15:38:47 Solving GTAP model in parallel using Doubly Bordered Block Diagonal ordering technique ∗ Pham Van Ha Prof. Tom Kompas ha.pham@anu.edu.au tom.kompas@anu.edu.au Crawford School of Public Policy ANU College of Asia & the Pacific Melbourne, 11 August 2014 ∗ Preliminary, not for citation DBBD solution PVH-TFK – 1 / 27
15:38:47 Contents INTRODUCTION INTRODUCTION GTAP MODEL AND THE CURRENT GTAP MODEL AND THE CURRENT SOLUTION METHOD SOLUTION METHOD DBBD MATRIX AND DBBD MATRIX AND DIRECT METHOD FOR SOLVING LINEAR SYSTEM DIRECT METHOD FOR SOLVING LINEAR SYSTEM GTAP MODEL AND DBBD FORM GTAP MODEL AND DBBD FORM NUMERICAL ANALYSIS NUMERICAL ANALYSIS CONCLUSION CONCLUSION REFERENCES REFERENCES DBBD solution PVH-TFK – 2 / 27
15:38:47 INTRODUCTION The Rationale GTAP MODEL AND THE CURRENT SOLUTION METHOD DBBD MATRIX AND DIRECT METHOD FOR SOLVING LINEAR SYSTEM INTRODUCTION GTAP MODEL AND DBBD FORM NUMERICAL ANALYSIS CONCLUSION REFERENCES DBBD solution PVH-TFK – 3 / 27
15:38:47 The Rationale INTRODUCTION Modelling regional economies is always a challenge to CGE modelling as ■ The Rationale they consist of multiple interacting economies (with similar structures). GTAP MODEL AND THE CURRENT There are 2 ways to model regional economies: the ‘bottom-up’ and ■ SOLUTION METHOD ‘top-down’ approaches. The ‘top down’ approach models the national DBBD MATRIX AND DIRECT METHOD FOR economy and solves it first. Regional variables are linked to macro national SOLVING LINEAR SYSTEM variables in a one way interface. On the other hand, the ‘bottom-up’ GTAP MODEL AND DBBD FORM approach builds a model for all regions in a complete national aggregate NUMERICAL model (Klein and Glickman, 1977). ANALYSIS ‘Bottom-up’ CGE models are more difficult to solve both in term of data ■ CONCLUSION requirement and computing time. REFERENCES The purpose of this research is to tackle the computational challenge of ■ bottom-up CGE models to solve a largest bottom-up regional CGE model, the GTAP model (Hertel, 1997). DBBD solution PVH-TFK – 4 / 27
15:38:47 INTRODUCTION GTAP MODEL AND THE CURRENT SOLUTION METHOD Overview of GTAP model GTAP model’s solution DBBD MATRIX AND GTAP MODEL AND THE DIRECT METHOD FOR SOLVING LINEAR SYSTEM CURRENT SOLUTION METHOD GTAP MODEL AND DBBD FORM NUMERICAL ANALYSIS CONCLUSION REFERENCES DBBD solution PVH-TFK – 5 / 27
15:38:47 Overview of GTAP model Figure 1: Graphical representation of GTAP model. INTRODUCTION GTAP MODEL AND THE CURRENT SOLUTION METHOD Overview of GTAP model GTAP model’s solution DBBD MATRIX AND DIRECT METHOD FOR SOLVING LINEAR SYSTEM GTAP MODEL AND DBBD FORM NUMERICAL ANALYSIS CONCLUSION REFERENCES Source: Brockmeier (2001) DBBD solution PVH-TFK – 6 / 27
15:38:47 GTAP model’s solution INTRODUCTION Currently, there are two software packages dedicated to solve the GTAP ■ GTAP MODEL AND (and CGE models in general) model: GAMS and GEMPACK. THE CURRENT SOLUTION METHOD The two software packages solve GTAP model by direct LU decomposition ■ Overview of GTAP model of its first order differential matrix. GAMS uses an iterative method to solve GTAP model’s solution a CGE model as a system of nonlinear equations (or constraints), DBBD MATRIX AND DIRECT METHOD FOR meanwhile, GEMPACK uses linear approximations (see SOLVING LINEAR SYSTEM Pham and Kompas, under review, for more details). GTAP MODEL AND DBBD FORM In the following sections, we will compare our method with MA48, the core ■ NUMERICAL engine behind GEMPACK, the fastest CGE model solver available on the ANALYSIS market. CONCLUSION REFERENCES DBBD solution PVH-TFK – 7 / 27
15:38:47 INTRODUCTION GTAP MODEL AND THE CURRENT SOLUTION METHOD DBBD MATRIX AND DIRECT METHOD FOR SOLVING LINEAR DBBD MATRIX AND DIRECT SYSTEM The DBBD matrix form and its corresponding METHOD FOR SOLVING LINEAR linear system DBBD matrix solution (a SYSTEM modified version of the solution method used by Yamazaki and Li (2011)) GTAP MODEL AND DBBD FORM NUMERICAL ANALYSIS CONCLUSION REFERENCES DBBD solution PVH-TFK – 8 / 27
15:38:47 The DBBD matrix form and its corresponding linear system INTRODUCTION A 1 C 1 x 1 y 1 GTAP MODEL AND A 2 C 2 x 2 y 2 THE CURRENT SOLUTION METHOD = ... ... ... ... (1) DBBD MATRIX AND DIRECT METHOD FOR A K C K x K y K SOLVING LINEAR SYSTEM B 1 B 2 ... B K D x d y d The DBBD matrix form and its corresponding linear system where A i ( i = 1 ...K ) and D are rectangular matrices. DBBD matrix solution (a modified version of the solution method used by Yamazaki and Li (2011)) GTAP MODEL AND Solution algorithm (following Yamazaki and Li, 2011): DBBD FORM NUMERICAL Solve A i u i = y i problem. ANALYSIS 1. CONCLUSION 2. Using the same LU decomposition solve the multiple right hand side REFERENCES problem: A i v i = C i . Solve the problem: ( D − � K i B i v i ) x d = y d − � K i B i u i . 3. Calculate x i = u i − v i x d . 4. DBBD solution PVH-TFK – 9 / 27
15:38:47 DBBD matrix solution (a modified version of the solution method used by Yamazaki and Li (2011)) INTRODUCTION The LU decomposition and linear equations solved in Steps 1 and 2 can be ■ GTAP MODEL AND done in parallel before the result can be fed back into the leading process in THE CURRENT SOLUTION METHOD Step 3. DBBD MATRIX AND Step 4 can be done in parallel before the result can be transmitted back to DIRECT METHOD FOR ■ SOLVING LINEAR the leading process to assemble the solution vector x . SYSTEM The DBBD matrix form The v i matrix is potentially a dense matrix, hence it should never been and its corresponding ■ linear system directly stored to conserve the memory. Instead of storing v i we store B i v i . DBBD matrix solution (a modified version of the For the last step, we define: solution method used by Yamazaki and Li (2011)) GTAP MODEL AND DBBD FORM NUMERICAL ANALYSIS A i v i x d = C i x d (2) CONCLUSION A i η i = C i x d (3) REFERENCES By solving Equation 3 for η i as a result of the matrix multiplication v i x d , we ■ again can avoid storing v i explicitly. DBBD solution PVH-TFK – 10 / 27
15:38:47 INTRODUCTION GTAP MODEL AND THE CURRENT SOLUTION METHOD DBBD MATRIX AND DIRECT METHOD FOR SOLVING LINEAR SYSTEM GTAP MODEL AND GTAP MODEL AND DBBD FORM DBBD FORM Bottom-up regional CGE models and DBBD form First order partial derivative matrix of CGE model GTAP model and DBBD direct matrix ordering technique Post ordering preparation matrix for parallel solution NUMERICAL ANALYSIS CONCLUSION REFERENCES DBBD solution PVH-TFK – 11 / 27
15:38:47 Bottom-up regional CGE models and DBBD form INTRODUCTION General form of a bottom-up regional CGE model. ■ GTAP MODEL AND THE CURRENT SOLUTION METHOD J J DBBD MATRIX AND � � 0 = f [ x ( r, s, i, ... ) , y ( s, i, ... )] DIRECT METHOD FOR ∀ r ∈ R, ∀ s ∈ S, ∀ i ∈ I... (4) SOLVING LINEAR SYSTEM j j GTAP MODEL AND J R J R DBBD FORM � � � � Bottom-up regional 0 = g [ x ( r, s, i, ... ) , y ( s, i, ... )] ∀ s ∈ S, ∀ i ∈ I... (5) CGE models and DBBD form r r j j First order partial derivative matrix of CGE model GTAP model and DBBD Equation 4 and Equation 5 represent intra-regional and inter-regional ■ direct matrix ordering technique equations, and x, y also represent intra and inter-regional variables. Post ordering preparation matrix for Regional set can have subset, but only one regional set will be chosen to ■ parallel solution classify equations and variables. NUMERICAL ANALYSIS CONCLUSION REFERENCES DBBD solution PVH-TFK – 12 / 27
15:38:47 First order partial derivative matrix of the non-linear regional CGE model INTRODUCTION GTAP MODEL AND 0 f x (“ r 1” ,s,i... ) ... f y ( s,i,... ) ∀ s ∈ S, ∀ i ∈ I... THE CURRENT SOLUTION METHOD 0 ... f x (“ rR ” ,s,i... ) f y ( s,i,... ) ∀ s ∈ S, ∀ i ∈ I... (6) DBBD MATRIX AND ∀ s ∈ S, ∀ i ∈ I... DIRECT METHOD FOR g x (“ r 1” ,s,i... ) ... g x (“ rR ” ,s,i... ) g y ( s,i,... ) SOLVING LINEAR SYSTEM GTAP MODEL AND DBBD FORM Bottom-up regional CGE models and DBBD form First order partial derivative matrix of CGE model GTAP model and DBBD direct matrix ordering technique Post ordering preparation matrix for parallel solution NUMERICAL ANALYSIS CONCLUSION REFERENCES DBBD solution PVH-TFK – 13 / 27
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