In this work we apply an efficient Isogeometric Analysis (IgA) scheme [2, 8] to parabolic initial-boundary value problems, which are frequently used to describe time evolution phenomena in physics, medicine, and so on.
The standard approach for these problems is to discretize separately in space and in time. For high dimensions, there are works which propose tensor methods in order to tackle the, so-called, curse of dimensionality, notably using the tensor-train format for a global space-time approximation [4, 5]. Recently, it was proposed in  to use IgA to discretize a pure heat conduction problem simultaneously in both space and time. In particular, the time variable was regarded as an extra spatial variable, and the problem was lifted in one dimension higher. Consequently, the space-time cylinder was parameterized by a NURBS volume and high-order and highly smooth splines were used to discretize the problem.
One issue in the efficiency of IgA is the increased cost of computations already for three-dimensional problems. When adding an extra dimension, the cost related to computing the discretized operator increases significantly . Indeed, the dependence of the computational complexity with respect to the dimension is exponential . In the recent work  a partial low-rank tensor decomposition was proposed for decoupling the integrals arising in isogeometric schemes, thereby accelerating their computation.
The present work combines  and . In particular, the scheme used in  is applied to a general parabolic problem with varying diffusion coefficient.
We revisit the analysis in  for the case in question, and provide the corresponding discretization error estimate in the appropriate norm.
Moreover, the fully varying coefficient requires highly accurate numerical integration in
The IgA space-time scheme in  is based on the space-time variational formulation presented in the books [10, 11]. In these works, the authors proved the uniqueness of the corresponding weak solution. Working in a different direction, in  a time discontinuous Galerkin space-time IgA scheme has been analyzed for solving simple parabolic problems (i.e., without varying coefficients).
The rest of the paper is organized as follows. We start by introducing our model problem, deriving the discrete variational form and the error estimates for spline discretizations in Section 2. In Section 3 we focus on the efficient computation of the matrix expressing the discrete operator and we bound the computational complexity in terms of the number of degrees of freedom and the polynomial degree of the discretization. We provide numerical results and computation times in Section 4. We conclude the paper and provide some future research directions in Section 5.
2 The Model Problem
Let Ω be a bounded Lipschitz domain in
For an integer
endowed with the norm
and also we introduce the subspace
According to the definition of
We equip the above spaces with the norms and seminorms
In what follows, positive constants c and C appearing in inequalities are
generic constants which do not depend on the mesh-size h.
2.2 The Model Parabolic Problem
as model problem, where
Using the standard procedure and integration by parts with respect to both x and t,
we can easily derive the following space-time variational formulation
of (2.1): Find
with the bilinear form
and the linear form
where note that the last integral in (2.3) is related to the initial conditions in (2.1).
For simplicity, we only consider homogeneous
Dirichlet boundary conditions on Σ. Also, in the rest of the paper, we will consider that
For our analysis, we make the following convenient assumption.
We assume that the solution u of (2.2) belongs to
2.3 B-Spline Spaces
In this subsection, we briefly present the B-spline spaces and the form of the B-spline parameterizations for the physical space-time patches (called also space-time subdomains). We refer to [2, 3], for a more detailed presentation.
We start by presenting the B-spline space for the univariate case. Let the integer p denote the
B-spline degree and the integer
of the unit interval
Now, let us consider the unit cube
In the frame of IgA, the representation of any volumetric domain is defined by a B-spline basis, see (2.4), and the associated control points, see
Given the associated control points
To keep notation simple, we denote the above space by
Since the parameterization
We assume that
2.4 Discrete Variational Forms
We denote by
The space-time IgA method for (2.1) can be formulated as follows: Find
2.4.1 Discretization Error Analysis
For simplifying the presentation, we derive the analysis for the case where
Let Assumption 2.1 hold and let V be the space defined there.
We define the space
The discrete bilinear form
Using Green’s formula
the fact that
The definition of
Now, since it holds that
The discrete bilinear form
where the constant
The proof can be given following the same steps as in [13, Lemma 4]. ∎
where the constants
Now, we can give the main discretization error estimate in terms of the discrete norm
where we immediately get
3 Matrix Assembly and Decoupling
Let us assume that the spatial domain Ω is described as the image of a regular B-spline parameterization
We obtain a parameterization
where instead of
In this section, we exploit the tensor product structure of the spline space to
vastly improve the computational complexity of computing the discrete bilinear form
3.1 Fast Assembly for Constant Diffusion Coefficients
In this case, we can exploit the tensor product structure of the B-spline basis and the
corresponding structure of the parameterization (3.1) directly,
in order to separate the integration in space and time.
For assembling the system matrix produced in (2.7), we need to compute the bilinear form
For the first term, the transformation of the integral yields
Transforming the third term leads to
Finally, the fourth integral becomes
We observe that these representations only consist of entries of the stiffness and mass matrices for space and time, as well as those of the matrix containing the mixed time derivatives.
The decomposition of the integrals implies that we can write the system matrix
3.2 Fast Assembly for Space-Time Dependent Diffusion
Next we consider the case where the diffusion coefficient ρ is a smooth function depending on
both x and t. Now,
the terms (3.4) and (3.5) no longer decompose directly as in the previous case. In order to decouple integration in space and time in this case, we use the partial tensor decomposition method presented in  to decompose the parametric diffusion coefficient
This decomposition of
We arrive at a Kronecker format representation of rank
If the diffusion coefficient is matrix-valued, let us say
3.3 Computational Complexity
In the following complexity analysis, we assume that the degrees of freedom and polynomial degrees in each
In the proposed method we compute d-variate and univariate integrals by element-wise Gauss quadrature exploiting the decomposition of the integrals. The complexity of computing each matrix
4 Numerical Examples
In this section we perform our experiments on a single patch.
For a multi-patch domain one can apply continuous or discontinuous discretization techniques and ultimately treat the problem patch-wise. Hence the methods developed in this work can be applied for handling the resulting local problems.
Typically, the global system matrix for a multi-patch discretization has sparse block structure. Each block has either tensor-product structure similar to (3.8), or is a very sparse block coming from interface coupling, which can be done in different ways.
For instance, the isogeometric tearing and interconnecting (IETI) method from  can be applied if the multi-patch discretization is continuous.
Otherwise a discontinuous Galerkin approach can be used as is described in  for constant coefficient
For our first numerical example, the space-time domain Q is the quarter annulus in space prolongated into the unit interval in time (see Figure 2). We consider the function
to be the exact solution of the problem. The right-hand-side f, the boundary data and the initial data are computed accordingly. Besides the system matrix K, we also compute the load vector using a partial low-rank approximation of the right-hand-side as presented in . The parameter θ is set to 1. The method was implemented using the G+Smo C++ library [12, 16]. The assembly was performed on a single 2GHz processor, using B-spline basis functions of degrees 3 and 4.
For solving the linear system, a parallel GMRES solver from Trilinos  is used
with tolerance set to
Table 1 shows the error convergence as well as the computation times for assembling the matrix in the case of a constant diffusion coefficient
Three-dimensional experiment on the example space-time domain (see Figure 2)with a constant diffusion coefficient.
|h||#DOF||Error||Error rate||Assembly time|
Next, we study the behavior of the method when we have a space-time dependent diffusion coefficient. We consider again the same exact solution given in (4.1), but the diffusion coefficient is defined as
As tolerance for the projection and truncation error in the low-rank approximation we chose
|h||#DOF||Error||Error rate||Assembly time|
For the cases of
For the cases of
In some cases it can be beneficial to avoid evaluating the sum of Kronecker products in (3.8) by using
the Kronecker format representation directly. Since matrix–vector multiplication can be implemented
easily for a matrix in this format, it can be used for solving the system iteratively. The main advantage is the reduction of the needed memory for storing the system matrix which allows us to assemble up to a very large number of degrees of freedom. The assembly is also very fast, since we only have to compute bivariate and univariate integrals.
Figure 4 shows the computation times for the bivariate and univariate integrals in the matrices of the right-hand side in (3.8).
The maximum number of degrees of freedom that were computed in this experiment are over 136 million for
In this example the space-time cylinder Q is the product of a volumetric shell shape (see Figure 5) in space and the unit interval in time. The diffusion coefficient is chosen to be
We note that in the four-dimensional case the overall complexity is no longer dominated by the sum of Kronecker products (3.8) but by the trivariate quadrature. For this reason, the computation of the matrix in Kronecker format is no longer significantly faster than the computation of the global matrix by performing the sum of Kronecker products in (3.8). However, the advantages stemming from the reduction in memory still apply to the four-dimensional case and we can assemble up to many more degrees of freedom using the same amount of memory.
Figure 7 shows the computation times for the
system matrix in Kronecker format. They are dominated by the
trivariate spatial integrals. For
We presented and analyzed a space-time IgA scheme for linear parabolic problems with varying coefficients. We employed low-rank approximation techniques to speed-up the computation of the discrete operator. In this way we were able to compute with several millions of degrees of freedom. However, a standing challenge is the solution of the linear system. Indeed, this system is non-symmetric and with a rapidly increasing bandwidth with respect to the polynomial degree. The GMRES solver that we used, combined with domain decomposition and ILU preconditioner performed reasonably good for a moderate number of degrees of freedom. Another challenging task is the extension of this approach to problems with discontinuous diffusion coefficients. In this case, the space-time cylinder Q could be described as a multi-patch space-time domain (cf. ) compatible with the discontinuities of the diffusion coefficient. The low-rank approximation would be applied patch-wise and appropriate discontinuous Galerkin techniques, i.e., numerical fluxes, would be needed for coupling the local patch-wise problems.
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