Methods, devices and computer program products providing for establishing a model for emulating a physical quantity which depends on at least one input parameter, and use thereof
The present invention proposes methods, devices and computer program products. To this extent, there is defined a set X including N distinct parameter values x_i for at least one input parameter x, N being an integer greater than or equal to 1, first measured the physical quantity Pm 1 for each of the N distinct parameter values x_i of the at least one input parameter x, while keeping all other input parameters fixed, constructed a Vandermonde matrix VM using the set of N parameter values x_i of the at least one input parameter x, and computed the model W for emulating the physical quantity P based on the Vandermonde matrix and the first measured physical quantity according to the equation W=(VM T *VM) −1 *VM T *Pm 1 . The model is iteratively refined so as to obtained a desired emulation precision. The model can later be used to emulate the physical quantity based on input parameters or logs taken from the field and thereby support device design optimization.
1. A method, comprising:
providing by a device for establishing a model W for emulating a physical quantity P which depends on at least one input parameter x,
defining by the device a set X comprising N distinct parameter values x_i for said at least one input parameter x, N being an integer greater than or equal to 1,
measuring by the device a first measuring of the physical quantity P, Pm 1 for each of the N distinct parameter values x_i of the at least one input parameter x, while keeping all other input parameters fixed,
constructing by the device a Vandermonde matrix VM using said set of N parameter values x_i of said at least one input parameter x, and
computing by the device the model W for emulating the physical quantity P based on the Vandermonde matrix and the first measured physical quantity Pm 1 according to the equation W=(VM T *VM) −1 *VM T *Pm 1 .
2. The method according to claim 1 , wherein the measuring comprises keeping all input parameters other than the at least one input parameter x fixed on their respective maximum value.
3. The method according to claim 1 , wherein the constructing comprises
setting a precision for the emulation, and
iteratively constructing the Vandermonde matrix VM from a first degree to a degree d that satisfies said set precision.
4. The method according to claim 3 , further comprising, for each iteration,
calculating an emulation result PE for said quantity by multiplying a constructed Vandermonde test matrix VMt with the computed model W according to the equation PE=VMt*W, wherein the test matrix is derived based on a matrix Xt=X+V, V=[N(0; 1);N(0; 1); . . . ;N(0; 1)] whose elements are drawn from a normal distribution with mean 0 and variance 1.
5. The method according to claim 4 , further comprising,
measuring a second measuring of the physical quantity P, Pm 2 for each of N distinct parameter values xt_i, independently of the first measuring of the physical quantity Pm 1 , and, for each iteration,
determining a maximum relative error between the emulation result PE and the second measured physical quantity Pm 2 according to the equation
error=max( Pm 2 −PE )/ Pm 2*100
and
incrementing the degree of the Vandermonde matrix if the determined maximum relative error exceeds said set precision.
6. The method according to claim 5 , further comprising
terminating the iterative construction of the Vandermonde matrix VM and storing the computed model W if the determined maximum relative error does not exceed said set precision.
7. The method according to claim 1 , further comprising
pre-evaluating whether the physical quantity P to be emulated depends on another parameter than said at least one input parameter x.
8. The method according to claim 7 , wherein the pre-evaluating comprises,
for a set of k parameters other than said at least one input parameter x,
measuring the physical quantity, for each of said k parameters when set to its respective maximum and minimum value, while keeping said at least one input parameter x and k−1 other parameters to its respective maximum value,
calculating the difference between the measured physical quantity for each of said k parameters when set to its maximum and to its minimum, respectively,
comparing each of the differences with a measurement accuracy of measurement equipments used, and
if any of said differences exceeds the measurement accuracy, evaluating that the corresponding parameter affects the physical quantity.
9. An apparatus, comprising
a device, provided for establishing a model W for emulating a physical quantity P which depends on at least one input parameter x,
the device comprising
an input module configured to receive a set X comprising N distinct parameter values x_i for said at least one input parameter x, N being an integer greater than or equal to 1,
a first measurement module configured to measure a first measuring of the physical quantity P, Pm 1 for each of the N distinct parameter values x_i of the at least one input parameter x, while keeping all other input parameters fixed,
a construction module configured to construct a Vandermonde matrix VM using said set of N parameter values x_i of said at least one input parameter x, and
a computer module configured to compute the model W for emulating the physical quantity P based on the Vandermonde matrix and the first measured physical quantity according to the equation W=(VM T *VM) −1 *VM T *Pm 1 .
10. The apparatus according to claim 9 , wherein the measurement module is configured to keep all input parameters other than the at least one input parameter x fixed on their respective maximum value.
11. The apparatus according to claim 9 , wherein the construction module is configured to
set a precision for the emulation, and
iteratively construct the Vandermonde matrix VM from a first degree to a degree d that satisfies said set precision.
12. The apparatus according to claim 11 , wherein said computer module is configured, for each iteration,
to calculate an emulation result PE for said quantity by multiplying a constructed Vandermonde test matrix VMt with the computed model W according to the equation PE=VMt*W, wherein the test matrix is derived based on a matrix Xt=X+V, V=[N(0; 1);N(0; 1); . . . ;N(0; 1)] having a normal distribution with mean 0 and variance 1.
13. The apparatus according to claim 12 , further comprising,
a second measurement module configured to measure a second measuring of the physical quantity P, Pm 2 for each of N distinct parameter values xt_i, while keeping all other input parameters fixed, independently of the first measured physical quantity Pm 1 ,
and,
a determination module configured to determine, for each iteration,
a maximum relative error between the emulation result PE and the second measured physical quantity Pm 2 according to the equation
error=max( Pm 2 −PE )/ Pm 2*100
and
to increment the degree of the Vandermonde matrix, if the determined maximum relative error exceeds said set precision.
14. The apparatus according to claim 13 , wherein
the determination module is configured to terminate the iterative construction of the Vandermonde matrix VM and to store the computed model W, if the determined maximum relative error does not exceed said set precision.
15. The apparatus according to claim 9 , further comprising
a pre-evaluation module configured to pre-evaluate whether the physical quantity P to be emulated depends on another parameter than said at least one input parameter x.
16. The apparatus according to claim 15 , wherein the pre-evaluation module is configured,
for a set of k parameters other than said at least one input parameter x,
to measure the physical quantity, for each of said k parameters when set to its respective maximum and minimum value, while keeping said at least one input parameter x and k−1 other parameters to its respective maximum value,
to calculate the difference between in the measured physical quantity for each of said k parameters when set to his maximum and to its minimum, respectively,
to compare each of the differences with a measurement accuracy of measurement equipment used, and
if any of said differences exceeds the measurement accuracy, to evaluate that the corresponding parameter affects the physical quantity.
17. A computer program product comprising a non-transitory computer readable storage medium storing computer-executable components which, when executed on a computer, are configured to at least:
provide for establishing a model W for emulating a physical quantity P which depends on at least one input parameter x,
define a set X comprising N distinct parameter value x_i for said at least one input parameter x, N being an integer greater than or equal to 1,
measure a first measuring of the physical quantity P, Pm 1 for each of the N distinct parameter values x_i of the at least one input parameter x, while keeping all other input parameters fixed,
construct a Vandermonde matrix VM using said set of N parameter values x_i of said at least one input parameter x, and
compute the model W for emulating the physical quantity P based on the Vandermonde matrix and the first measured physical quantity according to the equation W=(VM T *VM) −1 *VM T *Pm 1 .
18. An apparatus, comprising
a device configured to provide for emulating a physical quantity P, which depends on at least one input parameter x,
the device comprising
an extractor module configured to extract said at least one parameter x from an interface, and
an emulator module configured to emulate said physical quantity P by supplying said extracted parameter to a model W for emulation of the physical quantity obtained by an apparatus according to claim 9 .