Compressive analysis applied to radiation symmetry evaluation and optimization for laser-driven inertial confinement fusion

Computer Physics Communications 185 (2014) 459–471

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Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc

Compressive analysis applied to radiation symmetry evaluation and optimization for laser-driven inertial confinement fusion Yunbao Huang a , Shaoen Jiang b,∗ , Haiyan Li c,∗ , Qifu Wang a , Liping Chen a a

CAD Center of Huazhong University of Science & Technology, Wuhan, 430074, China

b

Laser Fusion Research Center, China Academy Engineering Physics, Mianyang, 621900, China

c

School of Naval Architecture & Ocean Engineering, Huazhong University of Science & Technology, Wuhan, 430074, China

article

info

Article history: Received 27 January 2013 Received in revised form 27 August 2013 Accepted 29 August 2013 Available online 19 September 2013 Keywords: Compressive analysis Radiation symmetry Optimization Inertial confinement fusion

abstract Having as symmetric a radiation drive as possible is very important for uniformly imploding the centrally located capsule in laser-driven Inertial Confinement Fusion (ICF). Usually, intensive computation is required to analyze and optimize the radiation symmetry in ICF. In this paper, a novel compressive analysis approach is presented to efficiently evaluate and optimize the radiation symmetry. The core idea includes (1) the radiation flux on the capsule for symmetry evaluation is transformed into frequency domain and weighted to obtain a sparse and orthogonal representation, (2) the sparse coefficients reflecting the radiation flux distribution are accurately and efficiently recovered from far less samples on the frequency domain, i.e. [0, 2π ) × [0, π] through ℓ 1-norm optimization, which greatly improves the efficiency of radiation symmetry evaluation and optimization for the design of physics experiments in the laser-driven ICF, and (3) the sparsity level to recover the sparse coefficients is adaptively determined with a one-dimensional optimization procedure for accurate and efficient compressive analysis. Finally, two examples on current laser facilities are utilized to demonstrate the evaluation accuracy, robustness and computation efficiency of compressive analysis approach. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Inertial Confinement Fusion (ICF) is a process in which nuclear fusion reactions are initiated by heating and compressing a fuel capsule containing a mixture of Deuterium and T ritium which can be got from seawater. Currently, research on ICF is very active and fruitful since ICF can potentially generate clean, safe, and economic energy without generating pollution. The indirect-laser-driven approach to ICF is believed to have promise [1–3], in this approach, the lasers are directed into a cylindrical cavity around the capsule, and converted into X-rays to radiate and drive the implosion of the capsule. The regions where laser energy deposits remain hotter than indirectly heated areas, and laser entrance and diagnostic holes also introduce additional asymmetries in the radiation flux on the capsule, which will result in asynchronous shock and asymmetric implosion. Hence, such radiation flux asymmetries should be kept at a lower level to get a uniform implosion, e.g. 2% has been reported in [4]. Thus, radiation symmetry evaluation and optimization is very important in the indirect-laser-driven ICF. The radiation symmetry evaluation is related to the laser– plasma interactions and transport of X-rays from the hohlraum

∗

Corresponding authors. Tel.: +86 02787541974. E-mail addresses: [email protected] (S. Jiang), [email protected] (H. Li).

0010-4655/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cpc.2013.08.024

wall to the capsule, which involves the solving of kinetic equations and hydrodynamics equations, which are very complex, and entail certain difficulties. In practice, simple mathematical models such as view-factor codes are used to compute the radiation flux on the capsule, especially for the preliminary design and optimization of thermonuclear target structure and shape [5,6], and play a complementary role to atomic and hydrodynamic codes [7,8]. To evaluate radiation symmetry on the capsule with the viewfactor model, we need to facet the surface patches of cylindrical hohlraum into quadrangular or triangular elements, and choose an appropriate analysis model such as equivalent energy model [7–12] to compute the radiation flux on the capsule, which forms non-linear equations of all the discrete elements. Then the radiation symmetry can be evaluated by transforming the radiation flux into the spherical harmonics (SPH) domain and calculating the symmetry parameters such as Root Mean Squared Error (RMSE) σ0 and Legendre non-uniformities (e.g. P1 , P2 , . . ., and P10 [4]). The radiation flux for any element on the capsule is related to all the elements in the view for the view-factor model. When faceting refinement increases, the evaluation accuracy of radiation symmetry tends to be higher. However, non-linear equations will mushroom greatly, which leads to a great challenge on efficiently finding an appropriate solution for such large-scale equations. To leverage the intensive computation of large scale equations solving, a two-section model is presented [13,14], in which the incident laser

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energy is uniformly distributed onto the discrete elements of cylindrical hohlraum by considering absorbing or losing energy, and only the radiation flux from the hohlraum to the capsule is computed for radiation symmetry evaluation. Such simplification enables a great reduction of intensive computation since there is no requirement on finding a solution for large-scale equations. Nevertheless, the radiation flux of all the elements on the capsule still needs to be computed and involves intensive computation when the discretization resolution is small enough to achieve comparable evaluation accuracy. In addition, the radiation symmetry is usually optimized to achieve a highly uniform radiation on the capsule by finding appropriate values of the target and physical parameters of incident lasers. Thus, such radiation symmetry evaluation needs to be iterated, and the computation time tends to sharply increase when the iteration times are multiplied. Hence, the radiation symmetry evaluation and optimization needs to be accelerated to enable the rapid design and optimization of ICF experiments. As described above, we need to transform the radiation flux into the frequency domain, i.e., spherical harmonics domain, and then evaluate the radiation symmetry. In such process, we find that the SPH coefficients are very sparse and no more than twenty orders or 21 × 21 = 441 of them are larger than zero, which are enough to accurately describe the radiation flux. Since the radiation flux over the spherical harmonics domain is sparse, it means that only a fraction of elements on the capsule may be enough to recover such sparse coefficients. Meanwhile, there are a series of simulation optimization approaches [15] such as gradient-based [16], random search [17], and kriging based response surface [18] to reduce iteration times. This motivates us to find a compressive analysis method so that it can be combined with available simulation based optimization approaches to reduce the computation time for efficient radiation symmetry evaluation and optimization. Recently, a novel theory named compressive sampling is proposed by Candès and Donoho in signal-processing field [19–21], in which a signal can be perfectly recovered from a fraction of samples, in far less than Nyquist sampling rate, by exploiting its sparsity or compressibility. This technique has gained many applications such as Medical Imaging [22], Analog-to-Information Conversion [23], and Computation biology [24], and computer graphics [25]. Since the radiation flux is sparse over the spherical harmonics domain, which means that sparse coefficients can be accurately recovered from a few compressed samples without intensive radiation computation of all the elements. In addition, compressed sampling and radiation recovering for radiation symmetry evaluation are very beneficial to radiation symmetry optimization since there needs repeated radiation flux computation. To efficiently evaluate and optimize the radiation symmetry on the capsule, we need to (1) elaborately choose a sparse and orthogonal basis to represent the radiation flux on the capsule, (2) determine an appropriate sparsity level for an unknown radiation flux and preplan enough samples to efficiently reconstruct the radiation, and (3) efficiently reconstruct sparse coefficients for radiation symmetry evaluation from compressed samples. In this paper, we present a novel compressive analysis approach for radiation symmetry evaluation and optimization in the laserdriven ICF, which achieves the following:

• The spherical harmonics is selected and weighted to sparsely and orthogonally represent the radiation flux on the capsule inside a cylindrical hohlraum, and only less than 5% elements are required to accurately recover sparse coefficients through ℓ 1-norm minimization (the maximal reconstruction error is no more than 3×10−5 ), which significantly improves the efficiency of radiation symmetry evaluation. • The sparsity level S can be adaptively determined for accurate radiation evaluation without any a-priori information on radiation distribution by transforming it into a one-dimensional unconstrained optimization problem to find an optimal sparsity S, so that enough number of samples are pre-planned.

• The compressive analysis combined with current simulation optimization approaches can be employed to efficiently and accurately find a near optimal solution for ignition target design in the laser driven ICF experiments (the distance between the optimal point and the true point is no more than 2 × 10−3 ). The remainder of this paper is organized as follows. Section 2 reviews related work on compressive analysis for radiation symmetry evaluation and optimization. Section 3 presents a two-section radiation symmetry evaluation model. Section 4 introduces the mathematic background of compressive analysis. Section 5 gives a compressive analysis framework for radiation symmetry evaluation and optimization. Section 6 validates and analyzes the efficiency of the presented approach. This paper concludes in Section 7. 2. Literature review Radiation symmetry evaluation and optimization is one of the most important issues in the indirect laser driven ICF [26,27] to achieve a more uniform radiation environment around the capsule, and is always an active research topic in the physics of plasma [7,13]. Usually, the radiated surfaces are first discretized into smaller elements, incident laser energy is then mapped on the elements for radiation computation, and finally, the radiation flux on the capsule is computed and transformed into the spherical harmonics domain to evaluate radiation symmetry. In such process, a lot of elements need to be generated to achieve a prescribed evaluation accuracy, which will result in a large number of non-linear equations and intensive radiation computation [11,12]. Therefore, radiated surfaces, laser-spots and diagnostic holes are often assumed to be rotationally symmetric and simplified to largely reduce the number of elements [11–14]. Typically, only one-dimensional models are built and the efficiency of radiation symmetry evaluation can be improved. However, the evaluation accuracy of radiation symmetry is limited, and the computed radiation flux is usually not consistent with the resulting distribution of ICF experiments. Hence, three-dimensional geometric elements based radiation flux computation has been discussed in [7,9,10]. Nevertheless, when the size of discrete element reduces by half, the number of elements will be doubled, and the resulting radiation flux computation will increase by a factor of 4, which may take more than half an hour for radiation symmetry evaluation, or one day for even single design parameter optimization. Therefore, a new analysis approach is very essential to accelerate the radiation symmetry evaluation and optimization. In the signal processing field, a new sampling scheme, i.e. compressive sampling, has been proposed in [19,20] to capture and represent compressible signals at a rate significantly below the Nyquist sampling rate [21]. Such scheme has been used in medical image processing [22], or A/D signal transformation [23], computing biology [24], and computer graphics [25]. To apply compressive sampling in radiation symmetry evaluation, we need to determine an appropriate sparse representation, sparsity level, enough samples pre-planning, and sparse coefficients reconstruction algorithm. Nevertheless, how to utilize compressive sampling in the radiation symmetry evaluation and optimization is new. In this paper, we first introduce the background of radiation symmetry evaluation, optimization and compressive analysis, then present a compressive analysis framework for radiation symmetry evaluation and optimization, and finally validate the approach with two examples running on current Shenguang laser facilities [28]. 3. Background of compressive analysis for radiation symmetry in the ICF In this section, radiation symmetry evaluation, optimization, and compressive analysis are discussed below to enable efficient

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inner surfaces of the cylindrical hohlraum, in which radiated surfaces are separated into laser-spot and non-laser-spot areas. Let SS0 be the incident radiation flux for the elements in laser-spot areas, Ss be the radiation flux for those elements, SR be the radiation flux of all elements, and SW be the radiation flux of the elements in nonlaser-spot areas. Then we have N 

SS0 AreaS0 = ηPL ,

SS = SR +

i =1

1 + αw 2

SS0 ,

and

(2)

SW = SR , Fig. 1. Configuration of a cylinder-to-sphere system.

evaluation and optimization of radiation symmetry on the fuel capsule in the cylindrical hohlraum. 3.1. Radiation symmetry evaluation and optimization on the capsule To evaluate the radiation symmetry on the capsule, we need to (1) recognize radiated surfaces and then discretize them into smaller elements for radiation analysis, (2) compute the incident laser spots on the hohlraum and map them onto discrete elements, and (3) calculate the radiation flux on the fuel capsule for symmetry evaluation and optimization. Since feature recognition and mesh generation techniques have been extensively researched [29], we here mainly discuss the model of radiation flux computation for the cylindrical cavity to spherical capsule system in ICF experiments. 3.1.1. Radiation energy computation With available discrete elements and mapping energy at laserspot areas, we can utilize a one-dimensional model to compute the radiation flux for each element [11,12]. However, there still include non-linear energy equivalent equations among discrete elements, and the computation efficiency is limited since solving the non-linear equations requires many iterations. Huang et al. [13,14] have presented a two-section radiation flux model to accelerate such computation, in which radiated surfaces are divided into laser-spot, re-emission and hole areas, an energy equivalent model is built to describe the energy distribution on such separated areas. The radiation flux and symmetry parameters on the capsule can then be easily computed through a view-factor function. The resulting radiation symmetry approaches the actual ICF experiments very well, and is selected as the radiation computation model in this paper. Fig. 1 gives a configuration of cylinder-to-sphere ICF implosion experimental system. High energy lasers are injected into the cylindrical hohlraum through two entrance holes with a diameter Leh, and laser spots are formed on the inner radiated surfaces defined by an incident angle α and a solid angle θ . The solid angle θ can be obtained from

θ = 2 arctan



1 2f



,

(1)

where f is the ratio of the length to width of the focus. Two holes defined by Ldh and Wdh are constructed on the cylindrical hohlraum for inner radiation temperature diagnostics. A spherical capsule is centered in the cylindrical hohlraum to receive the radiation energy. There are two kinds of experimental facilities in China now, Shenguang-II and Shenguang-III-YX, which have 8 laser spots and similar structure for the cylinder-to-sphere systems. Such laserspots can be mapped on the discrete elements of the hohlraum. With the laser spots definition, Huang [13,14] proposed a twosection energy equivalent model to compute radiation flux on the

where AreaS0 is the area of laser-spots, η is the conversion efficiency of laser-to-X-ray, PL is the energy of incident lasers, and αw is the albedo of cylindrical surfaces. The incident laser energy is separated into the absorbed energy at the laser-spot areas, the irradiated energy at non-laser-spot areas, the energy lost at entrance holes and diagnostic holes, and the energy absorbed by the spherical capsule. Such energy equivalence can be expressed as

ηPL =

1 + αw 2

ηPL [fh + fC (1 − αC )]

+ (1 − αw )

η PL 2

+ SR At (1 − fh ) · [fh + fC (1 − αC )]

(1 − αw ) SR At (1 − fh ) , (3) αw where fh = (Ah + Adh ) /At , fC = (AC ) /At , Ah is the area of en+

trance holes, Adh is the area of diagnostic holes, AC is the area of the spherical capsule, At is the total area of inner surfaces of cylindrical hohlraum, and αC is the albedo of the spherical capsule. From Eqs. (2) and (3), we can get the radiation flux SS at the laser-spot areas, and SR at non-laser-spot areas. Then, we can map SS and SR onto the discrete elements of the hohlraum, and compute the radiation flux on the spherical capsule from the cylindrical hohlraum through a view-factor function. 3.1.2. View-factor function for radiation flux computation on the capsule As shown in Fig. 2a, eight laser beams with the energy of 2 kJ are injected into a cylindrical hohlraum through two entrance holes in 1 ns. The dimension of the cylindrical hohlraum is 800 µm × 1380 µm, the diameter of the capsule centered in the cylindrical hohlraum is 230 µm. Then we can get SS and SR from Eqs. (2) and (3) and the radiation flux on the spherical capsule through a viewfactor function:

 V ri , rj =







    · nj · Pi − Pj ,  4 π Pi − Pj 

ni · Pj − Pi

(4)

where ri , rj are two elements of the spherical capsule and the cylindrical hohlraum, Pi , Pj are their centroids, and ni , nj are their normal (see Fig. 2b). With the view-factor function between two elements ri and rj , we can compute the radiation flux SC (ri ) for the element ri on the capsule from the cylindrical hohlraum by SC (ri ) =

n 

SH rj V (ri , rj )σ Aj ,

 

(5)

j =1

where SC (ri ) is the radiation flux of the element ri on the capsule, n is the  number of all the elements on the cylindrical hohlraum, SH rj is the radiation flux of the element rj on the cylindrical hohlraum which can be obtained from the results of Section 3.1.1, and σAj is the area of the element rj . 3.1.3. Radiation symmetry evaluation With the radiation flux of all the elements on the spherical capsule through Eqs. (5) and (6), we can evaluate the radiation symmetry i.e. the uniformity of the radiation flux received from the

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(a) Incident lasers and radiation flux on the capsule.

(b) Two elements for View-factor computation.

Fig. 2. Radiation flux on the spherical capsule from a cylindrical hohlraum.

cylindrical hohlraum, so that the capsule can be homogeneously  compressed and ignited. Let θj , ϕj be the spherical coordinate of rj , and SC θj , ϕj be the computed radiation flux of the element rj .





We can transform SC θj , ϕj into the spherical harmonics domain by





+∞  l 

SC θj , ϕj =





Clk Ylk

l=0 k=−l

   (θ , ϕ) 

,

(6)

(θj ,ϕj )

where Ylk (θ , ϕ) is the spherical harmonics, l is its order, and Clk is its coefficient. In Eq. (6), Ylk (θ , ϕ) and Clk can be obtained from

 Ylk (θ, ϕ) = and

Clk

2l + 1 (l − k) ! 4π

2π



(l + k) ! π



= 0

Plk (cos θ) eikϕ ,

SC (θ , ϕ)

(−l ≤ k ≤ l) (7)

Ylk

(θ , ϕ) sin θ dθ dϕ,

0

where Plk (cos θ ) are associated Legendre polynomials. Since the capsule is subdivided into a set of smaller elements {ri , i = 1, 2, 3, . . . , m} for its radiation flux computation, Clk can be calculated by Clk =

m 

SC (θi , ϕi ) Ylk (θi , ϕi ) sin θi ∆θ ∆ϕ,

(8)

i=1

where ∆θ and ∆ϕ are the minimal angles to subdivide the capsule over the domain Ω (θ , ϕ) = [0, π ] × [0, 2π ), and m is the number of discrete elements on the capsule.   With the spherical harmonics Ylk (θ , ϕ) and its coefficient Clk , we can evaluate the radiation symmetry for the spherical capsule through a Root Mean Squared Error (RMSE) σrms , and non-uniform radiation parameters σ1 , σ2 , . . . , σ10 . Such radiation symmetry parameters can be obtained from

 1 4π r 2

σrms = 

 2

IC − SC rj

j =1

m  l  

σAj ,

IC 1 4π r 2

σl =



m  

and (9)

  2 Clk Ylk θj , ϕj σAj

j=1 k=−l

,

IC

2 where IC = , and r is the radius of the j=1 SC rj σAj /4π r spherical capsule. Let n be the number of discrete elements on the cylindrical hohlraum, the radiation computation complexity of all the elements on the capsule will be O(m × n). We usually keep the minimal length of elements no more than 10um to obtain a comparable accuracy (2% as described in [4]) for the radiation symmetry evaluation. Then m = 144 × 72 = 10368, n = 138 × 144 + 2 × 80 × 144 = 42912, and the radiation computation times will be m × n ≈ 4.4 × 108 , which takes almost four minutes for the model in Fig. 2 on the DELL computer OPTIPLEX790. In addition, radiation symmetry from the cylindrical hohlraum to the capsule is related to their geometric structure and location, size and intensity of incident lasers, which may significantly

m

 

increase the radiation asymmetry and lead to a larger Rayleigh– Taylor instability on the capsule [4]. Therefore, we need to optimize the hohlraum-to-capsule system so as to reduce the radiation asymmetry as far as possible. 3.1.4. Radiation symmetry optimization Radiation symmetry optimization scans design parameters such as geometric parameters defining the hohlraum and the capsule, and the locations, size or intensity of incident lasers over the design domain to find an optimal geometric structure and laser configuration for the hohlraum-to-capsule system. Such optimization problems can be defined as

find: L , or D , . . . , Leh H C    arg min F (D, LH , DC , . . . , Leh)   s.t.: lmin ≤ LH ≤ lmax dmin ≤ DC ≤ dmax    · · · hmin ≤ Leh ≤ hmax ,

(10)

where D is the diameter of the cylindrical hohlraum, LH is the length of the cylindrical hohlraum along its axis, DC is the diameter of the capsule, and Leh is the diameter of the laser entrance holes. F (·) is the simulation box or process to evaluate the radiation symmetry as described above. In the laser driven ICF, D is usually a prescribed value according to the experimental feasibility of laser facilities (for example, the diameter D = 0.8 mm for Shenguang II laser facility, and D = 1 mm for Shenguang III-YX laser facility), and LH , DC , and Leh are the design variables to be determined. To get an optimal design for LH , DC , and Leh, we need to subdivide their intervals, and evaluate and compare the radiation symmetry parameters such as σrms , σ1 , σ2 , . . . , σ10 to obtain an optimal design point. Assume the length of a cylindrical hohlraum LH varies from 1 to 2 mm with a minimal subdivision interval 10 µm. If we need 100 radiation computations and symmetry evaluations to scan a single parameter, it takes more than 8 h to perform the scan, since the radiation flux computation on the capsule and symmetry evaluation for any value need more than 4 min (described in Section 3.1.3). In addition, more computation time is required if more design parameters are scanned simultaneously. Hence, an efficient approach is essential to compress or accelerate such analysis or optimization process.



3.2. Compressive sampling Compressive sampling was initiated in 2006 [19,20], whose core is to recover a sparse signal from a very few, non-adaptive, linear measurements by convex optimization. Assume a sparse signal x with N dimensions, which means that x ∈ RN . If x is S-sparse, we can find an orthogonal basis ϕ so that only a few coefficients (or c = ϕx) are larger than zero, which means that ∥c∥0 = S ≪ N. Given a sampling matrix A ∈ RM ×N , we can get M (S ≤ M ≪ N ) measurements y ∈ RM through A from x, which can be expressed as y = Ax.

(11)

Y. Huang et al. / Computer Physics Communications 185 (2014) 459–471

463

Due to c = ϕx, we can get y = Ax = Aϕ−1 c.

(12)

Since ∥c∥0 = S, the sparse vector c can be recovered from the M measurements by arg min ∥c∥0 , c

subject to y = Aϕ−1 c.

(13)

From Eq. (13), we can see that the optimization problem is NPhard [19,20] due to the combinatorial search in the sense of the ℓ0 -norm. Therefore, the ℓ1 -norm is often chosen as a substitute for the ℓ0 -norm to reconstruct the sparse vector c [19]. Then, such minimization problem can be formulated as arg min ∥c∥1 ,

subject to y = Aϕ−1 c.

(14)

Such formulation holds if the following conditions are satisfied: (1) the signal x is sparse over the orthonormal representation basis ϕ, and (2) the measuring matrix A is incoherent to the basis ϕ. If x does not show its sparsity, we need to reduce x by pursuing an appropriate orthonormal basis. There is some work on such sparse basis pursuit problem [30–32], in which the incoherence is validated through a Restricted Isometry Property (RIP) 1−ε ≤

∥Av∥2 ≤ 1 + ε, ∥v∥2

Fig. 3. Following chart of compressive analysis for radiation symmetry evaluation and optimization.

(15)

where v is any vector of the orthonormal basis matrix ϕ, and ε(ε ∈ (0, 1)) is a small and positive number, which describes the degree of coherence for the measuring matrix A and the basis matrix ϕ. A larger ε means there is strong coherence between A and ϕ. With the sparse basis matrix ϕ and incoherent measuring matrix A, we can get the measurement y and then recover the sparse vector c through a convex optimization 1

2

Aϕ−1 c − y + λ ∥c∥1 , (16) 2 2 where λ is a scalar, generally λ = 1. Such problems can be solved with convex optimization algorithms such as interiorpoint [33], iterative thresholding [34], and combinatorial search algorithms [35,36], i.e. Orthogonal Matching Pursuit (OMP). The OMP algorithm is very fast and often sub-linear, and is usually utilized to reconstruct sparse coefficients in compressive sampling. As described above, the radiation flux over the spherical harmonics domain has been shown to be sparse, and very limited samples are required for compressive sampling to efficiently recover the signal. It is possible to achieve an accurate and efficient radiation symmetry evaluation and optimization via compressive analysis of radiation symmetry. arg min

4. Compressive analysis for radiation symmetry evaluation and optimization In this section, compressive analysis for accelerating radiation symmetry evaluation and optimization is discussed. The overview of compressive analysis of radiation symmetry evaluation and optimization is shown in Fig. 3. As shown in Fig. 3, we select a simulation optimization approach such as simulated annealing [17,18] to generate points on design domain. For any design point, we can reconstruct CAD models of cylindrical hohlraums and capsules on parametric feature based modeling system such as SolidWorks, and recognize radiated surfaces in the hohlraum with feature recognition approaches as described in [29]. Then the elements can be uniformly generated and mapped with incident laser energy. From the generated elements and mapped laser energy, we can compressively analyze radiation symmetry and get an optimal design by comparing the resulting radiation symmetry of all generated points for the design of ICF experiments. In such procedure, compressive analysis of radiation symmetry is the key since the radiation flux of all discrete elements involves intensive computation, which is as described below.

Fig. 4. Sparsity of radiation for the capsule over the spherical harmonics domain.

4.1. Compressive analysis of radiation symmetry As discussed in Section 3.1.3, radiation symmetry evaluation includes three steps: (1) radiation flux computation of all the elements, (2) radiation flux transformation to get the coefficients over the spherical harmonics domain, and (3) radiation symmetry evaluation with calculated coefficients. Therefore, we select spherical harmonics as the basis for radiation symmetry evaluation and check its orthogonality, and then determine sampling and reconstruction methods in this section to efficiently recover the coefficients of spherical harmonics for compressive analysis of radiation symmetry. 4.1.1. Sparsity of the radiation flux and orthogonality of the spherical harmonics As described in Section 3.1.3, radiation symmetry evaluation is related to the coefficients over the spherical harmonics (SPH) domain for the spherical capsule inside the hohlraum. Therefore, we need to transform the radiation flux computed by a view-factor function into the SPH domain. For the radiation flux computed in Fig. 2, we transform it into the SPH domain, and get the SPH coefficients as follows. In Fig. 4, the radiation flux on the capsule is transformed into SPH domain, and (l + 1)2 SPH coefficients (l is the order and varies from 0 to 20) are extracted and employed to recover the radiation flux on the capsule by inverse transformation. The inverse transformation can be denoted as xl = ϕ−1 cl , where xl is the recovered radiation flux on the capsule from the SPH coefficient vector cl , the dimension of xl is the number of all the discrete elements on the capsule and is denoted as m, the dimension of ϕ is

T

m × (l + 1)2 , and c = C00 , C1−1 , C10 , C11 , . . . , Cll with dimen2 sion N = (l +  1) . We definea maximum recovering error as l   max xi − xˆ i , i = 1, 2, . . . , m (xˆ is the radiation flux of all the



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elements computed by the  view-factor function), and a root-mean-

2

squared error (RMSE) as − xˆ i /m. Then the maximum i=1 relative ratio of SPH coefficients is calculated by

N 

xli



 k   0   0   C  / C  , C  ̸= 0 l  00  0 l = 0 , 1 , 2 , . . . , l , − l ≤ k ≤ l . (17) ( ) 0, C0  = 0,

Two kinds of errors and the maximum relative rate are evaluated and shown in Fig. 4b for l from 0 to 20. From Fig. 4b, we can see that the maximum relative ratio of SPH coefficients is no more than 0.00035, and the recovering errors including the maximum error and RMSE approach zero (<1.0 × 10−5 ) when l = 20. Therefore, the radiation flux on the capsule is sparse over the SPH domain, and the SPH can be utilized as the sparse basis for compressive analysis of radiation symmetry. Let S bethe sparsity   level  of radiation flux, and is denoted as S = ∥c∥0 :  (l, k), Clk  ≤ δ  ≤ S, where δ is non-zero and a positive factor. In fact, S characterizes the degree to which a harmonic polynomial can be well approximated by its S most significant coefficients. As discussed in compressive sampling, the sparse basis needs to be orthonormal. Hence, we validate the orthonormality of ϕ by b



ϕi (x)ϕj (x)dx = δij

(18)

a

where ϕi and ϕj are the i, j-th column of the SPH matrix ϕ, [a, b] is the interval defining ϕ, and δij is the Kronecker delta function, and can be defined by

δij =



1 i=j 0 i= ̸ j.

(19)

Let ϕi = Ylk , we find that it is orthonormal when the spherical harmonics basis is weighted by a scalar sin θ , and have the following result π

 0

2π



T

radiation flux y = SC (r1 ) SC (r2 ) · · · SC (rM ) can be obtained with the view-factor function for radiation flux reconstruction.



max Clk

=

can apply uniform random sampling to get such M measurements on Ω since the uniform random sampling matrix A is incoherent to the orthogonal basis matrix ψ. And then M uniform random samples {(θi , ϕi ), i = 1, 2, . . . , M } from m discrete elements and their

′

Ylk (θ , ϕ) Ylk′ (θ , ϕ) sin θ dθ dϕ = δll′ δkk′ .

0

(20)

T

radiation flux y = SC (r1 ) SC (r2 ) · · · SC (rM ) computed by the view-factor function described in Section 3.1.1, we can formulate the coefficient vector c in terms of the recovering problem in the sense of ℓ 1-norm as



√

arg min ∥c∥1

√

subject to

sin θy = ψc,

(22)

where sin θ is a weighted diagonal matrix to form the orthonormal basis ψ from the spherical harmonics basis. Since the Orthogonal Matching Pursuit approach [35,36] (OMP) has a lower computation complexity and can be used to efficiently recover the sparse coefficient vector c, we select OMP as the recovering algorithm of c in this paper. Then, the OMP algorithm for the sparse coefficient vector c reconstruction can be described as below. OMP (A, y, ψ, S , e0 ) Input: a uniform random measuring matrix A with the dimension M × m, sparse basis matrix ψ with m × N dimensions, radiation flux y for M samples, sparsity level S , and a prescribed threshold e0 for the residual r between recovered flux from M samples and y. Output: a reconstructed sparse coefficient vector c. Initialize an index set I with ∅, circulation times with zero and sub-measure matrix ψS with zero set. Weighting the radiation flux y with the diagonal matrix √ √ sin θ, i.e. y = sin θ y. While ∥r∥ > e0 and i ≤ S

4.1.2. Uniform random sampling for compressive analysis As described above, to accurately recover the coefficient vector c, we should have: (1) the sparse basis ψ (θ , ϕ) is orthonormal, (2) the measuring matrix A is incoherent to the sparse basis matrix ψ formed by ψ (θ , ϕ) in Eq. (20), (3) enough points are sampled to accurately reconstruct the coefficient vector c for the sparse basis ψ (θ , ϕ). Since the sparse basis ψ (θ , ϕ) has shown to be orthonormal in Eq. (20), we need to determine a sampling approach and the number of samples over the domain Ω = {(θ, ϕ) ∈ [0, π ] × [0, 2π )} so that A is incoherent to ψ, and c can be accurately reconstructed. As discussed in [37], if M measurements {(θi , ϕi ), i = 1, 2, . . . , M } are randomly and independently sampled with a uniform measure on Ω , such measuring matrix A is incoherent to the sparse basis matrix ψ, and the required number of uniform random sampling measurements M should satisfy (21)

where m is the number of elements on the capsule, C is a smaller and positive universal constant. As shown in Fig. 2, when l = 20, there are N = (l + 1)2 = 441 SPH coefficients, and about 100 SPH coefficients are larger than a threshold δ (Let δ = 1.0−4 ). We can set S ≈ 100, and estimate the number of measurements M = 400 since m = 10368 (see in Section 3.1.3) and it works well in the practical experiments as discussed in [19]. Given M measurements, we

T

Identify: Choose the index J by comparing ψJ y.  Update: Update I with I ∪ {J }, and ψs with ψS ψJ , ˆ of y on the sub-space ψS by compute the projection w



√

Hence, we select ψ (θ , ϕ) = sin θ Ylk (θ , ϕ) as the basis [37] for compressive analysis so that we can accurately recover the sparse coefficient vector c.

M ≥ CS log (m) ,

4.1.3. Radiation flux reconstruction on the capsule based on ℓ 1minimization With M uniform random samples {(θi , ϕi ), i = 1, 2, . . . , M } obtained from the sampling matrix A, sparse basis matrix ψ, and their

2

ˆ = arg min y − ψS w2 , and then get the residual by w ˆ r = y − ψS w. 

i + +. End ˆ Ii for all the indices in I Let c = 0, and cIi = w Return c. Given the measuring matrix A, radiation flux y, sparse basis matrix ψ, and sparsity level S, we can reconstruct c from the above procedure, then calculate the radiation flux of all the elements on the capsule x by x = ψc and parameters σrms , and σ1 , σ2 , . . . , σ10 with Eq. (9) for radiation symmetry evaluation. In such a procedure, the sparsity level S is an a-priori parameter from the available radiation flux distribution, which may vary with different radiation flux distribution on the capsule due to laser inputs and the structure of the hohlraum-to-capsule system. Therefore, a fixed sparsity level S may under-estimate or over-estimate the radiation flux distribution, and result in a larger reconstruction error on the radiation flux on the capsule. 4.1.4. Adaptive selection of the sparsity level S for various radiation flux distributions The prescribed sparsity level S may lead to a large reconstruction error due to the variation of input laser parameters and geometric structure for the hohlraum-to-capsule system. Therefore,

Y. Huang et al. / Computer Physics Communications 185 (2014) 459–471

465

analysis. Then the optimal sparsity level adaptive determination problem can be defined as arg min ε = xm′ − xˆ m′  / xˆ m′  .





 



(25)

S

In Eq. (25), the sparsity level S is a positive number, and usually is smaller than the total number of uniform random sampling points M. Therefore, the optimal adaptive sparsity level determination problem is actually a one-dimensional unconstrained optimization problem. With a practical evaluation error ε , a near-optimal sparsity level S can be obtained from a one-dimensional unconstrained optimization method such as simplex, which can be described as

Fig. 5. Reconstruction errors ε v.s. the number of sampling points and sparsity.

we vary the sparsity S and the number of sampling points M so as to find an adaptive way to determine S by evaluating the variation of reconstruction error ε , which is denoted as

 ′  σ − σˆ  ∥∆σ∥ ε=   =   , σˆ  σˆ 

(23)

where σˆ is the vector formed by the radiation symmetry parameters σˆ rms and σˆ 1 , σˆ 2 , . . . , σˆ 10 , which can be computed from the radiation flux of all the elements without compressive analysis, and σ ′ is the vector reconstructed from compressive analysis for a given sparsity level S and M samples. For the example in Fig. 2, the reconstruction errors on various sparsity level S and M measurements are shown as follows. As shown in Fig. 5, we can see that (1) the reconstruction error ε decreases when the number of measurements M increases, (2) a smaller sparsity level S tends to get a smaller reconstruction error for a fixed number of measurements if S is not too small, and (3) an appropriate selection of the sparsity level S is very efficient since no more measurements are required (only 400 measurements are enough to reconstruct the radiation for the hohlraum-to-capsule system in Fig. 2). Therefore, the sparsity S needs to be adaptive so that the radiation flux on the capsule can be efficiently and accurately recovered from limited measurements. As described in Section 4.1.1, the SPH coefficients    are  sparse, and  of radiation the maximum rate of coefficients (l, k) : Clk  / C00  is less than 1.0 × 10−4 when the order l = 20. Then we get the total number of SPH coefficients N = (l + 1)2 = 441. Given the number of uniform random samples M, we formulate the problem for determining S adaptively as

 ′  σ − σˆ    , arg min ε = σˆ  S 

(24)

where σ ′ is the radiation symmetry from measurements M through ℓ 1-minimization described in Section 4.1.3, and σˆ is the true value of radiation symmetry. Since we cannot get the radiation flux of all the elements, it is not possible to get the true value of σˆ . One practical way is to evaluate ε with a residue of additional m′ points uniformly distributed    on  the capsule, which can be defined as ε = xm′ − xˆ m′  / xˆ m′ , xm′ = Am′ ϕ−1 ci , where i is the iteration index, ci is the reconstructed sparse coefficient vector, Am′ is the sampling matrix for m′ points which can be generated with a lowdiscrepancy based sampling approach [38], xˆ m′ is the radiation flux computed with the view-factor function for m′ points, and xm′ is the radiation flux reconstructed from M points with compressive

Adaptive OMP(A, y, ψ, m′ , e0 , ε0 ) Input: a measuring matrix ψ with N dimensions, radiation flux y for M samples, step length u = 1 for adaptive sparsity level search, a prescribed threshold e0 for the residual r between recovered radiation flux and y, and a prescribed threshold ε0 for adaptive sparsity S convergence. Weighting the radiation flux y with the diagonal matrix √ √ sin θ, i.e. y = sin θ y. Output: a reconstructed sparse vector c and reconstructed radiation flux x. Initialization i = 1, Si = 1, (ci , Ii ) = OMP (y, ψ, Si , e0 ), ri = y − Aci , c = 0 + cIi , and xi = ψ · c. Do u = 2 × u, Si+1 = Si + u. (ci+1 , Ii ) = OMP (y, ψ, Si+1 , e0 ) . ri+1 = y − Aci+1 , c = 0. xi+1 = ψ · ci+1 . If (ri+1 > ri ) i = i + 1. u = 2 × u, Si+1 = Si + u. Else i = i + 1. u = 0.5 × u, Si+1 = Si + u End     While (ε = xm′ − xˆ m′  / xˆ m′  ≥ ε0 ) c = 0 + cIi+1 , x = ψ · c Return c and the adaptive sparsity level S. From the above procedure, an adaptively determined sparsity level S and reconstructed x can be got through an iteration process. Since the OMP algorithm is almost linear when recovering the coefficient vector c, it means that c can be efficiently recovered by combining the adaptive sparsity level S determination and OMP algorithm without any a-priori knowledge. 4.2. Overview of compressive analysis for radiation symmetry evaluation Given the hohlraum-to-capsule system and incident lasers as shown in Fig. 6, we usually compute the radiation flux for each element on the capsule with the view-factor function described in Section 3.1.2, and then transform the radiation flux into the spherical harmonics domain for symmetry evaluation. In such process, the radiation flux needs to be computed for all the elements, and involves intensive computation to get an accurate radiation symmetry evaluation when the size of discrete elements is very small. The lower partition of Fig. 6 includes a follow chart for compressive analysis of radiation symmetry evaluation. Only M of the m elements, chosen by a uniform  and random measure, are re- quired. Then the radiation flux SC (r1 ) SC (r2 ) · · · SC (rM ) of such M elements is computed and utilized as the measurement vector y, and m′ samples and their radiation flux

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SC (r1 ) SC (r2 ) · · · SC (rm′ ) (denoted as xˆ m′ ) for the determination of adaptive sparsity level S. With the basis matrix ψ, measurement y, validation measurement xˆ m′ and diagonal weight √ matrix sin θ computed from the coordinate of M elements, we can efficiently and accurately recover the sparse vector c, radiation flux x and the parameters of radiation symmetry through OMP without radiation flux computation of all the elements. Relative to evaluating radiation symmetry, the compressive analysis approach is very efficient since only a fraction of elements are chosen as the seed points to recover the sparse coefficients for radiation symmetry evaluation.





4.3. Radiation symmetry optimization through compressive analysis The objective of radiation symmetry optimization is to get a highly uniform radiation on the capsule, which may significantly reduce the Rayleigh–Taylor instability in the ICF experiments. Changing the number of incident lasers or the geometric structure of cylinder-to-sphere system may lead to a large variation in the radiation symmetry. Therefore, an optimal design of hohlraum-tocapsule system is required for the ICF experiments to obtain better symmetry. Usually, a lot of design points are generated over the defining domain to find the optimal design point by computing and comparing their radiation symmetry, which thus leads to a time consuming process since the radiation symmetry evaluation for any design point needs 4 min. As described above, while compressive analysis can be employed to accelerate the radiation symmetry evaluation, it still needs a lot of iteration times to recover the sparse coefficients. Therefore, we can utilize a simulation optimization approach such as simulated annealing algorithm to reduce the number of required design points and the iteration times of radiation symmetry evaluation. As shown in Fig. 3, by selecting optimal parameters such as the diameter of the capsule, and setting their intervals, we can generate design points through sampling approaches such as Latin hypercube [39] to reconstruct a new CAD model of the hohlraum-to-capsule system. The lasers are directed to the inner surfaces of the hohlraum and mapped onto discrete elements. Then, M samples are selected to recover the sparse vector c and symmetry parameters such as σrms , and σ1 , σ2 , . . . , σ10 . We can then iterate such compressive analysis process to find the optimal design point. In such optimization process, only a fraction of design points are sampled for the optimal point determination, which is very beneficial for accelerating the radiation symmetry optimization process. 5. Experimental validation In this section, two examples on current SGII, and SGIII-YX laser facilities are selected and run on the computer Dell-OptiPlex 760 (Intel Dual-core CPUs with frequencies: 2.66 GHz and 1.97 GHz, memory: 2G) to validate the efficiency of the present approach. The inner surfaces of two models are represented and faceted into smaller elements. Then the radiation flux of all the elements (named the uniform sampling approach) is computed with the view-factor function for symmetry evaluation. We validate the efficiency and accuracy of compressive analysis approach by comparing them with uniform sampling approach, and additionally demonstrate the efficiency and accuracy in the design parameter optimization of geometric structure by combining it with simulation optimization approaches. 5.1. Implosion experiment target compressive analysis and optimization on the Shenguang II (SGII) laser facility As shown in Fig. 7a, a laser facility named Shenguang II built in 2001 is one of ICF experimental facilities, on which 8 lasers with 2 kJ energy are injected into the cylindrical hohlraum to heat

the centered capsule (Fig. 7b). The dimensions of such cylindrical hohlraum are: length L = 1380 µm, diameter DH = 800 µm, diameter of two entrance holes for laser injection Leh = 380 µm, diameter of two rectangular diagnostic holes on the cylindrical surfaces Dh = 330 µm. The diameter of the capsule DC = 230 µm. The CAD model of a hohlraum-to-capsule system and the laser spots can be obtained through a geometric modeling process on the CAD systems. Fig. 7b shows the represented surfaces for radiation flux computation (faces with red color boundaries). Then the surfaces are faceted with a resolution (length: 10 µm, angle: 2.5°) to compute the radiation flux on the capsule (Fig. 7d and e) with a uniform sampling approach and compressive analysis approach. The radiation symmetry is then evaluated, and the parameters are given in Table 1. Since the radiation flux of all the elements is calculated for the uniform sampling approach, we select the resulting parameters σrms , σ1 , σ2 , . . . , σ10 as the true value to validate the accuracy of compressive analysis approach and marked as σˆ rms , σˆ 1 , σˆ 2 , . . . , σˆ 10 . Then the maximal error and average error are given as: emax = max σrms − σˆ rms  , σ1 − σˆ 1  ,



eav g

 



    σ2 − σˆ 2  , . . . , σ10 − σˆ 10  and   10      σi − σˆ i  /11. = σrms − σˆ rms  +

(26)

i=1

From Table 1, we can see that (1) only 400 of 10 368 elements (144 × 72) are utilized to compute the radiation symmetry on the capsule for the compressive analysis approach, and evaluation is very accurate (the maximal error is less than 10−4 ), (2) computation time is reduced to 5% of that for the uniform sampling approach. In the experimental target design, the diameter of the cylindrical hohlraum DH is usually defined first for a given laser facility, e.g. DH = 800 µm for the SGII laser facility. Then the length of cylindrical hohlraum L can be optimized for the ICF experiments through compressive analysis and simulation optimization approach such as Simulated Annealing. The optimization result is given in Table 2. As shown in Table 2, the length of cylindrical hohlraum L for the implosion target on the SGII laser-facility is scanned over the interval [1.2 2], and 81 design points are sampled. For each sampled design point, we can compute the radiation flux of all the elements, or recover the radiation flux with the compressive analysis approach for symmetry evaluation. Compare the resulting σˆ rms and running time for radiation symmetry, we can see that the running time of compressive analysis is about 5% of that with the uniform sampling approach. In addition, simulation optimization such as the simulated annealing approach combined with compressive analysis can be very efficient and only 1.5% of the running time is required to achieve a comparable optimal point. 5.2. Implosion experiment target compressive analysis and optimization on the Shenguang III-YX (SGIII-YX) laser facility As shown in Fig. 8, Fig. 8a gives the SGIII-YX laser facility built in 2007, in which 8 lasers are injected to a hohlraum-to-capsule target with 10 kJ energy. The dimensions of such cylindrical hohlraum are: length L = 1700 µm, diameter DH = 1000 µm, the diameter of two entrance holes for incident lasers Leh = 700 µm, the diameter of two rectangular diagnostic holes on the cylindrical surfaces Dh = 400 µm. The diameter of the centered capsule DC = 320 µm. The radiated surfaces are faceted with resolution (length: 10 µm, angle: 2.5°) for radiation flux computation and symmetry evaluation. The resulting radiation symmetry is compared for compressive analysis and uniform sampling approaches in Table 3. From Table 3, it can be seen that (1) only 400 of 10 368 elements (144 × 72) are utilized to compute the radiation symmetry on the

Y. Huang et al. / Computer Physics Communications 185 (2014) 459–471

467

Fig. 6. Overview of compressive analysis for radiation symmetry evaluation.

Fig. 7. Compressive analysis of a hohlraum-to-capsule target on the SGII laser facility. (a) SGII laser facility, (b) recognized surfaces, (c) a hohlraum-to-capsule target with 8 injected lasers, (d) radiation flux computed on the target, (e) the unfolding view of radiation flux on the capsule.

Fig. 8. Compressive analysis of a hohlraum-to-capsule target on the SGIII-YX laser facility. (a) SGIII-YX laser facility, (b) recognized surfaces, (c) a hohlraum-to-capsule target with 8 injected lasers, (d) radiation flux computed on the target, (e) the unfolding view of radiation on the target.

capsule for the compressive analysis approach, and the maximal analysis error is no more than 1 × 10−4 , and (2) computation time is reduced to 4.7% for the compressive analysis approach. Using the diameter of the cylindrical hohlraum DH = 1000 µm, we also optimize the radiation symmetry to get an optimal length L. Compressive analysis and simulated annealing can be combined to efficiently optimize the radiation symmetry. The optimization result is compared and given as follows (Table 4).

coefficient vector c. The random characteristics may lead to variation in reconstruction accuracy, adaptively determined sparsity level S, and the location of the optimal design point. Therefore, we investigate the accuracy or robustness of radiation symmetry evaluation and optimization. To analyze the accuracy and robustness of compressive analysis, a relative error is computed from Eq. (23). The reconstruction accuracy or robustness on sparsity, sampling measurements and optimal design point are discussed in the following section.

6. Robustness and accuracy for compressive analysis of radiation symmetry

6.1. Reconstruction accuracy from compressive analysis compared with least squares

In the above compressive analysis approach, uniform random sampling is used to get M measurements for recovering the sparse

Given limited sampling points, we can reconstruct the sparse coefficient vector c with the least-squares approach, which can be

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Table 1 Evaluation efficiency of radiation symmetry through compressive analysis (‘H’ means the cylindrical hohlraum, ‘C’ means the capsule). Approaches

Size of discrete elements

σrms (eV)

σ1 , σ2 , . . . , σ10 (eV)

Maximal error (eV)

Average error (eV)

Running time (sec.)

Uniform sampling

H: 2.5°, 0.01 mm C: 2.5°

0.062541

{0.0016, 6.0948, 0.0, 1.37105, 0.0024, 0.0464, 0.00025, 0.1467, 0.0, 0.2096}×10−2

0

0

199.8

Compressive analysis

H: 2.5°, 0.01 mm C: 2.5°

0.062526

{0.0, 6.0935, 0.0, 1.37107, 0.0034, 0.0443, 0.0, 0.1461, 0.002, 0.2094}×10−2

2.5 × 10−5

1.18 × 10−5

10.1

Table 2 The optimization result of the length of cylindrical hohlraum L over the interval [1.2, 2]. Approaches

Optimal design point

σˆ rms

Scanning times

Running time (s)

Uniform sampling + optimal parameter scanning Compressive analysis+ optimal parameter scanning Compressive analysis + simulated annealing optimization

1.36 1.36 1.3598

0.061637 0.061638 0.061645

81 81 23

16103.5 830.4 245.7

Table 3 Evaluation efficiency of radiation symmetry through compressive analysis approach. Approaches

Size of discrete elements

σrms (eV)

σ1 , σ2 , . . . , σ10 (eV)

Maximal error (eV)

Average error (eV)

Running time (s)

Uniform sampling

H: 2.5°, 0.01 mm C: 2.5°

0.065864

{0.0, 6.066, 0.0, 2.252, 1.027, 0.058, 0.356, 0.202, 0.177, 0.405}×10−2

0

0

314.413

Compressive analysis

H: 2.5°, 0.01 mm C: 2.5°

0.065918

{0.0, 6.072, 0.012, 2.255, 1.022, 0.059, 0.351, 0.209, 0.173, 0.400}×10−2

8 × 10−5

2.8 × 10−5

14.889

Table 4 The optimization result of the length of cylindrical hohlraum L over the interval [1.2, 2]. Approaches

Optimal design point

σˆ rms

Scanning times

Running time (s)

Uniform sampling + optimal parameter scanning Compressive analysis + optimal parameter scanning Compressive analysis + simulated annealing optimization

1.7 1.7 1.6959

0.065864 0.065832 0.065823

81 81 15

25434.2 1189.6 223.5

denoted as c = Aϕ



 −1 −1

y.

(27)

The measurements y are from uniform random sampling over the SPH domain, the reconstructed sparse vector c may depend greatly on such sampled points. Since the least-squares approach is a usual way to reconstruct the coefficient vector c, we can compare the reconstructed errors to validate the accuracy of compressive analysis approach for the two models in Section 5. As shown in Fig. 9, 400 uniform random samples are run for 100 times to validate the recovering accuracy of compressive analysis by comparing the reconstruction errors with least squares approach. Analyzing the resulting reconstruction errors, we find that the reconstructed errors vary greatly from 0.3343 to 11.0643 for the least-squares approach as shown in Fig. 9a, the smallest reconstruction error (0.3343) is about 300 times the largest reconstruction error (1.09237 × 10−3 ) of compressive analysis approach for the example in Section 5.1, and the running time of least-squares approaches (10 h) is about 20 times of that for compressive analysis approach (30 min). For the second case, the reconstruction errors of radiation flux vary greatly from 0.4195 to 8.0119 for the leastsquares approach as shown in Fig. 9b, the smallest reconstruction error (0.4195) is about 100 times the largest reconstruction error (3.38434 × 10−3 ) of compressive analysis approach for the example in Section 5.2, and the running time of least-square (11 h) is about 14 times of that with compressive analysis (45 min). As compared in Fig. 10, instead of generating a larger variation on the reconstruction errors than least squares, the variation of the reconstruction errors of the compressive analysis approach is small, less than 0.924 × 10−3 for the example in Section 5.1, and 2.436 × 10−3 for the example in Section 5.2. Therefore, we can see

that compressive analysis approach is very efficient and robust for radiation symmetry evaluation with respect to accuracy and efficiency. 6.2. Reconstruction accuracy v.s. a-priori sparsity level S for compressive analysis As discussed in Section 4.1.3, the sparsity level S is an important factor in reconstructing the sparse coefficient vector c, and may affect the accuracy when recovering c. The reconstruction errors are analyzed and shown as follows. As shown in Fig. 11, the reconstruction errors show a similar variation on the number of sampling points and a-priori sparsity level S for both implosion targets on the SGII and SGIII-YX laser facilities. It includes (1) the reconstruction accuracy tends to be much better for lower sparsity levels, (2) the reconstruction error converges with the increase of samplings. Such variation of reconstruction errors means that it is important to select an appropriate sparsity level S, which can roughly reduce the required samplings to accelerate radiation symmetry analysis. Therefore, we can select the number of measurements M = 400 and choose an adaptively determined sparsity level S = 145 through a procedure described in Section 4.1.4, which can get a relative reconstruction accuracy 4 × 10−4 , and absolute accuracy 3 × 10−5 , which is accurate enough to recover the sparse coefficient vector c. 6.3. Sparsity level S adaptive determination v.s. uniform random sampling As described in Section 4.1.4, M uniform random samples are used to reconstruct c, and additional m′ uniformly distributed

Y. Huang et al. / Computer Physics Communications 185 (2014) 459–471

(a) The target on the SGII laser facility.

469

(b) The target on the SGIII-YX laser facility.

Fig. 9. Reconstruction accuracy of the least-squares and compressive analysis approaches.

(a) Implosion target on the SGII facility.

(b) Implosion target on the SGIII-YX facility.

Fig. 10. Reconstruction accuracy v.s. number of sampling points and the sparse level radiation.

(a) Implosion target on the SGII facility.

(b) Implosion target on the SGIII-YX facility.

Fig. 11. Reconstruction accuracy v.s. number of sampling points and the sparse level radiation.

samples are selected to determine the adaptive sparsity level S. The random sampling of M measurements may lead to variation in the adaptively determined sparsity level S. Therefore, we select the number of measurements M = 400 and m′ = 100, and then

run the sparsity level S adaptive determination program for 100 times to find an appropriate number over the interval [1, 400]. The adaptively determined sparsity levels S for the two examples are shown as follows.

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Y. Huang et al. / Computer Physics Communications 185 (2014) 459–471

(a) Sparsity levels v.s. running times for SGII facility.

(c) Sparsity levels v.s. running times for SGIII-YX facility.

(b) Reconstruction errors v.s. sparsity levels for SGII facility.

(d) Reconstruction errors v.s. sparsity levels for SGIII-YX facility.

Fig. 12. Sparsity level variation and its reconstruction errors v.s. uniform random sampling.

It can be seen from Fig. 12, the sparsity level S shows some variations (70 to 105 for the ignition target on the SGII facility, and 70 to 110 for the ignition target on the SGIII facility) due to the uniform random sampling for 100 running times. Nevertheless, the reconstruction errors compared with the radiation flux computed for all the elements are bounded by 1.2 × 10−3 (ignition target on the SGII facility) and 2.56×10−3 (ignition target on the SGII facility) for all the sparsity levels adaptively determined by the proposed algorithm. Therefore, the presented algorithm is efficient and can be used to accurately and adaptively determine the sparsity level S. 6.4. The location of optimal point v.s. uniform-random sampling As demonstrated in Sections 5.1 and 5.2, compressive analysis can be combined with simulation optimization algorithms such as simulated annealing to determine the optimal design point for the length of cylindrical hohlraum quickly, with the resulting optimal design point approaching the scanned optimal design point very well. Because the simulation optimization algorithm depends on the radiation symmetry evaluation, the variation of the presented compressive analysis due to uniform-random sampling may lead to alteration of the predicted optimal design point. Hence, we run the simulation optimization procedure combined with compressive analysis for 10 times and investigate the location of the optimal point for the targets on the SGII and SGIII-YX laser facilities as below. As shown in Fig. 13, for the two implosion targets on the SGII and SGIII-YX laser facilities, we run the simulation optimization

combined with compressive analysis approach for ten times, and find that the variation of resulting points is less than 1.2 × 10−3 for both targets. Therefore, we conclude that the compressive analysis is efficient and robust for optimizing design parameters, even though it has uniform-random sampling characteristics. 7. Conclusions In this paper, we introduce a novel compressive analysis approach for radiation symmetry evaluation and optimization of the experimental target design in the laser driven ICF, in which (1) the spherical harmonics is orthogonalized by multiplying a scalar to represent the radiation flux on the spherical capsule in the hohlraum, (2) the sparsity level without any a priori knowledge on radiation distribution is adaptively determined to efficiently and accurately recover the sparse coefficients, (3) ℓ 1-norm minimization is used to accurately reconstruct the sparse coefficients for radiation symmetry evaluation, and (4) such compressive analysis approach can be combined with available simulation optimization algorithms to efficiently and robustly find the optimal design point of the target design in current laser facilities. Finally, the compressive analysis approach is validated and analyzed with two experimental targets. It is found that the recovering errors of the compressive analysis are less than 10−3 , and requires only 5% of the running time of the uniform sampling approach. In addition, finding the optimal design point requires no more than

Y. Huang et al. / Computer Physics Communications 185 (2014) 459–471

(a) Implosion target on the SGII facility.

471

(b) Implosion target on the SGIII-YX facility.

Fig. 13. The variation of optimal design point for the uniform random sampling of compressive analysis.

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