FMM Preconditioner for Radiative Transport

Transcription

1 FMM Preconditioner for Radiative Transport SIAM CSE 2017, Atlanta Yimin Zhong Department of Mathematics University of Texas at Austin In collaboration with Kui Ren (UT Austin) and Rongting Zhang (UT Austin) March 1, 2017 Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

2 Introduction of Radiative Transport Radiative transport describes the propagation of particles through a medium which is affected by absorption, emission, scattering. It has a wide variety of applications in optics, astrophysics, remote sensing, nuclear engineering, biomedical imaging. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

3 Here we consider following steady-state equation of radiative transport in medium D Ă R d, upx, vq is the density of particles at x along unit direction v, ż v upx, vq ` pσ s pxq ` σ a pxqqu σ s pxq K pv, v 1 qupx, v 1 qdv 1 ` qpxq S d 1 where v P S d 1, x P D, σ a is absorption coefficient, σ s is scattering coefficient. K pv, v 1 q is scattering phase function, describes the probability of each particle altering direction from v 1 to v. qpxq is certain isotropic source. For simplicity, we call σ t σ s ` σ a be extinction coefficient and assume there is vanishing incoming boundary conditions. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

4 Numerical methods for Radiative Transport Solutions to radiative transport in general needs enormous of computing efforts, which is due to high dimensionality of the solution space. In recent decades, some numerical methods are developed for solution of radiative transport. 1 Approximation (P n, SP n methods) : only accurate for scattering dominated medium. 2 Discrete Ordinates Method : high dimensionality, ray effect. 3 Stochastic-based method(monte-carlo) : computational cost. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

5 Integral Representation for Isotropic case Under isotropic setting, K pv, v 1 q 1 VolpS d 1 q. Let Φpxq ż 1 VolpS d 1 upx, vqdv q S d 1 then equation is equivalent to a linear transport equation v u ` σ t u σ s Φpxq ` qpxq Rpxq Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

6 Integral Representation for Isotropic case Along v, it is 1D transport (ODE), the solution for this equation is known upx, vq ż τ px,vq 0 expp ż ρ 0 σ t px ρ 1 vqdρ 1 qrpx ρvqdρ Figure: Illustration for 1D transport along v Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

7 Integral Representation for Isotropic case Integrate upx, vq over S d 1 and take average, Φpxq ż 1 VolpS d 1 q S d 1 ż τ px,vq If we take y x ρv, and let 0 Epx, yq expp expp ż ρ 0 ż ρ 0 σ t px ρ 1 vqdρ 1 qrpx ρvqdρdv σ t px ρ 1 vqdρ 1 q, where E is equivalent to a line integral for σ t from x to y. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

8 Integral Representation for Isotropic case Put E back into the equation, we obtain Φpxq ż 1 VolpS d 1 q S d 1 ż τ px,vq where the dρdv actually fits polar coordinate. 0 Epx, yqrpyqdρdv, Figure: Illustration of polar coordinate Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

9 Integral Representation for Isotropic case We transform from polar coordinate back to Cartesian dy ρ d 1 dρdv, and replace R with σ s Φ ` q, observing ρ x y, Φpxq ż 1 Epx, yq VolpS d 1 q D x y d 1 pσ spyqφpyq ` qpyqqdy we obtain an integral equation about Φ. Observation 1: This equation does not contain angular variable v. Observation 2: This work does not require q to be isotropic. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

10 Integral Representation for Isotropic case In terms of operator, we can write as where integral operator Kf pi KΣ s qφ Kq (1) ż 1 Epx, yq VolpS d 1 qpyqdy (2) q D x y d 1 Remark: K is compact, thus I KΣ s is Fredholm. Since when q 0, this equation only admits zero solution, by Fredholm alternative, I KΣ s is invertible. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

11 Algorithm for Integral Equation Discretize the integral equation, take Ψ pφpx 1 q, Φpx 2 q,..., Φpx n qq t, then Ψ KpΣ s Ψ ` Qq where Σ s diagpσ s px 1 q,..., σ s px n qq, and Q pqpx 1 q,..., qpx n qq t. Finally we obtain pi KΣ s qψ pkqq where matrix K is K ij 1 Epx i, x j q VolpS d 1 q x i x j d 1 w ij where w ij are weights to control the order of accuracy. For singularity part, we handle that separately. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

12 Algorithm for Integral Equation For linear system pi KΣ s qψ pkqq, 1 Observe that when x i x j increases, K ij decreases dramatically, which means (far) off-diagonal parts of K could be small (low rank), thus Krylov subspace method (e.g. GMRES) might work well for such linear system. 2 During each iteration of GMRES, we have to apply operator I KΣ s multiple times, in general, if all entries are known, the matrix-vector multiplication is OpN 2 q complexity, which could be very expensive (running serially) when N is large. 3 To calculate the entries, we need to mention the computational cost of calculating Epx i, x j q. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

13 Algorithm for Integral Equation 1 Due to the low rank property of K, we can adopt interpolative fast multipole method (FMM), such as Chebyshev polynomial, to accelerate the application of operator pi KΣ s q. 2 OpNq time complexity of FMM for calculation of pi KΣ s qψ. Observation 3: For piecewise defined σ a and σ s, the calculation of Epx i, x j q does not change the time complexity due to FMM s hierarchical structure. Observation 4: The total time cost is OpM 2 Nq, M is iteration number. Observation 5: Since GMRES will re-use the same operator, thus caching all expensive information will be beneficial. Observation 6: It is symmetric operator, can reduce storage in half. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

14 Numerical Experiments in 2D We select two source functions as examples below. In following experiments, σ a 0.2 and σ s 2.0, 5.0, 10.0 respectively. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

15 Strong scaling for FMM Solution Figure: Left: Time (s) for preparation. Right: Time(s) for GMRES iterations Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

16 Error from FMM (first source) Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

17 Error from FMM (second source) Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

18 Anisotropic Case: An Intuitive Approach Since our formulation highly depends on integral formulation of the solution, thus anisotropic scattering might not be the most efficient. We consider Henyey-Greenstein phase function p HG pv v 1 q, that ż v upx, vq ` pσ s pxq ` σ a pxqqu σ s pxq p HG pv v 1 qupx, v 1 qdv 1 ` qpxq S d 1 where p HG 1 4π 8ÿ p2k ` 1qg k P k pv v 1 q k 0 and if we expand the formula further p HG pµ, θ, µ 1, θ 1 q 8ÿ m 0 l m 8ÿ χ m l Pl m pµqpl m pµ 1 q cos mpθ θ 1 q Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

19 Anisotropic Case: An Intuitive Approach Because cos mpθ θ 1 q cos mθ cos mθ 1 ` sin mθ sin mθ 1, we can see the decoupling of v and v 1 is completed here. Let Φ m l pµ, θq P m l pµq cos mθ, Ψ m l pµ, θq P m l pµq sin mθ and let ż Cl m pyq upy, µ, θqφ m l pµ, θqdµdθ żs 2 Sl m pyq upy, µ, θqψ m l pµ, θqdµdθ S 2 then multiply the integral formulation by Φ n r and Ψ n r on both sides, we can obtain an integral equation system on C n r and S n r. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

20 Anisotropic Case: An Intuitive Approach The system about pc n r, S n r q is written as, ż Cr n pxq ż Sr n pxq D D Epx, yq σ s pyq ÿ χ m l Φ n r pφ m l Cl m pyq ` Ψ m l Sl m pyqq ` Φ n r qpyq dy l,m Epx, yq σ s pyq ÿ χ m l Ψ n r pφ m l Cl m pyq ` Ψ m l Sl m pyqq ` Ψ n r qpyq dy l,m where all θ, µ can be transformed into Cartesian coordinate with simple algebra. Observation 7: This system is truncated at m L for some small L. Observation 8: This system is symmetric. Observation 9: The time cost is OpM 2 L 2 Nq, where M is iteration number. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

21 Numerical Experiments in 2D on δ-eddington Approximation We adopt one of our previous source function to run simulation under setting µ a 0.2 and µ s 5.0, and g 1 g{p1 ` gq ď 0.5, we take Figure: averaged solution, from left to right, g , Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

22 Numerical Experiments in 2D on δ-eddington Approximation The integral operator system is 3 by 3, but only 5 entries have to be calculated. Figure: averaged solution, from left to right, g , Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23

23 Summary and More 1 This method reduced the dimensionality of the problem. 2 For isotropic scattering media, we propose a fast algorithm for radiative transport equation, which can be used as a good preconditioner or approximation to the actual solution, the time complexity is OpNq. 3 This method does not require specific geometry or mesh. 4 And this method can extend to anisotropic scattering media with extra cost but solves in linear time and can be highly parallelized. 5 3D isotropic parallel solver is implemented. 6 Future work might be on optimization of algorithm, time dependent problems and non-intuitive method under anisotropic scattering, etc. Yimin Zhong (UT Austin) FMM Preconditioner for Radiative Transport March 1, / 23