Epistemic/Non-probabilistic Uncertainty Propagation Using Fuzzy Sets

Transcription

1 Epistemic/Non-probabilistic Uncertainty Propagation Using Fuzzy Sets Dongbin Xiu Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute University of Utah

2 Outline Introduction Motivation: epistemic uncertainty Fuzzy sets: the basics Numerical methods for propagating fuzzy sets (Limited) Existing methods Our new algorithm: efficiency and accuracy Examples

3 Problem Statement: 8 < : Setup v t (x, t, Z) =L(v), D (0,T] I z, B(v) [0,T] I z, v = v 0, D {t =0} I z. Solution: v(x, t, Z) : D [0,T] I z! R n v Quantity of interest (QoI): Q = f(z) =(q v)(z) Z are uncertain parameters : physical parameters or hyper-parameter QoI represents a complicated mapping from input to output Key task: forward propagation of uncertainty

4 UQ via Probabilistic Modeling 8 < : v t (x, t, Z) =L(v), D (0,T] I z, B(v) [0,T] I z, v = v 0, D {t =0} I z. Q = f(z) =(q v)(z) Probabilistic approach: Assumes Z are random variables To draw samples, or to construct orthogonal basis Various methods: Sampling: Monte Carlo or deterministic Generalized polynomial chaos: Stochastic Galerkin Stochastic collocation Etc., etc.

5 Parametric Epistemic Uncertainty Q = f(z) =(q v)(z) Probabilistic approaches require the distribution function of Z F Z (s) = Prob(Z apple s), s 2 R n z requires a large amount of information/data, and is often impossible at all (especially for dependence) Epistemic uncertainty: Uncertainty due to lack of knowledge Parametric form and model-form Reducible Our previous work: Allows the use of probabilistic method (e.g. polynomial chaos) Requirement of input probability distribution is much relaxed o Change-of-measure o Sharp estimation of solution bounds

6 Non-probabilistic Modeling via Fuzzy Sets Some uncertainties shall not be modeled probabilistically Maybe epistemic, maybe not Fuzzy sets vs. Classical/Crisp sets Key insight: Many things are inherently fuzzy.

7 Non-probabilistic Modeling via Fuzzy Sets Some uncertainties shall not be modeled probabilistically Maybe epistemic, maybe not Fuzzy sets vs. Classical/Crisp sets Key insight: Many things are inherently fuzzy. Example: It is raining. Yes

8 Non-probabilistic Modeling via Fuzzy Sets Some uncertainties shall not be modeled probabilistically Maybe epistemic, maybe not Fuzzy sets vs. Classical/Crisp sets Key insight: Many things are inherently fuzzy. Example: It is raining. No

9 Non-probabilistic Modeling via Fuzzy Sets Some uncertainties shall not be modeled probabilistically Maybe epistemic, maybe not Fuzzy sets vs. Classical/Crisp sets Key insight: Many things are inherently fuzzy. Example: It is raining. What about this? Is drizzling raining?

10 Fuzzy Sets: The Basics Fuzzy sets can be considered as a generalization of classical sets For every set A, it is associated with a membership function µ A (x) µ A :! [0, 1] denoting the likelihood of any element x belonging to A For classical sets, this is the indicator function, µ A (x) = 1, x 2 A, 0, x /2 A.

11 Fuzzy Sets: The Basics Fuzzy sets can be considered as a generalization of classical sets For every set A, it is associated with a membership function µ A (x) µ A :! [0, 1] denoting the likelihood of any element x belonging to A For classical sets, this is the indicator function, µ A (x) = 1, x 2 A, 0, x /2 A. For fuzzy sets, it is often not such a sharp transition.

12 More Fuzzy Sets Examples µ Example 1: A = {It is raining} Membership function prescription can be subjective; Depends on objective, risk, etc. Precipitation rate

13 More Fuzzy Sets Examples µ Example 1: A = {It is raining} Membership function prescription can be subjective; Depends on objective, risk, etc. Precipitation rate Example 2: A = {It is red} yes no no

14 More Fuzzy Sets Examples µ Example 1: A = {It is raining} Membership function prescription can be subjective; Depends on objective, risk, etc. Precipitation rate Example 2: A = {It is red} yes no no What about this?

15 More Fuzzy Sets Examples µ Example 1: A = {It is raining} Membership function prescription can be subjective; Depends on objective, risk, etc. Precipitation rate Example 2: A = {It is red} yes no no What about this? = +

16 RBG Colors Membership functions Membership functions are well accepted (non-subjective)

17 The Basics of Fuzzy Sets ea = {(x, µ ea (x)) x 2 X} µ ea (x) :X! [0, 1] Short-handed notation: Ã(x) Ã Support: supp(ã) ={x 2 X Ã(x) > 0} α-cut: Strong α-cut: [Ã] = {x 2 X Ã(x) } [Ã] + = {x 2 X Ã(x) > } [Ã] Extension principle: Given a function f : x! y supp(ã) f : Ã extension principle! B

18 The Basics of Fuzzy Sets: α-cuts α-cuts are standard crisp sets [Ã] + = {x 2 X Ã(x) > } Ã Decomposition principle: α-cuts completely characterize fuzzy sets ea = [ [ A] e + 2[0,1] Zadeh s extension principle: propagates α cuts via functions f : The mapped fuzzy set: Ã extension principle! B [Ã] supp(ã) eb = {(y, µ eb (y)) y = f(x),x2 X} where µ eb (y) = supx2f 1 (y) µ ea (x), if f 1 (y) 6= 0, 0, otherwise.

19 Our Setup PDE with parametric uncertainty: 8 < u t (x, t, ) =L(u), D (0,T] Z, B(u) [0,T] Z, : u = u 0, D {t =0} Z Solution: u(, ) :Z! R Modeling: We prescribe, or are given, a membership function over Z, and define a fuzzy set e Z Solution defines: e U = u(, e Z) supp( e Z)=Z Goal: To characterize the output fuzzy set Existing methods: Very limited Exhaustive sampling or interval analysis of α-cuts Application of the extension principle requires excessive simulation effort

20 Our Numerical Strategy o Define a tensor domain over the supports of each input fuzzy sets Z i =supp( i ) Z T = Z 1 Z d Remark: Z T can be (much) bigger than Z=supp(ξ) o Construct an accurate strong approximation (gpc, etc) in Z T n = ku u n k L p w (Z T ) 1 o Apply the extension principle to obtain an approximate output fuzzy set eu n = u n (, e Z) Note: The extension principle is explored on the surrogate --- no simulation effort

21 Accuracy Analysis: Setup The true output fuzzy set: eu = u(, e Z) defined via the function u over supp( e Z)=Z The approximate fuzzy set: eu n = u n (, e Z) defined via the numerical solution u n over Z T = Z 1 Z d n = ku u n k L p w (Z T ) 1 How to measure the difference between the two fuzzy sets? Needs a distance/norm/metric

22 Distance between Fuzzy Sets Consider two functions f and g, and the mapped fuzzy sets ef = f( e A) e G = g( e A) Let us measure the difference via their α cuts, because they are standard crisp sets, and they completely characterize the fuzzy sets Average: Maximum: D( e F, e G)= Z 0,1 dist [ F e ] +, [ G] e + d D 1 ( F, e G) e = max dist [ F e ] +, [ e G] + 2[0,1] Question: What distance to use for the standard sets?

23 Modified Hausdorff Distance Hausdorff distance: d H (X, Y ) = max sup x2x inf y2y d(x, y), sup y2y inf d(x, y) x2x

24 Modified Hausdorff Distance Hausdorff distance: d H (X, Y ) = max sup x2x inf y2y d(x, y), sup y2y inf d(x, y) x2x For practical usage, we make two modifications Use essinf Use L p norm 8 d H p([ F e ] +, [ G] e < Z +) = max : Z s2[ e A] + r2[ e A] + Remark: This distance is computable. essinf r2[ e A] + dp (f(s),g(r))ds essinf s2[ e A] + dp (f(s),g(r))ds! 1 p,! 9 1 p = ;

25 Accuracy Analysis: Result The true output fuzzy set: eu = u(, e Z) defined via the function u over supp( e Z)=Z The approximate fuzzy set: eu n = u n (, e Z) defined via the numerical solution u n over Z T = Z 1 Z d n = ku u n k L p w (Z T ) 1

26 Accuracy Analysis: Result The true output fuzzy set: eu = u(, e Z) defined via the function u over supp( e Z)=Z The approximate fuzzy set: eu n = u n (, e Z) defined via the numerical solution u n over Z T = Z 1 Z d n = ku u n k L p w (Z T ) 1 Theorem: Assum w C w 0, D( e U, e U n ; p) apple D 1 ( e U, e U n ; p) apple C 1/p w ku u n k L p w (Z T ) Accurate surrogate with independence input assumption leads to accurate output fuzzy sets, regardless of the true dependence in the inputs

27 Examples: 1D Illustrative example Membership functions Numerical solution Ũ 10 Error of Ũ n µ Err u n 27

28 2D Examples with Independent Inputs Membership function for inputs Exact solution: Numerical solution: µ Err u n 28

29 2D Examples with Dependent Inputs Membership function for inputs Exact solution: Numerical solution: µ Err u n 29

30 Summary There are other approaches for epistemic analysis Interval analysis Possibility theory Fuzzy set theory Probability boxes Etc Fuzzy sets theory is an important means for non-probabilistic analysis Computing output fuzzy set has been difficult Developed a surrogate-based output fuzzy set analysis Accuracy is guaranteed theoretically Theory is extended to mixed probabilistic and non-probabilistic inputs. Reference: Chen, He, Xiu, SIAM J. Sci. Comput Key feature: Only a one-time standard (stochastic) forward computation is required.