377 research outputs found
MATEX: A Distributed Framework for Transient Simulation of Power Distribution Networks
We proposed MATEX, a distributed framework for transient simulation of power
distribution networks (PDNs). MATEX utilizes matrix exponential kernel with
Krylov subspace approximations to solve differential equations of linear
circuit. First, the whole simulation task is divided into subtasks based on
decompositions of current sources, in order to reduce the computational
overheads. Then these subtasks are distributed to different computing nodes and
processed in parallel. Within each node, after the matrix factorization at the
beginning of simulation, the adaptive time stepping solver is performed without
extra matrix re-factorizations. MATEX overcomes the stiff-ness hinder of
previous matrix exponential-based circuit simulator by rational Krylov subspace
method, which leads to larger step sizes with smaller dimensions of Krylov
subspace bases and highly accelerates the whole computation. MATEX outperforms
both traditional fixed and adaptive time stepping methods, e.g., achieving
around 13X over the trapezoidal framework with fixed time step for the IBM
power grid benchmarks.Comment: ACM/IEEE DAC 2014. arXiv admin note: substantial text overlap with
arXiv:1505.0669
An Algorithmic Framework for Efficient Large-Scale Circuit Simulation Using Exponential Integrators
We propose an efficient algorithmic framework for time domain circuit
simulation using exponential integrator. This work addresses several critical
issues exposed by previous matrix exponential based circuit simulation
research, and makes it capable of simulating stiff nonlinear circuit system at
a large scale. In this framework, the system's nonlinearity is treated with
exponential Rosenbrock-Euler formulation. The matrix exponential and vector
product is computed using invert Krylov subspace method. Our proposed method
has several distinguished advantages over conventional formulations (e.g., the
well-known backward Euler with Newton-Raphson method). The matrix factorization
is performed only for the conductance/resistance matrix G, without being
performed for the combinations of the capacitance/inductance matrix C and
matrix G, which are used in traditional implicit formulations. Furthermore, due
to the explicit nature of our formulation, we do not need to repeat LU
decompositions when adjusting the length of time steps for error controls. Our
algorithm is better suited to solving tightly coupled post-layout circuits in
the pursuit for full-chip simulation. Our experimental results validate the
advantages of our framework.Comment: 6 pages; ACM/IEEE DAC 201
Adaptive rational Krylov methods for exponential Runge--Kutta integrators
We consider the solution of large stiff systems of ordinary differential
equations with explicit exponential Runge--Kutta integrators. These problems
arise from semi-discretized semi-linear parabolic partial differential
equations on continuous domains or on inherently discrete graph domains. A
series of results reduces the requirement of computing linear combinations of
-functions in exponential integrators to the approximation of the
action of a smaller number of matrix exponentials on certain vectors.
State-of-the-art computational methods use polynomial Krylov subspaces of
adaptive size for this task. They have the drawback that the required Krylov
subspace iteration numbers to obtain a desired tolerance increase drastically
with the spectral radius of the discrete linear differential operator, e.g.,
the problem size. We present an approach that leverages rational Krylov
subspace methods promising superior approximation qualities. We prove a novel
a-posteriori error estimate of rational Krylov approximations to the action of
the matrix exponential on vectors for single time points, which allows for an
adaptive approach similar to existing polynomial Krylov techniques. We discuss
pole selection and the efficient solution of the arising sequences of shifted
linear systems by direct and preconditioned iterative solvers. Numerical
experiments show that our method outperforms the state of the art for
sufficiently large spectral radii of the discrete linear differential
operators. The key to this are approximately constant rational Krylov iteration
numbers, which enable a near-linear scaling of the runtime with respect to the
problem size
Interpolation Based Parametric Model Order Reduction
In this thesis, we consider model order reduction of parameter-dependent large-scale dynamical systems. The objective is to develop a methodology to reduce the order of the model and simultaneously preserve the dependence of the model on parameters. We use the balanced truncation method together with spline interpolation to solve the problem. The core of this method is to interpolate the reduced transfer function, based on the pre-computed transfer function at a sample in the parameter domain. Linear splines and cubic splines are employed here. The use of the latter, as expected, reduces the error of the method. The combination is proven to inherit the advantages of balanced truncation such as stability preservation and, based on a novel bound for the infinity norm of the matrix inverse, the derivation of error bounds. Model order reduction can be formulated in the projection framework. In the case of a parameter-dependent system, the projection subspace also depends on parameters. One cannot compute this parameter-dependent projection subspace, but has to approximate it by interpolation based on a set of pre-computed subspaces. It turns out that this is the problem of interpolation on Grassmann manifolds. The interpolation process is actually performed on tangent spaces to the underlying manifold. To do that, one has to invoke the exponential and logarithmic mappings which involve some singular value decompositions. The whole procedure is then divided into the offline and online stage. The computation time in the online stage is a crucial point. By investigating the formulation of exponential and logarithmic mappings and analyzing the structure of sums of singular value decompositions, we succeed to reduce the computational complexity of the online stage and therefore enable the use of this algorithm in real time
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