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Epistemic/Non-probabilistic Uncertainty PropagationUsing Fuzzy SetsDongbin XiuDepartment of Mathematics, andScientific Computing and Imaging (SCI) InstituteUniversity of Utah

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

Setup Problem Statement:8 vt (x, t, Z) L(v), D (0, T ] Iz ,B(v) 0,@D [0, T ] Iz ,:v v0 ,D {t 0} Iz . Solution:v(x, t, Z) : D̄ [0, T ] Iz ! Rnv Quantity of interest (QoI): Q f (Z) (q v)(Z)Z are uncertain “parameters”: physical parameters or hyper-parameterQoI represents a complicated mapping from input to outputKey task: forward propagation of uncertainty

UQ via Probabilistic Modeling8 vt (x, t, Z) L(v),B(v) 0,:v v0 ,D (0, T ] Iz ,@D [0, T ] Iz ,D {t 0} Iz .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.

Parametric Epistemic UncertaintyQ f (Z) (q v)(Z) Probabilistic approaches require the distribution function of ZFZ (s) Prob(Z s), s 2 Rnz 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 relaxedo Change-of-measureo Sharp estimation of solution bounds

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.

Non-probabilistic Modeling via Fuzzy SetsYes 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.”

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

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?

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 µA(x)µA : ! [0, 1]denoting the “likelihood” of any element x belonging to A For classical sets, this is the indicator function, µA (x) 1,0,x 2 A,x2/ A.

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 µA(x)µA : ! [0, 1]denoting the “likelihood” of any element x belonging to A For classical sets, this is the indicator function, µA (x) For fuzzy sets, it is often not such a sharp transition. 1,0,x 2 A,x2/ A.

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

More Fuzzy Sets Examplesµ Example 1: A {It is raining}Precipitation rate Membership function prescription can be subjective; Depends on objective, risk, etc. Example 2: A {It is red}yesnono

More Fuzzy Sets Examplesµ Example 1: A {It is raining}Precipitation rate Membership function prescription can be subjective; Depends on objective, risk, etc. Example 2: A {It is red}yesWhat about this?nono

More Fuzzy Sets Examplesµ Example 1: A {It is raining}Precipitation rate Membership function prescription can be subjective; Depends on objective, risk, etc. Example 2: A {It is red}yesnonoWhat about this?

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

The Basics of Fuzzy Setse {(x, µ e(x)) x 2 X}AA Short-handed notation: Ã(x)µAe(x) : X ! [0, 1] Support: α-cut:supp(Ã) {x 2 X Ã(x) 0}[Ã] {x 2 X Ã(x) } Strong α-cut:[Ã] [Ã] {x 2 X Ã(x) }supp(Ã) Extension principle: Given a function f :f:Ãx!yextension principle! B̃Ã

The Basics of Fuzzy Sets: α-cuts[Ã] {x 2 X Ã(x) }Ã α-cuts are standard crisp sets Decomposition principle:α-cuts completely characterize fuzzy sets[ee A · [A] 2[0,1] Zadeh’s extension principle:propagates α–cuts via functionsf:Ã[Ã] extension principle! B̃supp(Ã) The mapped fuzzy set:e {(y, µ e (y)) y f (x), x 2 X}BB supx2f 1 (y) µAe(x), if f 1 (y) 6 0,where µBe (y) 0,otherwise.

Our Setup PDE with parametric uncertainty:8 ut (x, t, ) L(u), D (0, T ] Z,B(u) 0,@D [0, T ] Z,:u u0 ,D̄ {t 0} Z Solution:u(·, ) : Z ! R Modeling: We prescribe, or are given, a membership function over Z, anddefine a fuzzy set Zee Zsupp(Z) Solution defines:e u(·, Z)eU 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

Our Numerical Strategyo Define a tensor domain over the supports of each input fuzzy setsZi supp( i )ZT Z1 · · · ZdRemark: ZT can be (much) bigger than Z supp(ξ)o Construct an accurate strong approximation (gPC, etc) in ZT n kuun kLpw (ZT ) 1o Apply the extension principle to obtain an approximate output fuzzy seten un (·, Z)eUNote: The extension principle is explored on the surrogate --- no simulation effort

Accuracy Analysis: Setupe u(·, Z)e The true output fuzzy set: Ue Zdefined via the function u over supp(Z)en un (·, Z)e The approximate fuzzy set: Udefined via the numerical solution un over n kuZT Z1 · · · Zdun kLpw (ZT ) 1 How to measure the difference between the two fuzzy sets? Needs a distance/norm/metric

Distance between Fuzzy Sets Consider two functions f and g, and the mapped fuzzy setse g(A)eGeFe f (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:e D(Fe , G)Z0,1 e d dist [Fe ] , [G] e max dist [Fe ] , [G]e D1 (Fe , G) 2[0,1] Question: What distance to use for the standard sets?

Modified Hausdorff Distance Hausdorff distance:dH (X, Y ) max sup inf d(x, y), sup inf d(x, y)x2X y2Yy2Y x2X

Modified Hausdorff Distance Hausdorff distance:dH (X, Y ) max sup inf d(x, y), sup inf d(x, y)x2X y2Yy2Y x2X For practical usage, we make two modifications§ Use essinf§ Use Lp norm8! p1 Zpe ) maxdH p ([Fe ] , [G]essinfr2[A],e d (f (s), g(r))ds : s2[A]e ! p1 9Z pessinfs2[A]e d (f (s), g(r))ds ;e r2[A] Remark: This distance is computable.

Accuracy Analysis: Resulte u(·, Z)e The true output fuzzy set: Ue Zdefined via the function u over supp(Z)en un (·, Z)e The approximate fuzzy set: Udefined via the numerical solution un over n kuZT Z1 · · · Zdun kLpw (ZT ) 1

Accuracy Analysis: Resulte u(·, Z)e The true output fuzzy set: Ue Zdefined via the function u over supp(Z)en un (·, Z)e The approximate fuzzy set: Udefined via the numerical solution un over n kuZT Z1 · · · Zdun kLpw (ZT ) 1 Theorem: Assum w Cw 0,e, Uen ; p) D1 (Ue, Uen ; p) Cw 1/p kuD(Uun kLpw (ZT ) Accurate surrogate with independence input assumption leads to accurateoutput fuzzy sets, regardless of the true dependence in the inputs

Examples: 1D Illustrative exampleMembership functionsNumerical solution Ũ10µError of ŨnErrun27

2D Examples with Independent InputsMembership function for inputsExact solution:Numerical solution:Errµun28

2D Examples with Dependent InputsMembership function for inputsExact solution:Numerical solution:Errµun29

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. 2015. Key feature: Only a one-time “standard” (stochastic) forward computation is required.

Parametric Epistemic Uncertainty Epistemic uncertainty: Uncertainty due to lack of knowledge Parametric form and model-form Reducible Probabilistic approaches require the distribution function of Z F Z (s) Prob(Z s),s2 Rnz requires a large amount of information/data, and

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