Monte Carlo Methods & Virtual Photonics

Transcription

1 Monte Carlo Methods & Virtual Photonics Tuesday-Thursday 9 10:30 Room 3201, Nat. Sci. II Jerry Spanier P220 BLI jspanier@uci.edu Please help yourselves to refreshments at the back of the room and leave your name, affiliation and contact information on the sign-in sheet. L A M M P

2 Monte Carlo Methods & Virtual Photonics Purposes Provide basic instruction in MC methods for BLI, CCBS and MCSB Integrate M&C across LAMMP projects Advance plans for LAMMP Workshops on VTS/ATKs L A M M P

3 Specific Goals Provide overview of conventional + more advanced MC methods Outline conceptual framework for MC algorithm development from user perspective (VTS/ATK) Describe features of newer adaptive MC methods that converge geometrically L A M M P

4 Detecting Cancer Non-invasively At BLI we use a number of optical technologies to detect, diagnose and treat various subtle transformations in tissue, including those caused by cancer, wounds/burns, etc. The effective use of these methods requires accurate models of the interactions of light with tissue. L A M M P

5 L A M M P Tissue Pathology

6 L A M M P Histology: Squamous Epithelium Note irregular pattern

7 L A M M P Histology: Breast Carcinoma More irregularities, dark features

8 L A M M P BLI Optical Technologies MI

9 L A M M P Mathematical Models

10 Background Radiative transport theory provides a widely accepted gold standard for validation in biomedical optics For general geometries, fully heterogeneous descriptions of turbid media, such as tissue, Monte Carlo has unique capabilities Naive simulation of RTE is rarely effective (more sophisticated, not necessarily intuitive methods are needed) L A M M P

11 Handouts Brief reading list Outline of topics covered L A M M P

12 L A M M P Reading List 1. The Monte Carlo Method in Atmospheric Optics by G.I. Marchuk et. al., Springer Series in Optical Sciences, Springer- Verlag, Neutron Transport Theory by B. Davison, Oxford Univ. Press, Nuclear Reactor Theory by G.I. Bell & S. Glasstone, Krieger Pub. Co., Wave Propagation and Scattering in Random Media by A. Ishimaru, Academic Press 1978, IEEE Press, Monte Carlo Principles and Neutron Transport Problems by J. Spanier and E.M. Gelbard, original ed. 1969, reprint ed. Dover, 2008 (to appear) 6. Monte Carlo Particle Transport Methods: Neutron and Photon Calculations, I. Lux and L. Koblinger, CRC Press, 1991

13 Short Course Topics Outline Monte Carlo Basics Equations of Radiation Transport Diffusion & Higher Order Approximations Keys to Efficient Use of Monte Carlo The Virtual Tissue Model Adaptive Monte Carlo Methods L A M M P

14 L A M M P Monte Carlo Modeling Natural Transport Applications Atmospheric Optics Nuclear Reactor Design/Engineering Biomedical Optics Radiation Therapy Planning Financial Modeling Important Numerical Applications Estimation of Integrals Solution of Matrix Equations (e.g., linear systems arising from discretization of PDEs and/or integral equations Solution of Integral Equations The MC method is very robust and can be used for phenomenological descriptions that incorporate new knowledge quite easily. Monte Carlo simulations account for well over 50% of all computer time worldwide. Optimizing their use is important.

15 The Virtual Tissue System: First Look Visualization Optimization Analysis Solver Problem Definition L A M M P

16 L A M M P ATK/VTS Relationships

17 Tissue Primitives : Towards a Virtual Tissue Model L A M M P

18 Why Monte Carlo? MC is a versatile method for producing answers by computer using random sampling Estimate π: 4 (14/18) = 3.11 L A M M P

19 L A M M P Estimate Integrals The same principle can be used to estimate I b = a f( x) dx and higher order integrals f(x) a b

20 L A M M P Simple Random Walk What is average number of steps (each uniform on [0,1]) to pass through unit slab?

21 L A M M P The Exact Solution Let X = # steps to escape. X = n means (u 1,,u n-1 )εt n-1 &(u 1,,u n ) ε T n so Pr{X = n} = 1/(n-1)! 1/n! = 1/n(n-2)! = f(n) E[X] = nf ( n) = e = n= 2 and σ = T n

22 Model Problem Characteristics Solution is an infinite sum. The MC solution does not truncate this sum! Instead, it picks out every term with correct probability by sampling. Thus, the only error is statistical; i.e., because # of samples is finite. These qualities make the MC method quite special. L A M M P

23 L A M M P Monte Carlo Estimate Tchebycheff s inequality: Pr{ estimate(n) e > kσ(n)} 1/k 2 n 3.84(100) 2 (.77/2.72) 2 = 3100 to achieve relative error ~ 1% with probability.95. n = 1000 gives est(1000) = 2.737

24 Doing Better: Importance Sampling Exaggerate large value of X (sample from g(n) rather than f(n) based on uniform steps but count nf(n)/g(n) instead of n for X. Try h(u) = const. exp(-ku) instead of h(u) = 1. With n steps chosen this way, correct weight factor is [h(u 1 )h(u 2 ) h(u n )] -1 which is 4 times as efficient as before. n = 1000 gave est(1000) = L A M M P

25 Random Walk Processes Consider the general (operator) equation (1) φ = Ҝφ + s where Ҝ < 1 (or even Ҝ m < 1 for some integer m). Then under rather general circumstances, (1) has a unique solution φ = (I Ҝ) -1 s = (I + Ҝ + Ҝ 2 + )s. These simple ideas are basic and relevant to what we do later. L A M M P

26 L A M M P Two Important Special Cases If Ҝ is an nxn matrix, H, and φ, s are n- dimensional vectors, (1) becomes x = H x + s. MC methods can be applied to many important matrix problems. Okten has recently shown that such methods compete very favorably with the best deterministic matrix solvers; e.g., based on conjugate gradient methods.

27 L A M M P Okten s Findings From G. Okten, SIAM J. Sci. Comp. 27, pp : For 0.9 hij = ; bi = i/ N N+ i+ j time to converge is

28 More from Okten The MC method uses sequential correlated sampling (discussed in final class) to achieve geometric convergence in a few iterations. Table 7.4 shows the norm of the rel. residual and # adaptive stages to converge. L A M M P

29 L A M M P Table 7.4

30 L A M M P More About Matrix Problems The study of MC methods for solving matrix problems introduces most of the important ideas and methods used in solving continuous problems. The main difference is that the phase space for matrix problems consists of the integers {1, N}, N = order of matrix, while for continous problems, the phase space is continuous and multidimensional, in general.

31 Linear Integral Equations When Ҝ is an integral operator, K, and ψ, S are continuous functions, (1) becomes Ψ(P) = (K ψ)(p) + S(P) = K(P,Q) Ψ(Q) dq + S(P) which provides a model for all of our RTE applications. L A M M P

32 Duality and Reciprocity When solving (1) φ = Ҝφ + s, often we need only an accurate value of one or more functionals <s*, φ>, where (2) φ* = Ҝ*φ* + s*. It follows easily from the definitions of the operator adjoint operation * that (3) <s*, φ> = < φ*, s>. L A M M P

33 L A M M P Important Connections When we estimate either <s*, φ> or < φ *, s> by MC, we do so in the context of a probability space Р and an estimator (random variable) ξ: Р R so that E[ξ] = ξ dμ = <s*, φ> and Var [ξ] is as small as possible. Here, Р consists of all random walks on the phase space Γ (which is {1,..., N} for matrix problems or part of R d for d-dimensional integral equations).

34 L A M M P Further Connections Then φ = Ҝφ + s gives rise to a natural random walk process {s, Ҝ} where s (suitably normalized) is used to create initial collision points in Γ, and Ҝ (suitably normalized) is used to move from state to state in Γ as well as to terminate the random walks (in a finite number of steps with probability 1 if Ҝ m < 1).

35 Still More Connections Now define P n (x) = Pr density of making collision n at x for n = 1,2, Then P n (x) = [Ҝ P n-1 ](x), n > 1 and P 1 (x) = Pr{x 1 = x} = s(x). Define a collision estimator on the space Р of random walks on Γ by X(x) = number density of collisions at x εγ L A M M P

36 Completing the Connections Then if an abstract collision density is defined by P(x) = E[X(x)], we find that (*) P(x) = s(x) + [Ҝ P](x). The solution of (*) is always accomplished in practice through the Neumann series representation (I Ҝ) -1 s = (I + Ҝ + Ҝ 2 + )s based on the assumption that Ҝ m < 1 for some integer m. L A M M P

37 L A M M P The Final Connections Finally, the entire theory can be carried through for importance sampling; i.e., based on replacing the analog source s by ŝ and the operator Ҝ by ˆK under quite general conditions. The validity of importance sampling is established through. E[ξ] = ξ dμ = [ξdμ/dν] dν and the generalized derivative dμ/dν serves to correct the otherwise biased results.

38 Monte Carlo Methods and Virtual Photonics Acknowledgements: For co-sponsorship: Bruce Tromberg (BLI and LAMMP Director) and Arthur Lander (CCSB Director & Chair, Dev. & Cell Biology) NIH P-41-RR01192 (LAMMP), NSF/DMS , UCOP (LANL) The core Virtual Photonics group : (Vasan Venugopalan, Carole Hayakawa, Katya Bhan, David Cuccia, Albert Cerussi, Tony Durkin) Rong Kong, Martin Ambrose and the entire Friday morning M&C gang L A M M P