A Review of the Parallel Algorithms for Solving Multidimensional PDE Problems
Md. Rajibul Islam
This review study gives a widespread overview of the solutions of several engineering problems based on some multidimensional partial differential equations like parabolic, hyperbolic, elliptic, AGE, IADE equations. Different analytical methods of treatment as well as those of numerical methods are presented in this paper. Finally, some evaluation phases of several experiments in order to solve some current engineering problems and recommendations are demonstrated.
Key words: Partial differential equations, numerical analysis, parallel algorithms, engineering problems
Received: March 21, 2010;
Accepted: May 28, 2010;
Published: July 27, 2010
Many researchers have taken interests in developing finite difference methods
that could approximate the solution of a one dimensional parabolic diffusion
equation. Classical methods, however, have their own restrictions. Explicit
methods are simple but generally suffer the disadvantage of conditional stability
and low accuracy. Implicit methods, on the other hand, may possess unconditional
stability and higher accuracy. Their features, however, are less amenable to
parallelism (Smith, 1978).
Over the years, many highly refined iterative and alternating schemes have
been developed, in which many of them not only exhibit superior properties in
terms of stability, accuracy and rate of convergence, but they are also suitable
for parallel computing. One of the schemes which have been cited often is the
Alternating Group Explicit (AGE) method introduced by Evans
and Sahimi (1989c). It employs the fractional splitting strategy applied
alternately at intermediate time step on tridiagonal systems of the difference
scheme. The approach, which is second-order accurate in both time and space,
has been found to be stable, convergent and parallelizable.
Based on the AGE method, many new alternating schemes have been developed.
Baolina and Wenzhib (1994) presented the Alternating
Segment Crank-Nicolson method for the diffusion equation. The method is unconditionally
stable and has the obvious property of parallelism. Zhu
and Zhao (2007) designed a set of New Alternating Segment Explicit-Implicit
(NASEI) schemes that alternate between explicit and implicit segments at any
two consecutive time levels. The schemes are proven to be stable under reasonable
conditions, have truncation errors of third order in space and capable of parallel
computation. Zhen et al. (1994) developed a class
of Hopscotch algorithms for the finite difference solution of the diffusion
equation under consideration. The algorithm is convergent and efficient with
regards to parallel computing. Feng (2009) presented
a class of alternating group explicit iterative parallel method (AGI) by using
an unconditionally stable symmetry six-point implicit scheme of high accuracy.
Zhu et al. (2004) designed an explicit implicit
scheme for parabolic equations with discontinuous coefficients. The method is
intrinsically parallel. Baolina (1991) developed a class
of alternating schemes in three time levels, which are the unconditionally stable
AGE and the ASE-I (alternating segment explicit-implicit) methods. In the design
of these two methods, Saulyev asymmetric schemes (Saulyev,
1964) have been used. Tavakoli and Davami (2006)
applied a method which is based on domain decomposition concept and used the
asymmetric Saulyev schemes for internal nodes of each sub-domain and alternating
group explicit method for sub-domains interfacial nodes. The approach
is fully explicit, unconditionally stable and has merit in terms of accuracy.
Sahimi et al. (1993, 2001)
proposed an alternative to the AGE method, which is the Iterative Alternating
Decomposition Explicit (IADE) method. To approximate the solution of the diffusion
equation, the IADE scheme employs the fractional splitting of either the Mitchell
Fairweather (IADE-MF) variant (Mitchell and Fairweather,
1964) or the DYakonov (IADE-DY) variant (DYakonov,
1963) for a fixed acceleration parameter r>0 . Each variant is second-order
accurate in time and fourth-order accurate in space. By analyzing the results
of some numerical experiments based on the chosen variant for the IADE method,
Sahimi et al. (2001) concluded that the two-stage
IADE procedure has merit as an alternative iterative method with respect to
stability, accuracy and rate of convergence. As the method is fully explicit,
its feature can be fully utilized for parallelization.
The finite difference method is a well-established and conceptually simple
method that requires a point-wise approximation to the governing equations.
While the finite volume method is a further refined version of the finite differences
method and has become popular in computational fluid dynamics. The vertex-centered
finite volume technique is very similar to the linear finite element method
(Lewis et al., 2004). The basis concept of the
finite element method is that any solution domain can be divided into several
simple subdomains known as finite elements. Thus, the approximate solution of
the problem in the complete domain can be determined by assuming a simple form
of solution in each finite element (Rao, 2002; Gutpa
and Meek, 2003).
In Alias et al. (2009a), they focused on the
application of this method in solving the initial stages of crack propagation
problem which means the deformation due to the stress and strain of a material.
Propagation problems refer to time-dependent, transient and unsteady-state phenomenon.
The method was applied to evaluate the stress intensity factors for plates of
arbitrary shape using conventional finite elements (Cheung
et al., 1996). Fracture mechanics (Bui, 2006)
was used to investigate the failure of brittle materials, which was to study
material behavior and design against brittle failure and fatigue. The engineering
study of fracture mechanics (Stanley, 1977) does not emphasize
how a crack is initiated; the goal is to develop methods of predicting how a
In Alias et al. (2010a), the researchers discussed
the solution of two dimensional Partial Differential Equations (PDEs) using
some parallel numerical methods namely Gauss Seidel and Red Black Gauss Seidel.
The selected two-dimensional PDEs in order to solve the problem were parabolic
and elliptic type. Parallel Virtual Machine (PVM) is used in support of the
communication among all microprocessors of Parallel Computing System.
It is abundantly clear that many important scientific problems are governed
by partial differential equations according to Alias et
al. (2009a). The difficulty in obtaining exact solution arises from
the governing partial differential equations and the complexities of the geometrical
configuration of physical problems (Alias et al.,
2003a, b, 2008a, b).
For example, imagine a metal rod insulated along its length with no heat can
escape for its surface. If the temperature along the rod is not constant, then
heat conduction takes place. In such situations, the numerical method is used
to obtain the numerical solutions (Smith, 1965). These
partial differential equations may have boundary value problems as well as initial
value problems. In general, the transient particle diffusion or heat conduction
is Partial Differential Equations (PDE) of the parabolic type and Laplaces
equation for temperature, diffusion, electrostatic conduction is elliptic and
wave equation or transport equation is the PDE of hyperbolic type (Alias
et al., 2008a, 2009b; Evans,
1995). The parabolic partial differential equations are normally used in
such fields like molecular diffusion, heat transfer, nuclear reactor analysis
and fluid flow (Nakamura, 1993; Smith,
In Alias et al. (2009b), New Iterative Alternating
Group Explicit (NAGE) was introduced which is a powerful parallel numerical
algorithm for multidimensional temperature prediction. The discretization was
based on finite difference method of Partial Differential Equation (PDE) with
parabolic type. The critical 3-Dimensional temperature visualization involves
large scale of computational complexity. This computational challenge inspired
the authors to utilize the power of advanced high performance computing resources.
Incomplete blow-up is a condition under the quasilinear heat equation (Alias
et al., 2010b). The Porous Medium Equation (PME) with power source
are admitting incomplete blow-up. It is used as one of the filtration process
in the industry. Authors proposed a new variance of the Alternating Group Explicit
Scheme (AGE) algorithms to solve incomplete blow-up problem through High Performance
Mizoguchi (2005) presented multiple blow-ups to solve
a semilinear heat equation problem. Natalini et al.
(1996) presented an incomplete blowup of entropy solutions to first-order
quasilinear hyperbolic balance laws. They specified a general procedure to continue
solutions beyond the blowup time, which made use of monotonicity methods. The
continuations thus obtained were possibly unbounded and satisfied suitable generalized
entropy and Rankine-Hugoniot conditions. Then they proved the uniqueness of
continuations satisfying such conditions as well. Arrieta
and Bernal (2004) showed that blow-up occurred only on the boundary while
they analyzed the existence of solutions that blow-up in finite time for a reaction-diffusion
equation. Mizoguchi and Vazquez (2007) demonstrated
multiple blow-ups for semilinear heat equations at different places and different
times and also solutions for a semilinear heat equations II described by Mizoguchi
(2006). Nonlinear Volterra integral equations of the second kind with solutions
that blow-up or quench had analyzed by Roberts (2007).
SOME PROBLEM DEFINITIONS IN BRIEF
Here, some engineering and bioscience problems are explained briefly which were focused to solve through partial differential equations in many studies by the researchers:
Thermal control process on PCB: In the context of thermal control system
design, there are two major design approaches, that is, the system is either
passive or active controlled. When there is a need to deal with more sophisticated
system which requires high performance temperature controlling, the active thermal
control system design is better suited. In electronic engineering, complex semiconductor
devices are subjected to a number of tests during the manufacturing process
to determine device functionality and to insure future reliability. The first
test is usually at the wafer level. During this test the individual die on the
wafer are probed to determine die integrity and die parametric properties. This
quick test allows rejection of bad die and sorting of die for further testing
(Gardell, 1995). Then, burn-in test will follow after
the wafer level. The test thermally and electrically stresses the parts to accelerate
early life, or infant mortality, failures. The device junction temperatures
are typically held between 100 to 140°C to accelerate stress. Because the
parts are also subjected to higher than normal voltages, the power dissipation
levels can be very high, significantly higher than in normal operation (Tustaniwskyj
and Babcock, 2004). So, Ghaffar et al. (2008)
just focused on this part, where the problem under consideration is peak junction
temperature of semiconductor devices estimation.
Brain tumor growth: A brain tumor is a growth of abnormal cells or normal
cells in an inappropriate place in the brain. A primary brain tumor is one that
starts in the brain, rather than cancer in another part of the body that has
spread to the brain. Primary tumors can be grouped into non-cancerous (benign)
and cancerous (malignant). Malignant brain tumors are commonly called brain
cancer and they are usually invasive and life-threatening. Brain tumors also
may be metastatic or secondary brain tumors. These tumors are formed from cancer
cells that begin growing elsewhere in the body and travel to the brain, usually
through the bloodstream. The study of (Alias et al.,
2009b) was to visualize or capture the growth of brain tumor in three-dimensional
space and to develop or identify the three-dimensional brain tumor growth. The
aim was to identify the discretization of the mathematical models which will
be converted to standard form and to implement the algorithm to perform the
iterative methods from the discretization of the mathematical model. Angelis
and Preziosi (2000) described the evolution of tumor in vivo and
related to the boundary problem.
Breast cancer growth: Breast cancer is the most general disease among
women, except for non-melanoma skin cancers. Due to early detection and increased
awareness, resulting deaths have been decreasing recently. The second leading
reason of cancer death in women is breast cancer. The possibility that breast
cancer will be responsible for a womans death is about 1 in 33. Early
detection is the key to successful treatment. Alternative methods for tumor
recognition have been researched to couple with Thermal Simulation (Gonzalez,
2007), Microwave Imaging system through Space Time beamformer (Bond
et al., 2003; Gunnarsson, 2007) and 2D Finite-Difference
Time Domain (FDTD). So, the mathematical modeling could be advantage solutions
in terms of insights and predictions. The research of the Alias
et al. (2009b) focused on the study of elliptic equations, particularly
Helmholtzs wave equation and hyperbolic equations to monitor or predict
the cancer cell growth through computational modeling.
Temperature behavior of rubber materials: Heat transfer process occurs
due to the polymer flow as convection. The motion of fluid transfers an energy
along its flow path and thus convects heat during mould filling (Davis
et al., 2003). To predict the temperature behavior on rubber material
involving phase change processes, this prediction solving by the mathematical
simulation. A mathematical model was presented for the prediction of temperature
profiles and heat transfer rates during the blow moulding process (Edwards
et al., 1981). Darwis et al. (2009)
focused on the research to study the influence of operating conditions on cooling
time. The experimental attention to be focused on to using a chilled mould and
gas circulation to give enhanced cooling rates. Analytical data obtained on
a small laboratory at Lembaga Getah Malaysia as an exact solution and limited
to testing on an industrial production line for the manufacturing of large barrels
have been confirmed the validity of theoretical approach.
Food drying for preservation: It is very necessary to dry the tropical
fruits to a certain level after harvest. Drying processes are widely used in
food production especially fruits, but a scientific approach has not so widely
been applied, so rather empiric rules are often used to set up industrial production,
particularly in small-medium firms. The main objective of food drying process
is water removal up to particular moisture content in order to prevent food
from microbial spoilage and deterioration reactions and to increase the product
shelf life (Curcio, 2006). Drying is a process involving
simultaneous heat and mass transfer phenomena. Formulation of adequate mathematical
models to describe the transfer phenomena during dehydration fruit is very important
to optimize the process-leading in improvement product quality and reduction
process cost. Simulation results and information of drying kinetics of fruit
material such as time-temperature-moisture content distributions, as well as
theoretical approaches to moisture movement, is very essential for the prevention
of quality degradation and for the achievement of fast and effective drying.
Such information will be very useful to optimize production processes of tropical
fruits dried. Hence, the authors contribution of the study (Alias
et al., 2009b) is successful modified the mathematical simulation
in representing the actual process of dehydration in commercial foodstuff industry
in terms of heat and mass transfer inside tropical fruits material.
PARTIAL DIFFERENTIAL EQUATIONS TOWARD SOLUTIONS
Numerical methods/Parallel algorithms are utilized for solving large sparse problems which are based on domain decomposition methods. They are straight forward parallel implementation with fine grain approaches and highly convergent and accurate and also, well suited to implement on distributed, shared and hybrid memory architecture. Numerical methods/Parallel algorithms can able to solve grand challenge application for multidimensional problem. Here, the most implemented partial differential equations are elucidated which lead to solve many engineering problems as those presented in the previous section. In this review study, it can be seen that multi-dimensional partial differential equation had been considered for the application of numerical methods in several studies.
One- dimensional parabolic equation: Equation 1 shows the one-dimensional parabolic equation.
subject to initial condition,
and boundary condition
The finite difference discretization of Eq. 1 results in
Two-dimensional parabolic equation:
this is subject to the initial condition,
and U(x, y, t) subject to the boundary Ω which is ∂R with condition
The region R is a rectangle defines by:
The finite difference discretization of Eq. 2 results in,
Three-dimensional parabolic equation: Equation 3 shows
the three dimensional parabolic equation:
which subject to the initial condition below
Additionally, U(x, y, z, t) is subject to boundary Ω which is ∂R
with boundary condition,
and the discretization of Eq. 3 is as follows:
Formulation of IADE and AGE families
IADE methods: Six strategies of parallel algorithms are implemented to exploit
the convergence of IADE (Alias et al., 2003a;
Evans and Sahimi, 1989a, b).
In the domain decomposition strategy the IADE Michell-Fairweather which is fully
explicit is derived to produce the approximation of grid-i and not totally dependent
on the grid (i-1) and (i+1). In IADE Red Black and IADE SOR strategies, the
domain is decomposed into two different subdomains. The concept of multidomain
is observed in the IADE Multicoloring method. The decomposition of domain split
into several different groups of domain. On the vector iteration strategy, parallel
IADE is run in two sections (Alias et al., 2003a;
Hageman and Young, 1981). This method converges if the
inner convergence criterion is achieved for each section.
The objectives of the parallel algorithms are to minimize the communication
cost and computational complexity (Fig. 1a, b).
The sequential algorithm for IADE shown that the approximation solution for grid ui is depend on ui-1 and the approximation solution for um+1i is depend on um+2i. To avoid dependently situation, some parallel strategies is developed to create the non-overlapping subdomains.
IADE-New: On the strategy of incomplete block LU preconditioners on
slightly non-overlapping subdomains, the domain is decomposed into p processors
with incomplete subdomain (Alias et al., 2003a).
This strategy implemented the incomplete factorization with parameter of algebraic
boundary condition as follows,
Alternating Group Explicit(AGE) method: Based on the Douglas-Rachford
formula (Evans and Sahimi, 1989a, b),
the AGE fractional scheme involves the splitting of matrix A from system of
linear equations Au = f (Alias et al., 2003a,
b). A is split into the sum of its constituent symmetric
and positive definite matrices G1, G2, G3,
|| (a, b) IADE Algorithm to minimize the communication cost
and computational complexity
with diag(G1+G2) = diag(G3+G4)
The AGE fractional scheme is based on four intermediate levels, (k+1/4), (k+1/2),
(k+3/4) and (k+1). Using explicit (2x2) blocks for matrices (G1+G2)
and (G3+G4), we have a group of (2x2) block systems which
can be made explicit as follows:
AGE BRIAN method: BRIAN method is based on the AGE algorithm with Douglas-
Rachford variant and linear interpolation (BRIAN) concepts using the fractional
strategy (Douglas et al., 2003; Evans
and Sahimi, 1988). BRIAN algorithm (Alias et al.,
2003b) has been developed as an alternative to the parallel and sequential
algorithm of DOUGLAS method (Evans and Sahimi, 1989b).
The formula for BRIAN method for 2-dimensional problem leads to five intermediate
levels is as follows:
and with linear interpolation, we obtain,
AGE DOUGLAS algorithms: DOUGLAS Algorithms is based on the Douglas-Rachford
formula for AGE fractional scheme (Sahimi and Muda, 1989)
takes the form:
where, A is the sum of its constituent symmetric and positive define matrices G1, G2, G3 and G4,
where, a<0, c≥0 and b24ac = 0. The PDE is said to be parabolic if det(Z) = 0. The heat conduction equation and other diffusion equation are examples. The heat equation is:
where, K is a constant. Initial-boundary conditions are used to give.
where ux x = f (ux, uy, u, x, y) holds in Ω.
where, b24ac>0. The PDE is said to be hyperbolic if det(Z)<0. The wave equation is an example of a hyperbolic partial differential equation. The wave equation is:
where, β is a constant. Initial-boundary conditions are used to give:
u (x, y, t) = g(x, y, t) for x ∈ ∂ Ω,
u (x, y, 0) = v0 (x, y) in Ω
ut (x, y, 0) = v1 (x, y) in Ω
where, ux y = f (ux, ut, x, y) holds in Ω.
where, b24ac > 0. The PDE is said to be elliptic if Z is
a positive definite matrix with det(Z) = 0. Laplaces equation and Poissons
equation are examples. The Laplaces equation is: .
Boundary conditions are used to give the constraint u(x, y) on ∂Ω,
where, ux x + ux y = f (ux, uy, u, x, y)
EVALUATION PHASES OF EXPERIMENTAL SOLUTIONS
Partial differential equations occur from a variety of physical and engineering problems and assume a huge diversity of forms. Normally these forms are very complex, with nonlinearities, variable coefficients, high dimensionality, coupled equations of mixed type and irregular boundaries. Numerous constructive algorithms have been developed for solving these problems; nevertheless, the time and space complexities high and the class of problems to which all applies are limited. It has been seen in the previous sections that there exist some necessary conditions such that partial differential equations can be applied for solving those problems. Below are some phases that were implemented in order to evaluate the efficiency of those parallel experiments.
There are a master task and a number of worker tasks in the PVM implementation
of the modeling codes. Master task is responsible to divide the model domain
into sub domains and distribute them to worker tasks. Then, the worker tasks
perform time marching and communicate after each time step. Time execution,
speedup, efficiency, effectiveness and temporal performance were analyzed by
looking at the performance of the parallel algorithm (Islam
and Alias, 2010a, b).
Increasing number of processors significantly reduces the ratio but all the
methods that experimentally performed, represent the ability in maintaining
the condition where time for computation is always more than time consumed for
communication. This reflects the beneficial ability of the blends of methods
used with parallel algorithm that had been developed (Sahimi
et al., 1993; Foster, 1995). As more problems
need to be solved, each method results in higher time consumed for computation
rather than communication. The ratio between computation and communication is
known as granularity. High granularity reflects that computational cost dominating
the overall execution time. However, too high granularity will lead to loss
parallelism characteristics where the algorithm developed involved large size
of data passing between processors. Thus, best combination of parallel algorithm
and method being used will lead to better parallel performance evaluation where
there is balance between computation and communication cost.
The following definitions are used to measure the parallel strategies, speed up, efficiency, effectiveness and temporal performance. Where T1 is the execution time on one processor, Tp is the execution time on p processors and the unit of Lp is work done per micro second.
The execution time: Execution time is the amount of time needed for a complete run of a computer program routine. The time required for a computer to decode and perform a compiled instruction.
The Speedup: The Amdahls law states that the speed of a program is the time to execute the program while speedup is defined as the time it takes to complete an algorithm with one processor divided by the time it takes to complete the same algorithm with N processors. The formula of speedup for a parallel application is given:
||Execution time for a single processor and
||Execution time using p parallel processors
The efficiency: The efficiency of a parallel program is a measure of
processor utilization. Efficiency is defined as
the speed-up with N processors divided by the number of processors N. An efficiency of 100% means that all of the processors are being fully used all the time.
The effectiveness: Effectiveness is used to calculate the speedup and
the efficiency. The effectiveness is:
||No. of processors
||Execution time using p parallel processors
The temporal performance: Temporal performance is a parameter to measure the performance of a parallel algorithm which is:
|| Execution time using p parallel processors
Computation time and communication time ratio: Parallel execution time,
tpara is divided into two parts, computational time, tcomp
and communication time, tcomm. The tcomp is the time to
compute the arithmetic operations such as multiplication and addition operations
in the parallel algorithm. As all the processors doing the operation at the
same speed, calculation for the tcomm is depending upon for the size
of the message. The cost of communication comes from the two major phases in
sending a message: the start-up phase and the data transmission phase (Becker
et al., 2003). The total time to send K units of data for a given
system can be written as:
where, tcomm is time needed to communicate a message of K bytes, tstartup is sometimes referred as the network latency time. Tstartup is also referred to time to send a message with no data. It includes time to pack the message at source and unpack the message at the destination and to start a point-to-point communication.
The tdata is time to transmit units of information. It is also the transmission time to send one bytes of data. The tstartup and tdata are assumed as constants and measured in bits sec-1. Tidle is the time for message latency and time to wait for all the processors to complete the works. The evaluation of these communication costs via simple codes that time the send/recv messages.
The research focus on,
tpara = Time for parallel execution
tcomm1 = αtdata + βtstartup
α and β dependents on m and L.
Here, tcomm1 is the Communication time 1 which is obtained from the subtraction of idle time from communication time.
Communication cost for parallel processing is,
||Units of data that sending across processor
||No. of step overall the execution
Granularity analysis: Many metrics are used throughout the performance
evaluation of parallel programs (Bahi et al., 2008;
Cosnard and Trystan, 1995; Kwiatkowski,
1999). Perhaps the simplest and most intuitive metric of parallel performance
is the parallel runtime. It is the time from the moment when computation starts
to the moment when the last processor finishes its execution. The parallel run
time is composed as an average of three different components: computation time,
communication time and idle time (Kwiatkowski, 2006).
The computation time (Tcomp) is the time spent on performing computation
by all processors, communication time (Tcomm) is the time spent on
sending and receiving messages by all processors, the idle time (Tidle)
is when processors stay idle. The problem with parallel runtime is that it does
not account for the resources used to achieve the execution time. Specifically,
if one were to indicate that the parallel runtime of a program, which took 10s
on a serial processor, is 2s, we would have no way of knowing whether the parallel
program (and associated algorithm) performs well or not. The second metric is
scalability. The property of a program to adapt automatically to a given number
of processors is called scalability (Douglas et al.,
2003). Scalability is more sought after than efficiency (i.e., gain of computing
time by parallelism) on any specific architecture/topology. Another one is speedup.
Speedup is the ratio of the running time on a single processor to the parallel
running time on p processors (Douglas et al., 2003;
Alias et al., 1998, 2009b).
In other word, the ratio of two program execution times, particularly when times
are from execution on 1 and p nodes of the same computer.
We have presented a lengthy study review for parallel algorithms for solving multidimensional partial differential problems in different fields of engineering and physics. Some partial differential equations which lead to solutions have been illustrated. Also, various numerical and analytical methods of solution have been demonstrated briefly. And finally some phases that are implemented for solving the problems are presented.
For this review, we have not been concerned greatly with the outcomes found in those different researches in which partial differential equations were used to compare and analyze algorithms. The varied and composite problems natures in partial differential equations create this job mainly difficult and at this phase no commonly adequate measures of analytic or efficient techniques have been defined. This problem of methods and measurements have addressed and established a diversity of hopeful advances by numerous authors cited here.
We consider that significant appraisal of the comparative effectiveness of a variety of methods can only be prepared in the context of a universally established meaning of efficiency. Besides, significant analyses of classes of algorithms depend upon the aptitude to illustrate these classes theoretically. Outcomes from researches in which classes of algorithms are represented by a small number of haphazardly chosen members are of little value.
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