Epistemic/Non-probabilistic Uncertainty Propagation . - APAN Community

1y ago
9 Views
2 Downloads
4.30 MB
30 Pages
Last View : 1m ago
Last Download : 2m ago
Upload by : Joao Adcock
Transcription

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

Related Documents:

fractional uncertainty or, when appropriate, the percent uncertainty. Example 2. In the example above the fractional uncertainty is 12 0.036 3.6% 330 Vml Vml (0.13) Reducing random uncertainty by repeated observation By taking a large number of individual measurements, we can use statistics to reduce the random uncertainty of a quantity.

1.1 Measurement Uncertainty 2 1.2 Test Uncertainty Ratio (TUR) 3 1.3 Test Uncertainty 4 1.4 Objective of this research 5 CHAPTER 2: MEASUREMENT UNCERTAINTY 7 2.1 Uncertainty Contributors 9 2.2 Definitions 13 2.3 Task Specific Uncertainty 19 CHAPTER 3: TERMS AND DEFINITIONS 21 3.1 Definition of terms 22 CHAPTER 4: CURRENT US AND ISO STANDARDS 33

Dealing with Uncertainty: A Survey of Theories and Practices Yiping Li, Jianwen Chen, and Ling Feng,Member, IEEE Abstract—Uncertainty accompanies our life processes and covers almost all fields of scientific studies. Two general categories of uncertainty, namely, aleatory uncertainty and epistemic uncertainty, exist in the world.

73.2 cm if you are using a ruler that measures mm? 0.00007 Step 1 : Find Absolute Uncertainty ½ * 1mm 0.5 mm absolute uncertainty Step 2 convert uncertainty to same units as measurement (cm): x 0.05 cm Step 3: Calculate Relative Uncertainty Absolute Uncertainty Measurement Relative Uncertainty 1

Uncertainty in volume: DVm 001. 3 or 001 668 100 0 1497006 0 1 3 3. %. % .% m m ª Uncertainty in density is the sum of the uncertainty percentage of mass and volume, but the volume is one-tenth that of the mass, so we just keep the resultant uncertainty at 1%. r 186 1.%kgm-3 (for a percentage of uncertainty) Where 1% of the density is .

on radio propagation. This handbook also provides basic information about the entire telecommunications environment on and around Mars for propagation researchers, system . 1.2 Radio Wave Propagation Parameters. 4 2. Martian Ionosphere and Its Effects on Propagation (Plasma and Magnetic Field). 7

1 How Plant Propagation Evolved in Human Society 2 2 Biology of Plant Propagation 14 3 The Propagation Environment 49. part two. Seed Propagation. 4 Seed Development 110 5 Principles and Practices of Seed Selection 140 6 Techniques of Seed Production and Handling 162 7 Principles of Propagati

ASP.NET is a unified Web development model that includes the services necessary for you to build enterprise-class Web applications with a minimum of coding. ASP.NET is part of the .NET Framework, and when coding ASP.NET applications you have access to classes in the .NET Framework. You can code your applications in any language compatible with the common language runtime (CLR), including .