Final Report Summary - NUMERICS4NLP (Numerical Methods for Nonlocal Problems)
We are interested in studying instances of successful modeling by nonlocal theories of phenomena that cannot be captured by local theories. Our main applications of interest are peridynamics (PD) and nonlocal diffusion. PD is a nonlocal extension of continuum mechanics and is mainly used for the simulation of crack propagation. PD's success stems from utilizing an integral formulation instead of derivatives. A crack is inherently discontinuous, hence, its representation with an integral operator is more natural compared to a derivative based operator. This choice allows the treatment of cracks and failures with relative ease. PD has been successfully used in various engineering applications. Nonlocal diffusion has been used in a wide array of areas which include biological models, image processing, particle systems, coagulation, and phase transition.
We have made two fundamental contributions. Firstly, we discovered a way to incorporate local boundary conditions (BCs) to nonlocal theories. Secondly, we proved sharp condition number estimates for the discretized operator.
We started with a theoretical study of the PD governing operator on the unbounded domain of real coordinate space of d dimensions. This theoretical study was very fruitful, we obtained three crucial results. As the first result, we discovered that, the PD governing operator, which involves a convolution, is a bounded function of the classical (local) governing operator. Secondly, we proved the strong convergence of solutions through strong resolvent convergence, the first such result in the literature. As the third result, we proved that the solution could be expressed in terms of infinite series of Bessel functions. This practical series representation allows for straightforward numerical treatment of the solution with symbolic computation. By simply truncating the infinite series, one can obtain numerical approximation of the solution without running into artificial boundary reflections which are seen when unbounded domains are used. Furthermore, it can be used for benchmarking numerical computations.
Building on the first theoretical result, we generalized the standard integral based convolution to an abstract convolution operator which is defined by a Hilbert basis. This basis is induced by the classical operator with prescribed local BCs on bounded domains. The nonlocal operator becomes a function of the classical operator. By prescribing BCs to the classical operator, we construct a gateway to incorporate local BCs into nonlocal theories. Through the use of local BCs, it will be possible to solve important elasticity applications such as contact, shear, and traction, which PD cannot solve in its current state. In addition, we will be able to eliminate the long standing surface effects which are seen in PD due to employing nonlocal BCs.
Our approach presents a unique way of combining the powers of abstract operator theory with numerical computing. It utilizes operator theory tools, and hence, is not limited to PD. The abstractness of the theoretical methods used allows generalization to other nonlocal theories. When we studied PD and nonlocal diffusion, we guaranteed to satisfy BCs by using the Hilbert-Schmidt property of the abstract convolution operator. The main idea is based on obtaining a smooth piece in the solution so that the solution operator is a compact perturbation of a multiple of the identity operator. This gave rise to a fundamental insight into discontinuity propagation. Namely, in 1D, we proved that discontinuities of the solution remain stationary.
In applications, having reduced computation time is a desirable feature. In order to have a broader impact and to cover a larger area of applications , we developed scalable solvers. These solvers are in the form of domain decomposition methods that will lead to parallel computing. This is work in progress.
Sharp condition number estimates were the second fundamental contribution. The conditioning of an operator equation, which is represented by a single quantity called the condition number, is a measure that indicates the stability of the equation with respect to perturbations on the input data. A large sensitivity to such perturbations causes the equation to be unstable. Such equations are referred to as ill-conditioned. The mechanism, usually in the form of numerical methods, designed to correct ill-conditioning of the underlying equation is called preconditioning.
One of the goals of this project was to pave the way to constructing effective numerical methods, especially robust and scalable preconditioners, to solve nonlocal problems. We put great emphasis on the basis of designing and constructing such preconditioners. Condition number quantification is a critical first step in the study of preconditioners. Quantifying the condition number in terms of the involved parameters is a foundational step towards acquiring a fundamental understanding of the underlying equation. Effectiveness of an iterative solver is determined by the condition number of the preconditioned problem. A preconditioner is designed to lead to a well-conditioned problem. Therefore, sharpness of this quantification is a highly desirable feature of the analysis. If the quantification of the condition number is not sharp, then the designed method may target the incorrect component of the problem, which leads to an ineffective numerical method. Hence, our study lays the foundation of preconditioner research for nonlocal problems and will have a lasting impact in nonlocal applications.
Obtaining sharpness was an open problem and we achieved to obtain sharp quantifications of the extremal eigenvalues in all three parameters: size of nonlocality, mesh size, and regularity of the fractional Sobolev space. We accomplished sharpness both rigorously and numerically and this accomplishment was beyond state of the art.
We chose a scaling of the operator which was also used in the literature for other purposes. We discovered that this scaling promised a major advantage in the conditioning analysis. Namely, the dependence of the minimal eigenvalue, on the horizon size and regularity parameter, is eliminated, thereby, reducing dependence only to the mesh size. We also showed that when the nonlocality size goes to zero, assuming sufficient regularity of the solution, the nonlocal operator converged to the local one.
We used a nonlocal characterization of Sobolev spaces. In earlier publications, we had used a similar characterization for integrable kernels. We extended this characterization to singular kernel functions. This characterization enabled us to construct nonlocal Poincare inequalities through which the sharp quantification of the minimal eigenvalue has been obtained.
For the maximal eigenvalue quantification, first we tried bounds obtained by norm estimates in the spirit of Sobolev embedding theorems. The resulting quantifications turned out to be not sharp. But, we recovered the same estimates given in the literature in 1D. Our approach enabled us to extend the 1D results to higher dimensions. However, we could not correct the non-sharpness of the quantification by norm estimates. At that point, we decided to take a linear algebraic approach to determine the quantification of the maximal eigenvalue. This approach was the first in the literature and was quite successful. As a result, we were able to fully reveal sharp quantifications of the extremal eigenvalues.
Our results indicated that there were two terms involved in the quantification of the maximal eigenvalue: one positive and one negative. We discovered that the available quantifications in the literature were able to recover only the positive term. Hence, our sharp quantifications obviously were always smaller than the non-sharp ones.
Crack propagation is an important phenomenon in material science and structural mechanics. For instance, if aircraft manufacturers adopt PD modeling, our approach will be useful because they will be able to simulate important applications such as contact, shear, and traction. Ultimately, the numerical methods we developed can be applied to nonlocal problems in general.
We have made two fundamental contributions. Firstly, we discovered a way to incorporate local boundary conditions (BCs) to nonlocal theories. Secondly, we proved sharp condition number estimates for the discretized operator.
We started with a theoretical study of the PD governing operator on the unbounded domain of real coordinate space of d dimensions. This theoretical study was very fruitful, we obtained three crucial results. As the first result, we discovered that, the PD governing operator, which involves a convolution, is a bounded function of the classical (local) governing operator. Secondly, we proved the strong convergence of solutions through strong resolvent convergence, the first such result in the literature. As the third result, we proved that the solution could be expressed in terms of infinite series of Bessel functions. This practical series representation allows for straightforward numerical treatment of the solution with symbolic computation. By simply truncating the infinite series, one can obtain numerical approximation of the solution without running into artificial boundary reflections which are seen when unbounded domains are used. Furthermore, it can be used for benchmarking numerical computations.
Building on the first theoretical result, we generalized the standard integral based convolution to an abstract convolution operator which is defined by a Hilbert basis. This basis is induced by the classical operator with prescribed local BCs on bounded domains. The nonlocal operator becomes a function of the classical operator. By prescribing BCs to the classical operator, we construct a gateway to incorporate local BCs into nonlocal theories. Through the use of local BCs, it will be possible to solve important elasticity applications such as contact, shear, and traction, which PD cannot solve in its current state. In addition, we will be able to eliminate the long standing surface effects which are seen in PD due to employing nonlocal BCs.
Our approach presents a unique way of combining the powers of abstract operator theory with numerical computing. It utilizes operator theory tools, and hence, is not limited to PD. The abstractness of the theoretical methods used allows generalization to other nonlocal theories. When we studied PD and nonlocal diffusion, we guaranteed to satisfy BCs by using the Hilbert-Schmidt property of the abstract convolution operator. The main idea is based on obtaining a smooth piece in the solution so that the solution operator is a compact perturbation of a multiple of the identity operator. This gave rise to a fundamental insight into discontinuity propagation. Namely, in 1D, we proved that discontinuities of the solution remain stationary.
In applications, having reduced computation time is a desirable feature. In order to have a broader impact and to cover a larger area of applications , we developed scalable solvers. These solvers are in the form of domain decomposition methods that will lead to parallel computing. This is work in progress.
Sharp condition number estimates were the second fundamental contribution. The conditioning of an operator equation, which is represented by a single quantity called the condition number, is a measure that indicates the stability of the equation with respect to perturbations on the input data. A large sensitivity to such perturbations causes the equation to be unstable. Such equations are referred to as ill-conditioned. The mechanism, usually in the form of numerical methods, designed to correct ill-conditioning of the underlying equation is called preconditioning.
One of the goals of this project was to pave the way to constructing effective numerical methods, especially robust and scalable preconditioners, to solve nonlocal problems. We put great emphasis on the basis of designing and constructing such preconditioners. Condition number quantification is a critical first step in the study of preconditioners. Quantifying the condition number in terms of the involved parameters is a foundational step towards acquiring a fundamental understanding of the underlying equation. Effectiveness of an iterative solver is determined by the condition number of the preconditioned problem. A preconditioner is designed to lead to a well-conditioned problem. Therefore, sharpness of this quantification is a highly desirable feature of the analysis. If the quantification of the condition number is not sharp, then the designed method may target the incorrect component of the problem, which leads to an ineffective numerical method. Hence, our study lays the foundation of preconditioner research for nonlocal problems and will have a lasting impact in nonlocal applications.
Obtaining sharpness was an open problem and we achieved to obtain sharp quantifications of the extremal eigenvalues in all three parameters: size of nonlocality, mesh size, and regularity of the fractional Sobolev space. We accomplished sharpness both rigorously and numerically and this accomplishment was beyond state of the art.
We chose a scaling of the operator which was also used in the literature for other purposes. We discovered that this scaling promised a major advantage in the conditioning analysis. Namely, the dependence of the minimal eigenvalue, on the horizon size and regularity parameter, is eliminated, thereby, reducing dependence only to the mesh size. We also showed that when the nonlocality size goes to zero, assuming sufficient regularity of the solution, the nonlocal operator converged to the local one.
We used a nonlocal characterization of Sobolev spaces. In earlier publications, we had used a similar characterization for integrable kernels. We extended this characterization to singular kernel functions. This characterization enabled us to construct nonlocal Poincare inequalities through which the sharp quantification of the minimal eigenvalue has been obtained.
For the maximal eigenvalue quantification, first we tried bounds obtained by norm estimates in the spirit of Sobolev embedding theorems. The resulting quantifications turned out to be not sharp. But, we recovered the same estimates given in the literature in 1D. Our approach enabled us to extend the 1D results to higher dimensions. However, we could not correct the non-sharpness of the quantification by norm estimates. At that point, we decided to take a linear algebraic approach to determine the quantification of the maximal eigenvalue. This approach was the first in the literature and was quite successful. As a result, we were able to fully reveal sharp quantifications of the extremal eigenvalues.
Our results indicated that there were two terms involved in the quantification of the maximal eigenvalue: one positive and one negative. We discovered that the available quantifications in the literature were able to recover only the positive term. Hence, our sharp quantifications obviously were always smaller than the non-sharp ones.
Crack propagation is an important phenomenon in material science and structural mechanics. For instance, if aircraft manufacturers adopt PD modeling, our approach will be useful because they will be able to simulate important applications such as contact, shear, and traction. Ultimately, the numerical methods we developed can be applied to nonlocal problems in general.
