15 research outputs found
Threshold Accepting for Credit Risk Assessment and Validation
According to the latest Basel framework of Banking Supervision, financial institutions should internally assign their borrowers into a number of homogeneous groups. Each group is assigned a probability of default which distinguishes it from other groups. This study aims at determining the optimal number and size of groups that allow for statistical ex post validation of the efficiency of the credit risk assignment system. Our credit risk assignment approach is based on Threshold Accepting, a local search optimization technique, which has recently performed reliably in credit risk clustering especially when considering several realistic constraints. Using a relatively large real-world retail credit portfolio, we propose a new technique to validate ex post the precision of the grading system.credit risk assignment, Threshold Accepting, statistical validation
Complexity analysis of regularization methods for implicitly constrained least squares
Optimization problems constrained by partial differential equations (PDEs)
naturally arise in scientific computing, as those constraints often model
physical systems or the simulation thereof. In an implicitly constrained
approach, the constraints are incorporated into the objective through a reduced
formulation. To this end, a numerical procedure is typically applied to solve
the constraint system, and efficient numerical routines with quantifiable cost
have long been developed. Meanwhile, the field of complexity in optimization,
that estimates the cost of an optimization algorithm, has received significant
attention in the literature, with most of the focus being on unconstrained or
explicitly constrained problems.
In this paper, we analyze an algorithmic framework based on quadratic
regularization for implicitly constrained nonlinear least squares. By
leveraging adjoint formulations, we can quantify the worst-case cost of our
method to reach an approximate stationary point of the optimization problem.
Our definition of such points exploits the least-squares structure of the
objective, leading to an efficient implementation. Numerical experiments
conducted on PDE-constrained optimization problems demonstrate the efficiency
of the proposed framework.Comment: 21 pages, 2 figure
On the regularity of refinable functions
Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006.This work studies the regularity (or smoothness) of continuous finitely supported refinable
functions which are mainly encountered in multiresolution analysis, iterative interpolation
processes, signal analysis, etc. Here, we present various kinds of sufficient conditions on
a given mask to guarantee the regularity class of the corresponding refinable function.
First, we introduce and analyze the cardinal B-splines Nm, m ∈ N. In particular, we
show that these functions are refinable and belong to the smoothness class Cm−2(R). As
a generalization of the cardinal B-splines, we proceed to discuss refinable functions with
positive mask coefficients. A standard result on the existence of a refinable function in
the case of positive masks is quoted. Following [13], we extend the regularity result in
[25], and we provide an example which illustrates the fact that the associated symbol to
a given positive mask need not be a Hurwitz polynomial for its corresponding refinable
function to be in a specified smoothness class. Furthermore, we apply our regularity result
to an integral equation.
An important tool for our work is Fourier analysis, from which we state some standard
results and give the proof of a non-standard result. Next, we study the H¨older regularity
of refinable functions, whose associated mask coefficients are not necessarily positive, by
estimating the rate of decay of their Fourier transforms. After showing the embedding of
certain Sobolev spaces into a H¨older regularity space, we proceed to discuss sufficient conditions
for a given refinable function to be in such a H¨older space. We specifically express
the minimum H¨older regularity of refinable functions as a function of the spectral radius
of an associated transfer operator acting on a finite dimensional space of trigonometric
polynomials.
We apply our Fourier-based regularity results to the Daubechies and Dubuc-Deslauriers
refinable functions, as well as to a one-parameter family of refinable functions, and then
compare our regularity estimates with those obtained by means of a subdivision-based
result from [28]. Moreover, we provide graphical examples to illustrate the theory developed
Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
Many problems in computational science and engineering are simultaneously
characterized by the following challenging issues: uncertainty, nonlinearity,
nonstationarity and high dimensionality. Existing numerical techniques for such
models would typically require considerable computational and storage
resources. This is the case, for instance, for an optimization problem governed
by time-dependent Navier-Stokes equations with uncertain inputs. In particular,
the stochastic Galerkin finite element method often leads to a prohibitively
high dimensional saddle-point system with tensor product structure. In this
paper, we approximate the solution by the low-rank Tensor Train decomposition,
and present a numerically efficient algorithm to solve the optimality equations
directly in the low-rank representation. We show that the solution of the
vorticity minimization problem with a distributed control admits a
representation with ranks that depend modestly on model and discretization
parameters even for high Reynolds numbers. For lower Reynolds numbers this is
also the case for a boundary control. This opens the way for a reduced-order
modeling of the stochastic optimal flow control with a moderate cost at all
stages.Comment: 29 page
State-constrained Optimization Problems under Uncertainty: A Tensor Train Approach
We propose an algorithm to solve optimization problems constrained by partial
(ordinary) differential equations under uncertainty, with almost sure
constraints on the state variable. To alleviate the computational burden of
high-dimensional random variables, we approximate all random fields by the
tensor-train decomposition. To enable efficient tensor-train approximation of
the state constraints, the latter are handled using the Moreau-Yosida penalty,
with an additional smoothing of the positive part (plus/ReLU) function by a
softplus function. We derive theoretical bounds on the constraint violation in
terms of the Moreau-Yosida regularization parameter and smoothing width of the
softplus function. This result also proposes a practical recipe for selecting
these two parameters. When the optimization problem is strongly convex, we
establish strong convergence of the regularized solution to the optimal
control. We develop a second order Newton type method with a fast matrix-free
action of the approximate Hessian to solve the smoothed Moreau-Yosida problem.
This algorithm is tested on benchmark elliptic problems with random
coefficients, optimization problems constrained by random elliptic variational
inequalities, and a real-world epidemiological model with 20 random variables.
These examples demonstrate mild (at most polynomial) scaling with respect to
the dimension and regularization parameters.Comment: 29 page