! F (x)
variable X0
1
Figure 2: the distance between thecontinuous probability distributionand the discrete one is the shadedregion area.
The multi-parameter input case, d>1 4
• When X is a d-dimensional vector quantity, a di!e-rent definition of distance is more appropriate.
• Let X be a discrete random vector with probabilitydistribution f , approximating a continuous randomvector X. The distance between f and f can bedefined as the mean value of the error
!!!X ! X!!! re-
sulting from the substitution of X with X. We thusminimize
d(f, f) = E"|X ! X|
#.
random points with density f(x)
vari
able
X2
variable X1
x(1) x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
x(1), . . . , x(7) ! possible values of X
vari
able
X2
variable X1
• It can be shown that, in the case d = 1,
E!|X ! X|
"=
# xmax
xmin
$$$F (x)! F (x)$$$dx;
hence, the criterion for the multidimensional pro-blem is a generalization of that used in the one-dimensional case.
• d(f, f) is calculated through a Monte Carlo methodwhich involves the concept of Voronoi partitions.
• The procedure consists in searching for the discre-te random vector X that minimizes E
!|X ! X|
"; the
possible values x(1), . . . , x(N) of X and the corre-sponding weights w(1), . . . , w(N) generate the discreteapproximation f of the density f .
Figure 3: implementation of the multi-parameter inputcase. The blue points are a sample of X = (X1, X2); theorange points x(1), . . . , x(7) are the possible values of X,which is a discrete approximation of X. The orange linesdefine the Voronoi regions generated by the set of pointsx(1), . . . , x(7): the region associated to x(i) contains theblue points which are closer to x(i) than to any other ofthe orange points.
Testing stochastic quantization in simple cases 5
0.085
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
mea
n
real value
numerosity20 50 100 200 500 1000 2000
MC
SQ,N = 20
0.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
stan
dard
devi
atio
n
numerosity20 50 100 200 500 1000 2000
real valueMC
SQ,N = 20
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
95th
perc
enti
le
numerosity20 50 100 200 500 1000 2000
real valueMC
SQ,N = 20
• Let !(X) be a known analytical function. The pro-bability distribution of the output random variable Y
can be calculated exactly and compared with the ap-proximations produced by SQ and by Monte Carlo(MC) methods with variable numerosity.
• The case in the figure refers to
!(X1, X2) = X21X2
2 .
Figure 4: with the SQ method and only N = 20 simu-lations, we approximate the true values at a confidencelevel corresponding to N = 2000 MC simulations forthe mean or N = 200 MC simulations for the standarddeviation.
Application of SQ to volcanic conduit dynamics 6
• Application of SQ to a situation in which the outputprobability distribution cannot be explicitly calcula-ted, but quite complete statistical information aboutit can be obtained through MC simulations.
• One-dimensional steady model of magma flow in acilindrical conduit with fixed diameter and uniformtemperature [1].
• Random input quantities: diameter D of the conduitand total mass fraction wH20 of water.Random output quantity: logarithm of the mass flowrate m.
Figure 5: the correspondence between the distributionof mass flow rate found with 1000 MC simulations andSQ method is fully satisfactory when NSQ = 20.
Stochastic quantization
Model
!
Model
!
wH2O(%)
wH2O(%)
5
5
prob
abilit
yde
nsi
typr
obab
ilit
yde
nsi
ty
6 7
6 7
D(m)
0.2
0.71.2
1.20.7
0.2
D(m)0.5
9.5
9.5
0.5
6.5 7.5
7.56.5
prob
abilit
ydi
stri
buti
on
log10
!m(kg/s)
"
log10
!m(kg/s)
"
prob
abilit
ydi
stri
buti
on
SQ,NSQ = 20SQ,NSQ = 15SQ,NSQ = 10
MC
MC
CONCLUSIONSThe SQ method allows the introduction of uncertaintiesin the deterministic approach without requiring excee-ding CPU time. As a consequence, volcanic scenarioscan be estimated in the future by means of complexdeterministic models and taking into account the intrin-sic uncertainties involved in the definition of volcanicsystems.
Merging deterministic and probabilistic approaches to forecast volcanic scenarios
E. Peruzzo1, L. Bisconti2, M. Barsanti2,3, F. Flandoli3 and P. Papale2
1 Scuola Normale Superiore, Pisa, Italy2 Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Italy
3 Dipartimento di Matematica Applicata, Università di Pisa, Italye-mail:[email protected]
Abstract
We present the stochastic quantization (SQ) methodfor the approximation of a continuous probability den-sity function with a discrete one. This technique redu-ces the number of numerical simulations required to geta reasonably complete picture of the possible eruptiveconditions at a considered volcano. Finally we show theresults of a test using a one-dimensional steady modelof magma flow [1] as a benchmark.
Young Scientists'Outstanding Poster Paper
Contest
General Assembly
2009
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YSOPPReferences[1] P. Papale, Dynamics of magma flow in volca-
nic conduits with variable fragmentation e!ciencyand nonequilibrium pumice degassing, J. Geophys.Res., 106, 11043-11065, 2001
[2] S. Graf, H. Luschgy, Foundations of quanti-zation for probability distributions, Springer-Verlag,2000
